(Ideas in Context) Reviel Netz-The SHAPING of DEDUCTION in GREEK MATHEMATICS A Study in Cognitive History-Cambridge University Press (2003)
(Ideas in Context) Reviel Netz-The SHAPING of DEDUCTION in GREEK MATHEMATICS A Study in Cognitive History-Cambridge University Press (2003)
(Ideas in Context) Reviel Netz-The SHAPING of DEDUCTION in GREEK MATHEMATICS A Study in Cognitive History-Cambridge University Press (2003)
VOLUME XI
EDITED BY
Preface vii
Contributors 367
Index 371
Prefacee
v/7
viii Preface
In our opinion, in the decade since the 1974 workshop there has been
an upsurge of interest in an interdisciplinary approach to historical and
philosophical problems connected with mathematics. There have also been
recent developments internal to each discipline: historians of mathematics
have recognized the need for a more sophisticated historiography and for
greater attention to the social context in which the work of mathemati-
cians is set; philosophers of mathematics have moved away from the grand
foundational programs of the early twentieth century and the preoccupa-
tion with the demise of those programs.
For these reasons, we believed it was important once again to take ap-
praisal of our fields. Looking to the 1974 workshop as our model, we
brought together a distinguished group of historians, mathematicians, and
for the first time philosophers actively working in this area. Indeed, as
a measure of the continuity of this area, seven of the authors represented
in this volume were participants in the 1974 workshop. Our authors were
requested to keep in mind not only the historical, philosophical, and
mathematical significance of their topics, but also to devote special at-
tention to their methodological approach. These approaches came under
considerable discussion during the conference. We can only hope that our
proceedings will be as influential as those of the 1974 workshop.
W. A. and P. K.
AN OPINIONATED INTRODUCTION
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Philip Kitcher and William Aspray
An Opinionated Introduction
Each of the essays that follow could be sent forth as an orphan, with
only the most perfunctory comment, to make its own solitary way in the
world, and some of them would undoubtedly do well. However, we believe
that there are important reasons for setting them side by side, that each
gains from the company of the others. In our judgment, the present volume
offers a representation of the current state of history and philosophy of
mathematics. Hence we have felt encouraged—perhaps foolhardily—to
offer perspectives on the past, the present, and the future. This introduc-
tion attempts to sketch the history of the philosophy of mathematics in
the last century and the recent history of the history of mathematics. We
then endeavor to relate the individual studies not only to one another,
but also to the traditions of work in the history of mathematics and in
the philosophy of mathematics. We conclude with some speculations about
the present trends and possible futures. Quite plainly, any effort of this
kind must embody the authors' ideas, perspectives, and, maybe, prejudices
about what kinds of researches are important. So we have rejected the
plain title "Introduction" and announced our essay for what it is—an
opinionated introduction.
1. Philosophy of Mathematics: A Brief and Biased History
Philosophers of mathematics may argue about many things, but, until
recently, there has been remarkable agreement concerning when their
discipline began. Prevailing orthodoxy takes the history of the philosophy
of mathematics to start with Frege. It was Frege who posed the problems
with which philosophers of mathematics have struggled ever since, Frege
who developed modern logic and used it to undertake rigorous explora-
tion of the foundations of mathematics, Frege who charted the main
philosophical options that his successors were to explore. To be sure, there
3
4 Philip Kitcher and William Aspray
this perspective it does not matter, in the end, whether the axioms of
various bits of mathematics are (or are derivable from) laws of logic. What
counts is that they should be (or be derivable from) principles that em-
body the conventions that govern the primitive vocabulary.
The significance of this change for the credibility of the other programs
is that it provides a way to accommodate certain ideas of formalism and
to remove the sting of the intuitionist critique of classical mathematics.
One of the chief points of dispute between Hilbert and Frege had sur-
rounded Hilbert's early proposal that the primitive vocabulary of an ax-
iomatic system is implicitly defined by the axioms that are set down. On
Frege's conception of logic (which held no place for the notion of an
uninterpreted statement), this claim of Hilbert was absurd (see Resnik 1980
for a lucid discussion of the controversy.) In the wake of the develop-
ment of the semantics of logic (see Goldfarb 1979 and Moore, this volume),
Frege's conception of logic was abandoned and logicists came to adopt
Hilbert's proposal. Thus, at a time when the mature Hilbert program was
in difficulties, it appeared possible to salvage part of Hilbert's early motiva-
tions. Furthermore, instead of conceiving intuitionism to be a rival
mathematics, the emphasis on the conventionality of logic made it possi-
ble to see intuitionistic logic and mathematics as embodying a different
set of conventions. The system based on those conventions could be
tolerated as an interesting curiosity, something that could be placed
alongside classical mathematics but that could not pose any threat of
displacement (see Carnap 1937).
The new logicism represented both a return to Frege and a departure.
Because of Frege's opposition to the founding of mathematics on intui-
tion, his scorn for empiricism, and his emphasis that mathematical
statements are meaningful, the logical positivists could legitimately cite
him as a predecessor. However, their ideas about logic and semantics were
importantly different from those that Frege had espoused (see Dummett
1973; Sluga 1980; Ricketts 1985). Hence there was a genuine transition
between one version of logicism, on which the principles of logic were
seen as fundamental to all thought, to another, on which logic was true
by convention (Friedman, this volume, provides an illuminating account
of part of the transition).
The new form of logicism avoided awkward questions about how we
know the principles that prescribe to all thought, but it quickly became
vulnerable to a different challenge. In 1936, Quine published a seminal
10 Philip Kitcher and William Aspray
article containing two distinct arguments against the idea that logic and
mathematics are true by convention. The first of these arguments con-
tends that any part of our knowledge can be replaced by a system whose
principles are true by convention. Thus it appears (although Quine does
not hammer home the point) that the thesis that logic and mathematics
are true by convention cannot explain the epistemological status of these
disciplines, unless their epistemological status is the same as that of other
branches of inquiry. The second argument, adapted from an idea of Lewis
Carroll, is that logic could not be true by convention for the simple reason
that logic is required to extract the consequences of any conventions we
may lay down.
Quine's second argument seems to have been more immediately influen-
tial than his first, for it prompted a reformulation of the canonical doc-
trines of the new logicism. Although logicists continued to describe logic
and mathematics as conventional, they preferred to speak of the truth of
these disciplines as resting on semantical rules—rules that were explicitly
formulated in the construction of formal systems aimed at explicating prior
mathematical usage and that were taken to underlie that prior usage—
rather than thinking of truth as resulting from an explicit convention (see
Carnap 1939; Hempel 1945). Quine's first argument unfolded into a cri-
tique of the notion of semantical rule (and the related notion of truth by
virtue of meaning) designed to circumvent these reformulations. Thus in
(1953), in (1962), and in a host of later writings, Quine argued at length
that there are no statements true by virtue of meaning. One consequence
of these arguments was a muted (and underdeveloped) empiricism.
As logicism suffered under Quine's attack on its fundamental concep-
tion of logical truth, a different Fregean theme underwent a revival. Since
he did not believe the truths of logic to be true in virtue of meaning and
therefore as devoid of factual content (as some of the positivists main-
tained), Frege saw no tension between claiming that arithmetic is disguised
logic and declaring that numbers are objects. Logical positivism
downplayed the latter declaration, and the positivists sometimes main-
tained that, as statements that are true in virtue of meaning, truths of logic
and mathematics are devoid of any reference. Quine's scrutiny of the con-
cept of truth by virtue of meaning led him to pose the ontological ques-
tions: what, if anything, is mathematics about? Accepting Frege's analysis
of the logical forms of arithmetical statements and adopting the seman-
AN OPINIONATED INTRODUCTION 11
which has been pursued in his own subsequent work and (in different ways)
in Gottlieb (1980) and Field (1980, 1982, 1984, 1985).
However, the most influential study of the new orthodoxy in philosophy
of mathematics was another article by Benacerraf (1973). Benacerraf con-
structed a dilemma, designed to show that our best views about mathe-
matical truth do not fit with our ideas about mathematical knowledge.
Start from the relatively unproblematic theses that there are some math-
ematical statements that are true and that some of these are known by
some people. Then a task of philosophy is to provide an account of the
truth of those statements that are true and an account of how we know
the mathematics that we know. On our best accounts of truth in general
and of the form of mathematical statements (accounts that derive from
Frege and Tarski), true statements of mathematics owe their truth to the
properties of and relations among mathematical objects (paradigmatical-
ly sets). It appears that such objects are outside space and time—for there
seem to be too few spatiotemporal objects to go around, and, in any case,
the truths of mathematics seem independent of the fates of particular
spatiotemporal entities. According to Benacerraf, on our best account of
human knowledge, knowledge requires a causal connection between the
knower and the objects about which the knower knows. Since mathemati-
cal objects are outside space and time, there can be no such connection
between them and human beings. Hence, on the best accounts of truth
and knowledge, mathematical knowledge turns out to be impossible after
all.
Since the early 1970s, much of the research in the philosophy of
mathematics has been devoted to evaluating the merits of neo-Fregeanism
in the light of the studies that I have briefly reviewed. Particularly impor-
tant has been the task of finding an appropriate response to Benacerraf's
dilemma. Some writers (notably Steiner 1975; Maddy 1980; Kim 1982)
have argued that neo-Fregeanism is unthreatened by the dilemma: they
hold that a proper understanding of the conditions on human knowledge
will allow for us to have knowledge of objects that are outside space and
time. Others (Lear 1977; Jubien 1977) have deepened the dilemma and
have proposed revisions of neo-Fregeanism to accommodate it. In addi-
tion, some writers have proposed approaches to the ontology of
mathematics that question the thesis that mathematical objects are sets.
The less radical of these is structuralism, the doctrine that mathematics
describes the properties of mathematical structures. More radical is the
nominalism of Chihara, Gottlieb, and Field.
AN OPINIONATED INTRODUCTION 75
obvious to the reader who is following the story for the first time. Although
our tale begins with the researches of mathematicians, originating with
questions that seem to arise from the mathematics of the late nineteenth
century, and although, with the discovery of the paradoxes, the task of
providing foundations for mathematics seems to assume some mathe-
matical importance, the distance between the philosophical mainstream
and the practice of mathematics seems to grow throughout the twentieth
century. Philosophy of mathematics appears to become a microcosm for
the most general and central issues in philosophy—issues in epistemology,
metaphysics, and philosophy of language—and the study of those parts
of mathematics to which philosophers most often attend (logic, set theory,
arithmetic) seems designed to test the merits of large philosophical views
about the existence of abstract entities or the tenability of a certain pic-
ture of human knowledge. There is surely nothing wrong with the pursuit
of such investigations, irrelevant though they may be to the concerns of
mathematicians and historians of mathematics. Yet it is pertinent to ask
whether there are not also other tasks for the philosophy of mathematics,
tasks that arise either from the current practice of mathematics or from
the history of the subject.
A small number of philosophers (including one of us) believe that the
answer is yes. Despite large disagreements among the members of this
group, proponents of the minority tradition share the view that philosophy
of mathematics ought to concern itself with the kinds of issues that oc-
cupy those who study other branches of human knowledge (most obviously
the natural sciences). Philosophers should pose such questions as: How
does mathematical knowledge grow? What is mathematical progress?
What makes some mathematical ideas (or theories) better than others?
What is mathematical explanation? Ideally, such questions should be ad-
dressed from the perspective of many areas of mathematics, past and pre-
sent. But, because the tradition is so recent, it now consists of a small
number of scattered studies, studies that may not address the problems
that are of most concern to mathematicians and historians or explore
episodes or areas within mathematics that most require illumination.
If the mainstream began with Frege, then the origin of the maverick
tradition is a series of four papers by Lakatos, published in 1963-64 and
later collected into a book (1976). Echoing Popper, Lakatos chose the title
Proofs and Refutations, and the choice is, in part, apt. One obvious theme
of the essays is the discussion of a segment of the history of mathematics
18 Philip Kitcher and William Aspray
led to a natural alliance between the fledgling history of science and the
scientific disciplines and to a unified approach that "explored the filia-
tion and intellectual contexts of successful ideas" (Thackray 1983, 17).
In the 1950s the first professional historians of science entered the
universities, bringing with them an increase in the number and sophistica-
tion of historical studies of science. Most of these studies were internalist,
but a separate "externalist" tradition developed, inspired by the work of
Robert Merton. The externalist studies traced the social structure of the
scientific community and the relation of this community to the external
world. Internalist and externalist approaches were seldom employed in
the same study.
In this period the rise of the history of science profession had little im-
pact on the history of modern mathematics, except perhaps to publicize
the importance and legitimacy of historical study and to stimulate the
mathematical community to produce greater numbers of collected works,
monographs, and smaller studies. Modern mathematics received almost
no attention from historians of science in this period for at least two
reasons. First, the number of practitioners was small and they tended to
be scientific generalists, or at least to focus on a wide range of scientific
activities in a single chronological period. Mathematics, especially after
the mid-eighteenth century, was considered to have a content and method
distinct from the other sciences, and even today this belief creates a gulf
between historians of modern mathematics and historians of other modern
sciences. Second, following Sarton and Merton, early historians of science
were attracted to the great revolutions in science—particularly to the Scien-
tific Revolution. Thus, the little energy of early historians of science for
mathematics was consumed in the study of topics from the sixteenth
through the early eighteenth centuries, particularly the rise of algebra, the
beginnings of analytic geometry, and, most of all, the calculus.
In the 1960s and early 1970s the history of science profession grew at
a rapid pace, stimulated in part by the substantial government support
to science and its cultural study. As the number of practitioners increased,
subspecialties in histories of the individual sciences emerged. History of
science programs—Harvard and Wisconsin in particular—produced their
first historians of modern mathematics: Michael Crowe, Joseph Dauben,
Judith Grabiner, Thomas Hawkins, Uta Merzbach, and Helena Pycior.
Since then, a small but continuous stream of young historians with train-
ing in both history of science and mathematics has entered the field.
AN OPINIONATED INTRODUCTION 23
The line drawn between internal and external history of science was
too tenuous to resist fading. In the 1970s and 1980s, historians of science
have balanced their internalist studies with social studies and have advanced
a more sophisticated historiography incorporating both internal and ex-
ternal factors. These studies increasingly consider social factors when ex-
amining not only the institutions of science, but also the form and con-
tent of scientific ideas.
Social histories of modern mathematics are relatively uncommon, prob-
ably because in comparison with other sciences mathematics is regarded
as least affected by factors beyond its intellectual content. Yet mathemati-
cians have long recognized the importance of communities such as those
in Gottingen, Paris, Berlin, and Cambridge in sponsoring particular styles
of research and producing certain kinds of research mathematicians; a
number of studies of institutions, their educational programs (Biermann
1973), and the individuals who shaped them (Reid 1970, 1976) have
appeared.
For Sarton, the study of national science made no sense because history
of science was the study of scientific ideas, which knew no national bound-
aries. But mathematicians have long understood that there are national
differences, both in the subjects studied and the way in which they are
approached. One famous example is the geometric approach to calculus
in vogue in eighteenth- and early-nineteenth-century Britain in contrast
to the analytic approach of Leibniz favored on the Continent. But con-
trasts can be drawn in nineteenth-century mathematics, as well, for ex-
ample, between German and British approaches to applied mathematics,
between the Italian and French work in projective geometry and the Ger-
man work in transformational geometry, and in the German dominance
of the arithmetization of analysis. The appreciation of national styles of
mathematics has resulted in a few studies, for example of the introduc-
tion of Continental methods into British analysis (Enros 1979). American
mathematics has come under close scrutiny, partly through the interests
of American mathematical societies and further encouraged by the na-
tion's bicentennial celebration in 1976 (Tarwater 1977). In the 1980s,
American mathematics has become an active area of research.
Although mathematicians and historians have come to understand the
value of studying professional societies, journals, prizes, institutions, fund-
ing agencies, and curricula, they have considerably less appreciation for
the study of the social roots of the form and content of mathematics. This
24 Philip Kitcher and William Aspray
is evidence of the firmly seated belief that mathematicians but not their
ideas may be affected by external factors. This attitude is slowly begin-
ning to change, as Daston (this volume) and others are able to demonstrate
the interplay between social factors and mathematical ideas. Another group
assaulting this belief are the feminist historians (J. LePage, E. Fee, E.
F. Keller), who variously are trying to establish cognitive differences be-
tween male and female mathematicians and to explain why mathematics
has long been considered a male vocation.
Historical studies conducted before the 1960s sometimes projected con-
temporary standards of proof, rigor, problem definition, and discipline
boundary onto their mathematical subject. The effect was to "explain"
the subject in anachronistic concepts and terminology, to select topics for
study only insofar as they had a connection to more recent developments,
and to praise or damn these efforts on the basis of whether they anticipated
(took a step toward) the current state of mathematical knowledge. Thus,
the focus was on the great "successes" and sometimes the "blunders"
of the greatest mathematicians. To accomplish this they studied the great
men (there were almost no women mathematicians) without considering
the lesser, but able practitioners that comprised the wider mathematical
community. These studies of the great ideas were accompanied by anec-
dotal biographies of the great men, in the worst cases amounting to no
more than hagiographic tributes. These authors relied principally upon
their mathematical acumen and personal experiences to evaluate the
published corpus of the great mathematicians, instead of examining a wider
range of published and unpublished documents.
One historian of science has noticed a similar trend in historical writing
about the sciences (Thackray 1983, 33):
Discipline history by scientists has usually been based on an in-
dividualistic epistemology, in keeping with the image of the scientist
as one voyaging through strange seas of thought, alone. There are also
individualistic property relations in science, giving importance to the
adjudication of rival claims. One result has been an historical interest
in questions of priority—of who first exposed "error" and established
"right" answers, or who developed successful instruments and
techniques.
With the advent of a professional history of science, a new and more
sophisticated historiography has arisen and is being put into practice in
the history of mathematics. This historiography measures events of the
AN OPINIONATED INTRODUCTION 25
past against the standards of their time, not against the mathematical prac-
tices of today. The focus is on understanding the thought of the period,
independent of whether it is right or wrong by today's account. The
historiography is more philosophically sensitive in its understanding of
the nature of mathematical truth and rigor, and it recognizes that these
concepts have not remained invariant over time. This new historiography
requires an investigation of a richer body of published and unpublished
sources. It does not focus so exclusively on the great mathematicians of
an era, but considers the work produced by the journeymen of mathematics
and related scientific disciplines. It also investigates the social roots of
mathematics: the research programs of institutions and nations; the im-
pact of mathematical patronage; professionalization through societies,
journals, education, and employment; and how these and other social fac-
tors shape the form and content of mathematical ideas.
The new historiography has not been universally adopted. Historical
works of the older style continue to be written by mathematicians and
some historians. Perhaps because of the perceived differences in method
and content between mathematics and the other sciences and because of
the slow rate at which the history of science profession has produced
scholars interested in modern mathematics, historiographic change has
been relatively slow in coming. However, one area of considerable activ-
ity is the preservation and use of archival materials. The mathematics com-
munity has a long tradition of publishing collected papers of eminent
mathematicians. This tradition continues today, but attention is also be-
ing given to the publication of collections of unpublished manuscripts and
correspondence such as those of Wiener, Godel, and Russell that are now
in production. Many fine European archival collections relating to
nineteenth-century mathematics (e.g., at Institute Mittag-Leffler, the Berlin
Akademie der Wissenschaften der DDR, the West Berlin Staatsbibliothek
Preussischen Kulturbesitz, and others in Cambridge, Freiburg, and Got-
tingen) have been in existence for many years, but they have received lit-
tle attention over the years. Recently, historians have used these materials
to great effect in preparing new interpretations and more accurate accounts
of classic events—for example, Dauben (1979) on Cantor, Moore (1982)
on the set-theoretic paradoxes, and Hawkins (1984) on the Erlanger Pro-
gram m. Others have used these sources to pioneer new areas—for exam-
ple, the work of Cooke (1984), Koblitz (1983), and Kochina (1981) on
Kovalevskaya.
26 Philip Kitcher and William Aspray
Institutions have been founded in the last decade with a major objec-
tive of collecting and preserving important archival materials on
mathematics. These include the Archives of American Mathematics at the
Humanities Research Center of the University of Texas, the Charles Bab-
bage Institute for the History of Information Processing Archives at the
University of Minnesota, the Bertrand Russell Archives at McMaster
University, and the Contemporary Scientific Archives Center at Oxford
University. For more information on archival resources in repositories in
the United States, see Merzbach (1985).
The new professionalism in history of mathematics is reflected in the
formation and growth of specialist societies and journals in the 1970s
and 1980s. These include the Canadian and British societies for the his-
tory of mathematics and the journals Historia Mathematica, Archive for
History of Exact Science, History and Philosophy of Logic, Annals of
the History of Computing, Mathematical Intelligencer, and Bolletino
di storia della scienze matematiche. Joseph Dauben, Ivor Grattan-
Guinness, and Kenneth May have made noteworthy contributions to
professionalization.
The professionalization of the history of mathematics has stimulated
the production of a rich set of publications. The remainder of this section
presents a brief survey of the literature of the 1970s and 1980s. For reasons
of space and manageability, the survey is restricted to full-length studies.
This does not do justice to the contributions of some scholars, like Thomas
Hawkins or Helena Pycior, who contribute primarily through journal ar-
ticles. For a fuller discussion of both book and journal literature see
Grattan-Guinness (1977), Jayawardene (1983), and Dauben (1985).
Historians of mathematics have long appreciated the value of
bibliographies. Dauben (1985, p. xxii) lists nine bibliographies produced
in the nineteenth century, and his list does not include important ones
appearing in the journal literature, notably the bibliographies produced
between 1877 and 1900 by Moritz Cantor in Zeitschrift fur Mathematik
und Physik. In the twentieth century there have been four major
bibliographies of the history of mathematics: Sarton (1936), Loria (1946),
May (1973), and Dauben (1985). The two of greatest research value to-
day are the last two. May's bibliography is intended to be comprehen-
sive, and therefore lists everything of which he knew, whereas Dauben's
is selective, critical, and annotated. Sarton and May in particular saw their
AN OPINIONATED INTRODUCTION 27
over the last ten decades, with little activity in the 1940s (because of the
war, presumably) and a slight increase in the 1960s and 1970s. Volumes
published in recent years seem to be more sensitive to historical concerns,
for example in being less likely to introduce anachronistic modernization
in notation. In fact, many of the recent collections have simply photo-
reproduced the articles as they originally appeared in print. In the last
ten years a number of the older of these collected works have been re-
printed, demonstrating their enduring value. Although collected works
are primarily published to meet mathematical research needs—at least in
the case of modern mathematicians, whose collected works are published
in far greater (absolute) numbers than those of "mathematicians" of the
sixteenth through eighteenth centuries—they also serve the historical com-
munity. Thus the number published and kept in print is likely to increase.
Dauben (1985) also identifies about 75 mathematicians (active before
World War I) whose correspondence has been published. Of these a
disproportionately high percentage (in comparison to the number of
mathematical practitioners at different periods of time) flourished prior
to the nineteenth century, perhaps indicating the unavailability of more
recent correspondence. It may also result from a lessening in the number
and quality of letters over the last century with the improved opportunities
for mathematical communication through journals and professional
meetings, easier travel, and more recently the widespread use of the
telephone and the computer. Or perhaps an explanation is provided by
the fact that early mathematicians were often involved in a range of scien-
tific activities, that history of science (especially of the Scientific Revolu-
tion) matured more rapidly than history of modern mathematics, and that
historians place greater value on correspondence than scientists do. If the
history of modern mathematics follows the trend in the history of science,
the number of volumes of collected correspondence will grow and will
have increasingly sophisticated annotation and analysis.
Biographies of eminent mathematicians have been popular among
mathematicians and mathematics educators since early in the century, when
history was first perceived as an effective introduction to the culture of
mathematics. Before that time, mathematical biographies had appeared
only occasionally, often written by younger contemporaries of the
biographical subject, such as Koenigsberger (1904) on Jacobi. Although
full-length biographies were written in the early twentieth century, most
biographical writings were semipopular sketches emphasizing the personal
and the anecdotal. Some of these were very popular, serving pedagogic
AN OPINIONATED INTRODUCTION 29
3. The Essays
We have divided the essays in this volume into four sections with the
aim of bringing together papers that address similar topics or share com-
mon themes or approaches. The first section consists of three studies that
tackle the traditional area of concern to philosophers of mathematics—
logic and the foundations of mathematics. The second contains essays that
articulate the historian's enterprise in different ways, either by displaying
the structure of some particular episode of the history of modern
mathematics or by using a historical example to comment on how that
enterprise should be conducted. In the third section, historians and
philosophers of mathematics explore the two fields in attempts to find
illumination of one by the other. Finally, the fourth section comprises
two studies of the interactions between mathematics and the broader social
context.
In "Poincare Against the Logicists," Warren Goldfarb considers Poin-
care's criticisms of the logicist program. On Goldfarb's account, Poincare
was not primarily guilty of missing the point of the work of Frege, Russell,
Couturat, and others. His objections were founded in a quite different
conception of the philosophy of mathematics. The root of the difference
32 Philip Kitcher and William Aspray
a lot of mathematics—in fact, he hints, far more than they do. Unless
they are trained beyond "the current undergraduate curriculum and first-
year graduate courses," there is a danger that they will overlook those
episodes in the history of mathematics that are really significant and con-
centrate on peripheral issues. However, Askey does not simply suggest
that history of mathematics should be turned over to professional
mathematicians, as a recreation in which they can indulge when they take
themselves to be on the verge of their dotage. History has its own stan-
dards and methodological canons, and Askey, despite his keen interest
in history, is quite modest about his knowledge of these. The heart of his
paper is thus a plea for cooperation and development. Mathematicians
can contribute "protohistory" (our term, not his) by drawing the atten-
tion of historians to problems and episodes that are mathematically signifi-
cant. The mathematician, playing protohistorian, will not attempt any
coherent treatment of these episodes. The goal will simply be to assemble
"mathematical facts" whose normal form may be the attribution of a rela-
tion of kinship between the writings of a past mathematician and some
(perhaps sophisticated) piece of contemporary mathematics. Once con-
fronted with these suggestions of kinship, the historian must go to work,
following the canons and standards of the discipline. However, part of
Askey's message is that doing the work properly may involve a great deal
of further study in mathematics.
Historians are likely to view protohistory as incomplete in two respects.
The more obvious deficiency, touched on in the last paragraph, is that
the mathematician's recognition of kinship needs to be scrutinized from
the perspective of an understanding of the concepts and standards in force
in the historical epoch under study. In addition, we should not assume
that the most significant historical problems concern work that has any
straightforward connection with (or offers any anticipation of) problems
and methods of current interest. Askey seems to us tacitly to appreciate
the point when he couches his discussion in terms of the writing of history
that mathematicians will find interesting.
Askey illustrates his general proposal by recounting some of his own
historical research on series identities. After contending that some of the
historical attributions commonly made by professional mathematicians
seem erroneous, Askey recounts the reactions of historians to his work.
His tale seems clearly to be one of missed communication. The technical
discussions of series identities and hypergeometric series are dismissed as
AN OPINIONATED INTRODUCTION 39
relation of mathematics to the external world. She concludes that, for the
mathematicians of the Enlightenment, "abstract" mathematics occupied
one end "of a continuum along which mathematics was mixed with sen-
sible properties in varying proportions." "Mixed" mathematics occupied
much of the energy of eighteenth-century mathematicians, and yet, as
Daston notes, there was a sense that, in introducing a greater "mixture"
of sensible ideas, mathematicians ran the risk of error and "retrogression."
Daston's first illustration concerns the art of conjecture, and her discus-
sion centers on one celebrated problem and its impact. The problem is
the St. Petersburg paradox—a paradox in virtue of the fact that applica-
tion of the art of conjecture to a contrived game offers a recommenda-
tion that is intuitively unacceptable. Daston argues that we can only see
the St. Petersburg paradox as a paradox—that is, as the eighteenth-century
discussants saw it—if we recognize the status of the art of conjecture as
a piece of mixed mathematics. The ability of the problem to threaten the
credentials of probability theory must strike us as absurd if we approach
the situation from the perspective of our modern distinction between pure
and applied mathematics. We can consign the puzzle to economics. Our
eighteenth-century predecessors could not.
The second example concerns the use of probability in a legal context,
the probability of judgments. Daston explains how the mathematical com-
munity set itself the task of deciding on the optimal design of a tribunal
of judges, where the criterion for optimality consisted in minimizing the
risk of error. The hope, bizarre as it now seems, was that, by treating
judges as akin to dice, the practice of legal judgment could be "reduced
to a calculus." Daston relates the fate of this discipline, showing how the
understanding of legal reasoning became divorced from the calculus of
probabilities, so that a branch of "mixed mathematics" came to be viewed
as a faulty combination of an unassailable piece of pure mathematics
(probability theory) and a misguided application.
Finally, she presents us with one of the success stories in the "applica-
tion" of probability theory, the development of actuarial mathematics.
Here Daston seeks to understand why the use of probability theory in in-
surance took so long to become established, why the early ventures in ac-
tuarial mathematics were undertaken with such extreme caution, why an
antistatistical attitude was displaced so slowly. The kernel of her answer
is that the phenomena to be discussed seemed insusceptible to proper
mathematical analysis because it appeared that statistical treatment would
AN OPINIONATED INTRODUCTION 41
blur subtle distinctions that experienced insurers would be able to use ad-
vantageously in their decisions.
Taken together, the three examples show how the eighteenth-century
notion of mixed mathematics differed from our concept of applied
mathematics, how the distinction of a pure discipline from its applica-
tions can be achieved, and how mathematical theories are sometimes
dependent on the fates of their applications. It seems to us that Daston's
essay raises interesting philosophical questions concerning the traditional
topic of the relation of mathematics to reality and that her treatment of
the examples she has chosen offers some fruitful suggestions for address-
ing those issues. Moreover, her study shows the effect of the social con-
text on the development of a branch of mathematics. In this way, Daston
touches on themes akin to those pursued by Grabiner and Aspray.
If Daston demonstrates how the detailed study of the history of
mathematics can have significance for philosophical issues, then it seems
to us that Howard Stein's essay shows how attention to philosophical ques-
tions can shed considerable light on episodes in the history of mathematics.
Stein is explicitly concerned to trace the main foundational programs of
the early twentieth century to mathematical roots in the nineteenth cen-
tury. He begins from the thesis that mathematics underwent a "second
birth" in the nineteenth century and that it is a primary task for philosophy
to understand this transformation. After reviewing the early-nineteenth-
century developments in algebra, analysis, and geometry, Stein identifies
several "pivotal figures"—Dirichlet, Riemann and Dedekind—the latter
two greatly influenced by Dirichlet's teaching. With the filiations to the
early nineteenth century in place, he then begins a more detailed account
of some late-nineteenth-century developments.
The account starts with a problem, a problem that occupies anyone
who ponders Dedekind's dual status as a respected mathematician and
as a figure in the "foundations of mathematics." Why did Dedekind write
his monograph on the natural numbers (Was sind und was sollen die
Zahlen?)! Stein's answer consists in a careful tracing of the connections
between the project of this monograph and Dedekind's earlier work in
number theory, work that shows the influence of both Gauss and Dirichlet.
The account culminates in the contention that the famous supplement to
Dirichlet's lectures in number theory not only served as a major source
in the history of algebraic number theory, but was also the origin of some
of Dedekind's deepest philosophical ideas. Stein uses this contention to
42 Philip Kitcher and William Aspray
as the key to solving all kinds of social problems. These enthusiastic ex-
trapolations provoked continued criticism from outsiders, who objected
even to the early achievements of Darwinian evolutionary theory. Final-
ly, the more thorough internal critique from practicing biologists and social
scientists trimmed away the excesses of social Darwinism while preserv-
ing the genuine biological successes.
Grabiner suggests that we are currently in the middle of the last stage
in the debate about artificial intelligence. She outlines the early successes
of computer science and relates the confidence with which its advocates—
such as Simon, Newell, and McCarthy—predicted dramatic results both
in constructing machines that would perform all kinds of intelligent tasks
and in shedding light on the nature of human intelligence. Grabiner cites
the philosophers Dreyfus and Searle as prominent representatives of the
external critics, who announce the "triviality" of the entire venture. The
role of internal critic is filled by Weizenbaum, who has attempted to chart
the limits of artificial intelligence research instead of dismissing it entirely.
While Grabiner is concerned with the ways in which mathematical ideas
(and scientific ideas) are elaborated, disseminated and criticized in the
social context, William Aspray focuses on the impact of social structure
on the development of mathematics. Aspray considers one of the most
prominent success stories in the history of American mathematics, the
development of mathematics at Princeton in the early decades of the twen-
tieth century. He begins by giving a detailed account of the conditions
under which professional mathematicians worked at the end of the nine-
teenth century. After showing how heavy teaching loads and few incen-
tives to research were the order of the day, Aspray describes how Wilson's
presidency at Princeton University initiated the building of a modern
mathematics department.
A key figure in the building was Fine. Originally trained as a classicist,
Fine had pursued his mathematical education in Germany and had become
impressed with the high academic quality of the German universities.
Returning to Princeton, first as a professor and later as dean, Fine devoted
considerable energies to reorganizing the structure of appointments and
the commitments to research. Veblen was also instrumental in the develop-
ment of a research community in mathematics at the university, and Aspray
traces the ways in which mathematical research was fostered by Fine and
Veblen.
The implementation of the program would have been impossible
AN OPINIONATED INTRODUCTION 47
without funding from a wealthy patron, and Fine was fortunate to secure
support from a former classmate. One of the important consequences of
the influx of money was the opportunity to construct a building for the
mathematics department, Fine Hall, designed by Veblen to facilitate ex-
change of ideas among mathematicians. Aspray documents the significance
of this physical facility in attracting promising young mathematicians either
to come to Princeton or to return there. He also recounts the founding
of the Institute for Advanced Study and shows how cooperation between
the institute and the university further promoted Princeton as a center
for mathematical research. The exodus of brilliant mathematicians from
central Europe in the 1930s combined with the attractiveness of Princeton
as a haven to create an extraordinary mathematical community.
Although Aspray does not exhibit in detail how the institutional fac-
tors he describes actually influenced the development of any particular
mathematical field, his central message is abundantly clear. It is surely
hard to believe that the ideas of the great mathematicians who were at
Princeton in the 1930s were unaffected by their frequent exchanges with
one another. Those exchanges were made possible by a number of historical
contingencies: the drive of a university president, accidents in the educa-
tion of a dean, the lucky business success of a classmate, an unwonted
understanding of the importance of architecture, the emergence of an evil
dictatorship. We do not ordinarily think of the course of mathematics
as being affected by such chances. Aspray reminds us that they may easi-
ly leave their mark.
4. Common Themes and Possible Futures
We would like to conclude by pointing to some connections (and con-
trasts) among the essays and by offering some brief speculative comments
about the possible future of history of mathematics, philosophy of
mathematics, and history-and-philosophy of mathematics. Some common
themes are already indicated in our division of the articles. Other connec-
tions crisscross the groupings we have imposed.
A. Reading the past through the categories of the present. Many of
the writers are concerned to warn against the dangers of Whig history.
This is especially obvious in the essays of Dauben and Daston, but it is
implicit in the studies of Edwards, Moore, Crowe, and Kitcher as well.
Askey's paper serves as an important reminder that sensitivity to the
possibility that the conceptions of the past may not be those of the pre-
48 Philip Kitcher and William Aspray
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AN OPINIONATED INTRODUCTION 57
61
62 Warren Goldfarb
Poincare begins his attack on the new logic with the avowed aim of
showing that the new logicians have not eliminated the need for intuition
in mathematics. By showing this, he says, he is vindicating Kant (1905,
815). This avowal is misleading, for in Poincare's hands the notion of
intuition has little in common with the Kantian one. The surrounding Kan-
tian structure is completely lacking; there is no mention, for instance, of
sensibility or of the categories. Indeed, in (1900), his address to the Paris
Congress, Poincare explicitly separates mathematical intuition from im-
agination and the sensible. Thus there is no hint of an epistemological
framework in which the notion functions. For Poincare, to assert that
a mathematical truth is given to us by intuition amounts to nothing more
than that we recognize its truth and do not need, or do not feel a need,
to argue for it. Intuition, in this sense, is a psychological term; it might
just as well be called "immediate conviction."2 As a result of the unstruc-
tured nature of the notion, Poincare's argument in (1905) can be purely
negative. If a mathematical proposition is convincing, that is, it seems
self-evident to us, and the purported logical proofs of it are insufficient,
then, tautologously, intuition in Poincare's sense is what is at work.
Such a sense of "intuition" underlies another of Poincare's criticisms,
namely, that the logicists have done nothing but rename the chapter in
which certain truths are to be listed: what was called "mathematics" is
now called "logic." These truths, he says, "have not changed in nature,
they have changed only in place" (1905, 829). This is merely a renaming,
he claims, for since the axioms the logicists use are not merely "disguised
definitions," they must rest on intuition. He emphasizes this even with
respect to truth-functional logic. Here Poincare is simply ignoring a cen-
tral philosophical point of logicism. Frege and Russell take logic to con-
sist of those general principles that underlie all rational discourse and all
rational inference, in every area, about matters mathematical or empirical,
sensible or nonsensible. Consequently, the truth of these principles could
not stem from any faculty with a specialized purview; certainly, their truth
does not play any particular role in constituting the world of experience.
If one persists, as Poincare does, in claiming that intuition is needed, then
one has in fact robbed "intuition" of all content.3
Poincare also complains that logicism errs insofar as "in reducing
mathematical thought to an empty form.. .one mutilates it" (1905, 817).
64 Warren Goldfarb
good one but it has force only against a naive version of Hilbert's posi-
tion. Early on, Hilbert saw that if axioms are to endow talk of their ob-
jects with warrant, a consistency proof is needed. He was unclear about
the status of such a proof at first. Subsequently, he came to see that one
cannot hope to ground mathematics without residue in this way; rather,
the metamathematical reasoning employed has to be taken for granted.
His formalist program rests on the idea that the residue would be finitary
mathematics, a small part of number theory. Finitary mathematics includes
some amount of induction—although Hilbert was never explicit about the
matter—namely, induction on decidable number-theorectic predicates (that
is, quantifier-free induction). There is no petitio since there is no claim
that all number-theoretic principles are to be legitimized from scratch.
To be sure, Poincare's arguing in this manner is explicable. Hilbert's
sketch in (1904) of the application of his early ideas of implicit definition
to number theory is quite unclear. The distinction between mathematics
and metamathematics does not become rigorous until nearly two decades
later, and the question of limitation of means available in the metatheory
does not exist in 1904, even in embryo. More to the historical point,
Couturat (1904), to which Poincare is responding, does not distinguish
Hilbert's notion of implicit definition from the Frege-Dedekind-Russell
strategy for explicit definition of the numbers. Hence it is not surprising
that Poincare thinks of the project of reducing arithmetic to logic as be-
ing carried out by implicit definition. In point of fact, the logicists ob-
jected strongly to the idea of implicit definition. Frege and Russell (and,
later on, Couturat) insist that existence is not proved by consistency; rather,
consistency is vouchsafed by showing existence. Hence this form of the
argument does not touch logicism.
Another form is more relevant. In a sarcastic passage (1905, 823), Poin-
care examines definitions of particular numbers given in Peano-writing
by Burali-Forti (1897). He notices, in the definition of zero, the use of
the concept "in no case," and, in the definition of one, the concept that
would be rendered in ordinary language "a class that has only one
member." He concludes, "I do not see that the progress is considerable."
Similarly, he cites phrases used in introducing logical machinery, such as
"the logical product of two or more propositions," as showing that a
knowledge of number must be presupposed. In general,
It is impossible to give a definition without enunciating a phrase, and
difficult to enunciate a phrase without putting in it a name of a number,
66 Warren Goldfarb
II
Poincare first enunciates the vicious circle principle in (1906), and his
subsequent foundational writings focus almost exclusively on it. By 1906,
Poincare had gained a far more extensive acquaintance with the new logic.
In particular, he had read Whitehead (1902), in which the logicist con-
struction of the numbers is outlined, and Russell (1905), in which Russell
discusses various paradoxes and gives the famous delineation of routes
to avoid them—namely, the zigzag theory, the theory of limitation of size,
and the no-classes theory.
Poincare introduces the vicious circle principle in a discussion of
Richard's paradox, the paradox of the set of definable decimals. This set
is denumerable, and from an enumeration of it a Cantor-diagonal decimal
can be formed. That decimal is not among the enumerated ones but is,
it seems, definable and hence is in the set. Poincare draws the vicious cir-
cle principle from Richard's own solution of the paradox (Richard 1905).
The set E of definable decimals, Poincare says, must be construed as com-
posed just of the decimals that can be defined "without introducing the
notion of the set E itself. Failing that, the definition of E would contain
a vicious circle" (1906, 307). He then proposes as a general principle that
only those definitions that do not contain a vicious circle should be taken
to determine sets.
An analysis of the Richard paradox as resting on some type of circularity
is extremely plausible, since the paradox exploits a notion of definability
that allows definitions containing reference to the notion of definability
itself. Most contemporary solutions of the paradox recognize this and pro-
ceed by stratifying notions of definability. On such views, the circle that
would otherwise exist is peculiar to semantic or intensional notions like
"definability." Poincare's analysis, however, has a different cast. He
focuses his challenge not on the notion of definability but on the set of
decimals specified by invoking the notion.7 This leads Poincare to apply
the analysis more widely; he descries the same circularity in the set-theoretic
paradoxes as well.
Thus Poincare immediately goes on to claim that the vicious circle prin-
ciple solves the Burali-Forti paradox of the order type of the set of
ordinals—a paradox that Poincare, like Russell but unlike Burali-Forti
himself, recognized from the first to be a true paradox. He says, "One
introduces there the set E of all ordinal numbers; that means all ordinal
72 Warren Goldfarb
numbers that one can define without introducing the notion of the set
E itself" (1906, 307). Thus paradox is blocked. The extension of the vicious
circle analysis to this paradox is a bold step, for there is a real distinction
between this case and that of the Richard paradox. An item qualifies for
membership in the Richard set only if there is a sequence of words that
defines it. Since the truth or falsity of the latter condition may depend
on the composition of the Richard set, a circle is apparent. In contrast,
an item qualifies for membership in the Burali-Forti set if and only if it
is an ordinal, and this condition in no way depends on the Burali-Forti
set. Nonetheless, Poincare alleges a vicious circle: one that arises if the
condition is applied universally. Therefore, he takes the vicious circle prin-
ciple to restrict the range of candidates for membership in the set, name-
ly, to items that can be given independently of the set.
Poincare's treatment of the Burali-Forti paradox may have been in-
fluenced by a lesson Russell drew from it: "There are some properties
such that, given any class of terms all having such a property, we can
always define a new term also having the property in question. Hence we
can never collect all the terms having the said property into a whole"
(Russell 1905, 144). For Russell, this lesson does not constitute an analysis
or solution of the paradox; it is only an indication of what is to be analyzed.
Poincare can be seen as taking it in a more definitive manner: a specifica-
tion of a set must be interpreted as simply not applicable to any "new"
term producible from that set.
So far, then, Poincare is using the vicious circle principle to bar from
membership in a set anything that in some sense presupposes that set. In
this form, the principle can also be used to block the Cantor and Russell
paradoxes. This is not yet, however, the full strength of the principle that
Poincare urges. (Indeed, under a suitable interpretation of "presupposes,"
both the simple theory of types and the so-called iterative conception of
set wind up abiding by a restriction of this sort.)
The full force of Poincare's principle first emerges in his use of it to
criticize the logicist definition of number. A finite number is defined as
a number that belongs to every inductive set, that is, every set that con-
tains 0 and contains n + 1 whenever it contains n. Poincare claims that
to avoid a vicious circle, the inductive sets invoked in the definition can-
not include those specified by reference to the set of finite numbers. If
such a restriction is adopted, then mathematical induction cannot be de-
duced for such sets; many basic laws of arithmetic become unprovable,
POINCARE AGAINST THE LOGICISTS 73
and the logicist claim is defeated. Now, the vicious circle principle, so con-
strued, restricts not just the candidates for membership in the set being
defined, but also the range of the quantifiers in the definition. This yields
the notion of predicativity in its modern sense: quantifiers occurring in
specifications of sets must be construed as not including in their range
of variation the set that is being defined, or anything defined by reference
to that set.8
Poincare makes one further application of the vicious circle principle
in (1906). Earlier (1905, 25-29), he had pointed out that the then extant
proof of the Schroder-Bernstein theorem invoked mathematical induction
by utilizing an inductive construction of a sequence of sets. This belied
the claim that cardinal number theory can be developed without special
mention of the finite numbers. In 1906 Zermelo sends Poincare a different
proof: instead of an inductive construction, the intersection U of all sets
with a certain property P is formed; the theorem is obtained by showing
that a particular set defined from U has property P, and thus is among
the sets intersected. Poincare invokes the vicious circle principle to block
the argument, for it rules out from the sets intersected any set in whose
definition the intersection U itself figures. Here again, the principle is taken
to restrict the range of quantifiers in the definition of a set.
The restriction Poincare urges can be put, in the formal mode, thus:
the quantifiers in a specification of a set cannot legitimately be instan-
tiated by names that contain reference to that set. In (1906) he gives a
general reason for his restriction, namely, that a purely logical proof must
start from identities and definitions, and proceed in such a way that when
definienda are replaced by definientia, one obtains an "immense
tautology" (1906, 316). Impredicative definitions block such a replace-
ment; hence they cannot be allowed in logic. In the polemical setting of
1906, this remark cuts no ice: logicists and set theorists did not think that
their proofs must reduce to tautologies. On the technical level, however,
it is a tremendously prescient remark. Although adherence to a predicativ-
ity constraint does not ensure reducibility to tautology in a literal sense,
in many settings it does yield conservative extension results, which fail
once impredicative definitions are allowed.
Poincare's proposal met much criticism. In particular, Russell and
Zermelo both found the same flaw in it: the very formulation of the vicious
circle principle, they charge, violates the principle.
It is precisely the form of definition said to be predicative that con-
74 Warren Goldfarb
tains something circular; for unless we already have the notion, we can-
not know at all what objects might at some time be determined by it
and would therefore have to be excluded. (Zermelo 1908, 191)
That is, if the specification "class of $s" is taken as meaning "class of
things that have $ but do not presuppose the class of 4>s," then the
specification involves the notion of the class of $s, and so a vicious circle
has been introduced. Russell repeats the point often, and likens it to an
attempt to avoid insulting a person with a long nose by remarking "When
I speak of noses, I except such as are inordinately long" (Russell 1908,155).
Now, as was pointed out above, for Poincare the vicious circle princi-
ple amounts to a restriction on legitimate instantiations for the quantified
variables in a specification of a set. In their criticism, Zermelo and Russell
are assuming that such a restriction must arise from explicit limitations
on the quantifiers, limitations that are part of the content of the specifica-
tion. It is those explicit limitations that introduce the circle. Poincare never
answered this objection. For him, one presumes, a specification of a set
stands as is, without explicit reference to the cases that are ruled out. The
effect of the vicious circle principle comes in the application of the
specification. The restriction on instantiations need not be grounded in
an explicit anterior fixing of the quantifiers' ranges.
The difference between Poincare and the logicians here is fundamen-
tal. For Russell and Zermelo, inference is a matter of universal and subject-
neutral logical laws. An inference seemingly licensed by such a general
law can be blocked only by so construing the content of the statement
concerned as to make the law inapplicable. In this sense, logic is the ar-
biter of content. But, as we may surmise from section I, Poincare does
not share this conception of universal logical laws; hence he does not
recognize any need for a notion of fixed content to ground restrictions
on inference.
Such a denial is unintelligible from Russell's and Zermelo's point of
view and is unsatisfactory from most contemporary ones. Contemporary
predicative systems arrange their entities in a hierarchy; a definition with
quantifiers ranging over entities of level at most n in the hierarchy
automatically defines something at level n + 1. As a result, the defined
entity need not be invoked to restrict the ranges of the quantifiers in its
definition.
POINCARE AGAINST THE LOGICISTS 75
apply to all of mathematics. Underlying this step is the idea that all en-
tities that mathematics legitimately treats must be definable. Indeed, Poin-
care urges, "the only objects about which it is permissible to reason are
those which can be defined in a finite number of words (1909b, 464). Yet
since, for Poincare, definability has no formal analysis, no prior limit can
be put on the objects that may, one day, be subject to our reasoning. Poin-
care's rejection of the notion of a fixed range of quantifiers is explicit
here. A universal theorem does not relate to all objects, imaginable or
not, in such a range; rather, it asserts only that each particular case of
the theorem—each case defined in a finite number of words that will be
considered by the mathematicians or by succeeding generations of
mathematicians—can be verified (1909b, 480).
This construal of generality, and the insistence on definability, reflect
Poincare's repeated denial of the existence of the actual infinite. "Every
proposition about the infinite should be the translation, the abbreviated
statement, of propositions about the finite" (1909b, 482). Poincare is not
here calling for some kind of finitist reduction, however. As a result of
the lack of predetermined limits to "definability," the array of proposi-
tions about the finite that are abbreviated by a given proposition about
the infinite is open-ended rather than fixed.
From all this, the view of mathematics that moves Poincare may be
surmised. Mathematics is a matter of our thinking and verbalizing
mathematically. There is nothing beyond our words and thoughts that can
anchor or ground the discipline. The idea that our classifications are what
we reason about, is a direct rejection of the tenet, central to logicism (but
not limited to it), that logic applies to a realm of fixed content. Once that
tenet is rejected, logic cannot be thought of as the basic rationality-framing
subject that Frege and Russell took it to be; indeed, logic loses all claim
to priority. It is no wonder, then, that Poincare is blind to the advances
of the new logic and resists the purely logical sense of presupposition.
Similarly, Poincare's view of the vicious circle principle as governing ap-
plications of definitions, rather than as constraining the content of those
definitions, becomes explicable. The immutability of our classifications
is to be secured by our subsequent behavior: we must not do anything
that would cause mutation. Immutability is not secured by reference to
a logical structure present from the start.
The priority of the finite in Poincare also stems from his psychologism.
The finitude of our minds is, presumably, the basis for his insistence that
POINCARE AGAINST THE LOGICISTS 79
Notes
1. This marks a distinction between Poincare's writings on geometry and those on foun-
dations. (I use the latter term throughout this paper to mean foundations of analysis and
number theory.)
2. Poincard's psychological rendering of Kantian terms is explicit in an earlier article,
where he says that mathematical induction is "an affirmation of a power of the mind" and
that "the mind has a direct intuition of this power" (Poincare 1894, 382).
3. For more on this theme, see Goldfarb (1982). In that article, I also note that the ques-
tion of whether mathematics is analytic or synthetic in Kant's sense is not an issue for the
logicists, since the new logic renders the distinction as Kant drew it philosophically insignifi-
cant. (Frege in [1884] redefines "analytic"; Russell in [1902] baldly asserts that logic is syn-
thetic in Kant's sense.) Poincare is unaware of this. He represents the logicists as claiming
that mathematics is analytic in Kant's sense, and he often speaks of his own aim, against
the logicists, as being that of showing this claim false.
4. In an article on the teaching of mathematics (1899), Poincare stresses the importance
of imparting intuition, in just this sense.
5. A similar point is made by Russell in explicit reply to Poincare (Russell 1910, 252).
6.1 have in mind here particularly the approach urged by Kitcher (1984). I do not mean
to include Quine's very different sort of naturalism.
7. On this point Poincar6 was quickly criticized by Giuseppe Peano (1906). The distinc-
tion Peano suggests, between semantic paradoxes and set-theoretic paradoxes, was given
canonical form in Ramsey (1925).
8. A terminological note: Originally Russell used "predicative" for properties that define
classes, i.e., those $ such that [x|$x] exists. PoincarS proposes in (1906) that properties
are predicative only if they contain no vicious circles. In later writings, Poincar6 simply defines
"predicative" to mean containing no vicious circles. Russell follows this later usage, which
is the one that has come down to us.
9. The justifications for Russell's ramified theory of types are discussed in detail in
Goldfarb (forthcoming).
10. The distinction between greatest lower bound and minimum became a well-known
issue in late-nineteenth-century analysis, through critical scrutiny of Dirichlet's fallacious
argument for the Dirichlet Principle, which relied on conflation of the two. The problem
of finding a correct proof for the principle received much attention, and Poincar6 himself
contributed to its solution. Monna (1975) gives an excellent historical account of this issue.
11.1 am grateful to Burton Dreben, Peter Hylton, and Thomas Ricketts for helpful sug-
gestions and comments.
References
Burali-Forti, Cesare. 1897. Una questione sui numeri transfinit. Rendiconti del Circolo
Matematico di Palermo 11: 154-64. Translation in (van Heijenoort 1967), pp. 105-11.
Couturat, Louis. 1904. Les principes des math&natiques. Revue de Metaphysique et de Morale
12: 19-50, 211-40, 341-83, 664-98, 810-44.
1906. Pour la logistique (Reponse & M. Poincard). Revue de Metaphysique et de Morale
14: 208-50.
POINCARE AGAINST THE LOGICISTS 81
82
LOGICAL TRUTH AND ANALYTICITY IN CARNAP 83
nap's claim to reconcile Frege and Hilbert appears hollow indeed. What
he has actually done, it seems, is thrown away all that is most interesting
and characteristic in both views.
Such an evaluation would be both premature and fundamentally un-
fair, however. To see why, we must look more closely at the centerpiece
of Carnap's philosophy—his conception of analytic truth—and how that
conception evolves from Frege's while incorporating post-Fregean ad-
vances in logic: in particular, advances due to Hilbert and Godel.
The first point to bear in mind is the familiar one that Frege's con-
struction of arithmetic is not simply the embedding of a special math-
ematical theory (arithmetic) in a more general one (set theory). Frege's
Begriffsschrift is not intended to be a mathematical theory at all; rather,
it is to function as the logical framework that governs all rational think-
ing (and therefore all particular theories) whatsoever. As such, it has no
special subject matter (the universe of sets, for example) with which we
are acquainted by "intuition" or any other special faculty. The principles
and theorems of the Begriffsschrift are implicit in the requirements of any
coherent thinking about anything at all, and this is how Frege's construc-
tion of arithmetic within the Begriffsschrift is to provide an answer to
Kant: arithmetic is in no sense dependent on our spatiotemporal intuition
but is built in to the most general conditions of thought itself. This, in
the end, is the force of Frege's claim to have established the analyticity
of arithmetic.
But why should we think that the principles of Frege's new logic delimit
the most general conditions of all rational thinking? Wittgenstein's Trac-
tatus attempts to provide an answer: this new logic is itself built in to any
system of representation we are willing to call a language. For, from Witt-
genstein's point of view, the Begriffsschrift rests on two basic ides: Frege's
function/argument analysis of predication and quantification, and the
iterative construction of complex expressions from simpler expressions via
truth-functions. So any language in which we can discern both func-
tion/argument structure—in essence, where there are grammatical
categories of intersubstitutable terms—and truth-functional iterative con-
structions will automatically contain all the logical forms and principles
of the new logic as well. Since it is plausible to suppose that any system
of representation lacking these two features cannot count as a language
in any interesting sense, it makes perfectly good sense to view the new
logic as delimiting the general conditions of any rational thinking what-
LOGICAL TRUTH AND ANALYTICITY IN CARNAP 85
soever. For the new logic is now seen as embodying the most general con-
ditions of meaningfulness (meaningful representation) as such.10
Carnap enthusiastically endorses this Wittgensteinian interpretation of
Frege's conception of analyticity, and he is quite explicit about his debt
to Wittgenstein throughout Logical Syntax (§§14, 34a, 52) and throughout
his career.11 Yet, at the same time, Carnap radically transforms the con-
ception of the Tractatus, and he does this by emphasizing themes that
are only implicit in Wittgenstein's thought. It is here, in fact, that Car-
nap brings to bear the work of Hilbert and Godel in a most decisive
fashion.
First of all, Carnap interprets Wittgenstein's elucidations of the no-
tions of language, logical truth, logical form, and so on as definitions in
formal syntax. They are themselves formulated in a metalanguage or
"syntax-language," and they concern the syntactic structure either of some
particular object-language or of languages in general:
All questions of logic (taking this word in a very wide sense, but ex-
cluding all empirical and therewith all psychological reference) belong
to syntax. As soon as logic is formulated in an exact manner, it turns
out to be nothing other than the syntax either of some particular
language or of languages in general. (§62)
This syntactic interpretation of logic is of course completely foreign to
Wittgenstein himself. For Wittgenstein, there can be only one language—
the single interconnected system of propositions within which everything
that can be said must ultimately find a place; and there is no way to get
"outside" this system so as to state or describe its logical structure: there
can be no syntactic metalanguage. Hence logic and all "formal concepts"
must remain ineffable in the Tractatus.12 Yet Carnap takes the work of
Hilbert and especially Godel to have decisively refuted these Wittgenstein-
ian ideas (see especially §73). Syntax (and therefore logic) can be exactly
formulated; and, in particular, if our object-language contains primitive
recursive arithmetic, the syntax of our language (and every other language)
can be formulated within this language itself (§18).
Secondly, Carnap also clearly recognizes that the linguistic or "syn-
tactic" conception of analyticity developed in the Tractatus is much too
weak to embrace all of classical mathematics or all of Frege's Begriff-
sschrift. For the two devices of function/argument structure (substitution)
and iterative truth-functional construction that were seen to underly Frege's
distinctive analysis of predication and quantification do not lead us to
the rich higher-order principles of classical analysis and set theory. As
86 Michael Friedman
Alas, however, it was not meant to be. For Godel's results decisively
undermine Carnap's program after all. To see this, we have to be a bit
more explicit about the details of the program. For Carnap, a language
or linguistic framework is syntactically specified by its formation and
transformation rules, where these latter specify both axioms and rules of
inference. The language in question is then characterized by its
consequence-relation, which is defined in familiar ways from the underly-
ing transformation rules. Now such a language or linguistic framework
will contain both formal and empirical components, both "logical" and
"physical' rules. Language II, for example, will not only contain classical
mathematics but classical physics as well, including "physical" primitive
terms (§40)—such as a functor representing the electromagnetic field—
and "physical" primitive axioms (§82)—such as Maxwell's equations. The
task of defining analytic-for-a-language, then, is to show how to distinguish
these two components: in Carnap's technical terminology, to distinguish
L-rules from P-rules, L-consequence from P-consequence (§§51, 52).
How is this distinction to be drawn? Carnap proceeds on the basis of
a prior distinction between logical and descriptive expressions (§50). In-
tuitively, logical expressions include logical constants in the usual sense
(connectives and quantifiers) plus primitive expressions of arithmetic (the
numerals, successor, addition, multiplication, and so on). Given the
distinction between logical and descriptive expressions, we then define the
analytic (L-true) sentences of a language as those theorems (L- or P-
consequences of the null set) that remain theorems under all possible
substitutions of descriptive expressions (§51). In other worlds, what we
might call "descriptive invariance" separates the L-consequences from
the wider class of consequences simpliciter. But how is the distinction be-
tween logical and descriptive expressions itself to be drawn? Here Car-
nap appeals to the determinacy of logic and mathematics (§50): logical
expressions are just those expressions such that every sentence built up
from them alone is decided one way or another by the rules (L-rules or
P-rules) of the language. That is, every sentence built up from logical ex-
pressions alone is provable or refutable on the basis of these rules. In the
case of descriptive expressions, by contrast, although some sentences built
up from them will no doubt be provable or refutable as well (in virtue
of P-rules, for example), this will not be true for all such sentences—for
sentences ascribing particular values of the electromagnetic field to par-
LOGICAL TRUTH AND ANALYTICITY IN CARNAP 89
ticular space-time points, for example. In this way, Carnap intends to cap-
ture the idea that logic and mathematics are thoroughly a priori.
It is precisely here, of course, that Godelian complications arise. For,
if our consequence-relation is specified in terms of what Carnap calls
definite syntactic concepts—that is, if this relation is recursively
enumerable—then even the theorems of primitive recusive arithmetic
(Language I) fail to be analytic; and the situation is even worse, of course,
for full classical mathematics (Language II). Indeed, as we would now
put it, the set of (Godel numbers of) analytic sentences of classical first-
order number theory is not even an arithmetical set, so it certainly cannot
be specified by definite (recursive) means. Carnap himself is perfectly aware
of these facts, and this is why he explicitly adds what he calls indefinite
concepts to syntax (§45). In particular, he explicitly distinguishes (recur-
sive and recursively enumerable) d-terms or rules of derivation from (in
general nonarithmetical) c-terms or rules of consequence (§§47, 48).
Moreover, it is here that Carnap is compelled to supplement his "syn-
tactic" methods with techniques we now associate with the name of Tar-
ski: techniques we now call "semantic." In particular, the definition of
analytic-in-Language-H is, in effect, a truth-definition for classical
mathematics (§§34a-34d). Thus, if we think of Language II as containing
all types up to a> (all finite types), say, our definition' of analytic-in-
Language-II will be formulated in a stronger metalanguage containing
quantification over arbitrary sets of type o> as well. In general, then, Car-
nap's definition of analyticity for a language of any order will require
quantification over sets of still higher order. The extension of analytic-
in-L for any L will therefore depend on how quantifiers in a metalanguage
essentially richer than L are interpreted; the interpretation of quantifiers
in this metalanguage can only be fixed in a still stronger language; and
so on (§34d).14
But why should this circumstance cause any problems for Carnap? After
all, he himself is quite clear about the technical situation; yet he never-
theless sees no difficulty whatever for his logicist program. It is explicitly
granted that Godel's Theorem thereby undermines Hilbert's formalism;
but why should it refute Frege's logicism as well? The logicist has no special
commitment to the "constructive" or primitive recursive fragment of
mathematics: he is quite happy to embrace all of classical mathematics.
Indeed, Carnap, in his Principle of Tolerance, explicitly rejects all ques-
tions concerning the legitimacy or justification of classical mathematics.
90 Michael Friedman
do not, properly speaking, have meaning at all. They are not words like
others for which a "theory of meaning" is either possible or necessary.
Indeed, for Wittgenstein, "there are no...'logical constants' [in Frege's
and Russell's sensel" (5.4). Rather, for any language, with any vocabulary
of "constants" or primitive signs whatsoever, there are the purely com-
binatorial possibilities of building complex expressions from simpler
expressions and of substituting one expression for another within such
a complex expression. These abstract combinatorial possibilities are all
that the so-called logical constants express: "Whenever there is com-
positeness, argument and function are present, and where these are pres-
ent, we already have all the logical constants" (5.47). Thus, for Wittgen-
stein, logical truths are not true in virtue of the meanings of particular
words—whether of 'and', 'or', 'not', or any others—but solely in virtue
of "logical form" the general combinatorial possibilities common to all
languages regardless of their particular vocabularies.
Now this conception—that logical truths are true in virtue of "logical
form," and not in virtue of "meaning" in anything like Quine's sense—
is essential to the antipsychologism of the Tractatus. For, if logic depends
on the meanings of particular words—even "logical words" like 'and',
'not', and so on—then it rests, in the last analysis, on psychological facts
about how these words are actually used. It then becomes possible to con-
test these alleged facts and to argue, for example, that a correct theory
of meaning supports intuitionistic rather than classical logic, say. For Witt-
genstein, this debate, in these terms, simply does not make sense. Logic
rests on no facts whatsoever, and certainly not on facts about the mean-
ings or usages of English (or German) words. Rather, logic rests on the
abstract combinatorial possibilities common to all languages as such. In
this sense, logic is absolutely presuppositionless and thus absolutely
uncontentious.
The problem for this Tractarian conception has nothing at all to do
with the Quinean problem of truth-in-virtue-of-meaning or truth-in-virtue-
of-language. Rather, the problem is that the logic realizing this conception
is much too weak to accomplish the original aim of logicism: explaining
how mathematics—classical mathematics—is possible. Frege's Begriffs-
schrift cannot provide the required realization, because of the paradoxes;
and neither can Russell's Principia, because of the need for axioms like
infinity and reducibility. The Tractatus itself ends up with a conception
of logic that falls somewhere between truth-functional logic and a ramified
92 Michael Friedman
From our present, post-Quinean vantage point, the triviality and cir-
cularity of these suggestions is painfully obvious; but it was never so for
Carnap. He never lost his conviction that the notion of analytic truth,
together with a fundamentally logicist conception of mathematics, stands
firm and unshakable. And what this shows, finally, is that the Fregean
roots of Carnap's philosophizing run deep indeed. Unfortunately,
however, they have yet to issue in their intended fruit.
Notes
1. I am indebted to helpful suggestions, advice, and criticism from Warren Goldfarb,
Peter Hylton, Thomas Ricketts, and the late Alberto Coffa.
2. R. Carnap, "Reply to E. W. Beth on Constructed Language Systems," in The
Philosophy of Rudolf Carnap, ed. P. Schilpp. (La Salle: Open Court, 1963), p. 932.
3. R. Carnap, "Intellectual Autobiography," in Philosophy of Rudolf Carnap, p. 47.
4. Ibid.
5. E. W. Beth, "Carnap's Views on the Advantages of Constructed Systems over Natural
Languages in the Philosophy of Science," in Philosophy of Rudolf Carnap, pp. 470-71.
6. "Reply to E. W. Beth," pp. 927-28.
7. See, for example, Carnap's "Intellectual Autobiography," pp. 4-6; and the preface
to the second edition of The Logical Structure of the World, trans. R. George (Berkeley:
University of California Press, 1967), pp. v-vi.
8. R. Carnap, The Logical Syntax of Language, trans. A. Smeaton (London: Routledge,
1937). References are given parenthetically in the text via section numbers.
9. Cf. also Beth, "Carnap's Views on Constructed Systems," pp. 475-82; and Carnap,
"Reply to E. W. Beth," p. 928.
10. For the relationship between Frege and the Tractatus, see especially T. Ricketts, "Frege,
the Tractatus, and the Logocentric Predicament," Nous 19 (1985): 3-15.
11. See especially his "Intellectual Autobiography," p. 25.
12. Tractatus Logico-Philosophicus, 4.12-4.128; this is the basis for the "logocentric
predicament" of note 10 above. Further references to Wittgenstein in the text are also to
the Tractatus. See also W. Goldfarb, "Logic in the Twenties: the Nature of the Quantifier,"
Journal of Symbolic Logic 44 (1979): 351-68.
13. For the rejection of set theory, see 6.031. Frege's and Russell's impredicative defini-
tion of the ancestral is rejected at 4.1273; the axiom of reducibility is rejected at 6.1232-6.1233.
Apparently, then, we are limited to (at most) predicative analysis—and even this may be
too much because of the doubtful status of the axiom of infinity (5.535). The discussion
of mathematics and logic at 6.2-6.241 strongly suggests a conception of mathematics limited
to primitive recursive arithmetic.
14. Cf. Beth, "Carnap's Views on Constructed Systems," who takes this situation to
undermine both Carnap's logicism and his formalism—and, in fact, to require an appeal
to some kind of "intuition." Carnap's ingenuous response in his "Reply to E. W. Beth,"
pp. 928-30, is most instructive.
15.1 am indebted to Thomas Ricketts and especially to Alberto Coffa for pressing me
on this point.
16. W. V. Quine, "Two Dogmas of Empiricism," in From a Logical Point of View (New
York: Harper, 1963), pp. 22-23.
17. See note 13 above.
18. See T. Ricketts, "Rationality, Translation, and Epistemology Naturalized," Jour-
nal of Philosophy 79 (1982): 117-37.
19. R. Carnap, Introduction to Semantics (Cambridge, Mass.: Harvard University Press,
1942); parenthetical references in the text are to section numbers.
Gregory H. Moore
1. Introduction
To most mathematical logicians working in the 1980s, first-order logic
is the proper and natural framework for mathematics. Yet it was not always
so. In 1923, when a young Norwegian mathematician named Thoralf
Skolem argued that set theory should be based on first-order logic, it was
a radical and unprecedented proposal.
The radical nature of what Skolem proposed resulted, above all, from
its effect on the notion of categoricity. During the 1870s, as part of what
became known as the arithmetization of analysis, Cantor and Dedekind
characterized the set n of real numbers (up to isomorphism) and thereby
found a categorical axiomatization for n. Likewise, during the 1880s
Dedekind and Peano categorically axiomatized the set M of natural
numbers by means of the Peano Postulates.1 Yet in 1923, when Skolem
insisted that set theory be treated within first-order logic, he knew (by
the recently discovered Lowenheim-Skolem Theorem) that in first-order
logic neither set theory nor the real numbers could be given a categorical
axiomatization, since each would have both a countable model and an
uncountable model. A decade later, Skolem (1933, 1934) also succeeded
in proving, by the construction of a countable nonstandard model, that
the Peano Postulates do not uniquely characterize the natural numbers
within first-order logic. The Upward Lowenheim-Skolem Theorem of Tar-
ski, the first version of which was published as an appendix to (Skolem
1934), made it clear that no axiom system having an infinite model is
categorical in first-order logic.
The aim of the present article is to describe how first-order logic grad-
ually emerged from the rest of logic and then became accepted by
mathematical logicians as the proper basis for mathematics—despite the
opposition of Zermelo and others. Consequently, I have pointed out where
95
96 Gregory H. Moore
What happened circa 1880, in the work of Peirce and Frege, was not that
the notion of quantifier was invented but rather that it was separated from
the Boolean connectives on the one hand and from the notion of predicate
on the other.
Peirce's contributions to logic fell squarely within the Boolean tradi-
tion. In (1865), Peirce modified Boole's system in several ways, reinter-
preting Boole's + (logical addition) as union in the case of classes and
as inclusive "or" in the case of propositions. (Boole had regarded A + B
as defined only when A and B are disjoint.)
Five years later, Peirce investigated the notion of relation that Augustus
De Morgan had introduced into formal logic in (1859) and began to adapt
this notion to Boole's system:
Boole's logical algebra has such singular beauty, so far as it goes,
that it is interesting to inquire whether it cannot be extended over the
whole realm of formal logic, instead of being restricted to that simplest
and least useful part of the subject, the logic of absolute terms, which,
when he wrote [1854], was the only formal logic known. (Peirce 1870,
317)
Thus Peirce developed the laws of the relative product, the relative sum,
and the converse of a relation. When he left for Europe in June 1870,
he took a copy of this article with him and delivered it to De Morgan (Fisch
1984, xxxiii). Unfortunately, De Morgan was already in the decline that
led to his death the following March. Peirce did not find a better recep-
tion when he gave a copy of the article to Stanley Jevons, who had
elaborated Boole's system in England. In a letter of August 1870 to Jevons,
whom Peirce described as "the only active worker now, I suppose, upon
mathematical logic," it is clear that Jevons rejected Peirce's extension of
Boole's system to relations (Peirce 1984, 445). Nevertheless, Peirce's work
on relations eventually found wide currency in mathematical logic.
It was through applying class sums and products (i.e., unions and in-
tersections) to relations that Peirce (1883) obtained the notion of quanti-
fier as something distinct from the Boolean connectives. By way of ex-
ample, he let Ijj denote the relation stating that / is a lover of j. "Any
proposition whatever," he explained,
is equivalent to saying that some complexus of aggregates and products
of such numerical coefficients is greater than zero. Thus,
Z,.2.V>o
means that something is a lover of something; and
THE EMERGENCE OF FIRST-ORDER LOGIC 99
(1883, 200-201)
When Peirce returned to the subject of quantifiers in (1885), he treated
them in two ways that were to have a pronounced effect on the subse-
quent development of logic. First of all, he defined quantifiers (as part
of what he called the "first-intentional logic of relatives [relations]") in
a way that emphasized their analogy with arithmetic:
Here, in order to render the notation as inconical as possible, we may
use £ for some, suggesting a sum, and II for all, suggesting a pro-
duct. Thus £,-#/ means that x is true of some one of the individuals
denoted by / or
That is, to say that things are identical is to say that every predicate
is true of both or false of b o t h . . . . If we please, we can dispense with
the token q, by using the index of a token and by referring to this in
the Quantifier just as subjacent indices are referred to. That is to say,
we may write
(1885, 199)
In effect, Peirce used a form of Leibniz's principle of the identity of in-
discernibles in order to give a second-order definition of identity.
Peirce rarely returned to his second-intensional logic. It formed chapter
14 of his unpublished book of 1893, Grand Logic (see his [1933, 56-58]).
He also used it, in a letter of 1900 to Cantor, to quantify over relations
while defining the less-than relation for cardinal numbers (Peirce 1976,
776). Otherwise, he does not seem to have quantified over relations.
Moreover, what Peirce glimpsed of second-order logic was minimal. He
appears never to have applied his logic in detail to mathematical problems,
except in (1885) to the beginnings of cardinal arithmetic—an omission that
contrasts sharply with Frege's work.
THE EMERGENCE OF FIRST-ORDER LOGIC 101
You write: "I call axioms propositions that are true but are not proved
because our knowledge of them flows from a source very different from
the logical source, a source which might be called spatial intuition. From
the truth of the axioms it follows that they do not contradict each
other." I found it very interesting to read this sentence in your letter,
for as long as I have been thinking, writing, and lecturing on these
things, I have been saying the exact opposite: if the arbitrarily given
axioms do not contradict each other with all their consequences, then
they are true and the things defined by the axioms exist. For me this
is the criterion of truth and existence. (Hilbert in [Frege 1980a}, 39-40)
Hilbert returned to this theme repeatedly over the following decades.
Frege, in a letter of 6 January 1900, objected vigorously to Hilbert's
claim that the consistency of an axiom system implies the existence of a
model of the system. The only way to prove the consistency of an axiom
system, Frege insisted, is to give a model. He argued further that the crux
of Hilbert's "error" was in conflating first-level and second-level con-
cepts.5 For Frege, existence was a second-level concept, and it is precise-
ly here that his system of logic, as found in the Fundamental Laws, dif-
fers from second-order logic as it is now understood.
What is particularly striking about Hilbert's axiomatization of geometry
is an axiom missing from the first edition of his book. There his Axiom
Group V consisted solely of the Archimedean Axiom. He used a certain
quadratic field to establish the consistency of his system, stressing that
this proof required only a denumerable set. When the French translation
of his book appeared in 1902, he added a new axiom that differed fun-
damentally from all his other axioms and that soon led him to try to
establish the consistency of a nondenumerable set, namely the real
numbers:
Let us note that to the five preceding groups of axioms we may still
adjoin the following axiom which is not of a purely geometric nature
and which, from a theoretical point of view, merits particular attention:
Axiom of Completeness
lem can be solved, and his conviction that the consistency of a set S of
axioms implies the existence of a model for S (1900b, 264-66). When he
gave this address, his view that consistency implies existence was only an
article of faith—albeit one to which Poincare subscribed as well (Poin-
care 1905, 819). Yet in 1930 Godel was to turn this article of faith into
a theorem, indeed, into one version of his Completeness Theorem for first-
order logic.
In 1904, when Hilbert addressed the International Congress of
Mathematicians at Heidelberg, he was still trying to secure the founda-
tions of the real number system. As a first step, he turned to providing
a foundation for the positive integers. While discussing Frege's work, he
considered the paradoxes of logic and set theory for the first time in print.
To Hilbert these paradoxes showed that "the conceptions and research
methods of logic, conceived in the traditional sense, do not measure up
to the rigorous demands that set theory makes" (1905, 175). His remedy
separated him sharply from Frege:
Yet if we observe attentively, we realize that in the traditional treat-
ment of the laws of logic certain fundamental notions from arithmetic
are already used, such as the notion of set and, to some extent, that
of number as well. Thus we find ourselves on the horns of a dilemma,
and so, in order to avoid paradoxes, one must simultaneously develop
both the laws of logic and of arithmetic to some extent. (1905, 176)
This absorption of part of arithmetic into logic remained in Hilbert's later
work.
Hilbert excused himself from giving more than an indication of how
such a simultaneous development would proceed, but for the first time
he used a formal language. Within that language his quantifiers were in
the Peirce-Schroder tradition, although he did not explicitly cite those
authors. Indeed, he regarded "for some x, A(x)" merely as an abbrevia-
tion for the infinitary formula
A(\) o. A(2) o. A(3) o. ... ,
where o. stood for "oder" (or), and analogously for the universal quanti-
fier with respect to "und" (and) (1905, 178). Likewise, he followed Peirce
and Schroder (as well as the geometric tradition) by letting his quantifiers
range over a fixed domain. Hilbert's aim was to show the consistency of
his axioms for the positive integers (the Peano Postulates without the Prin-
108 Gregory H. Moore
We can assume that any law of the algebra [of logic] which holds
whatever finite number of elements be involved holds for any number
of elements whatever.... This also resolves our difficulty concerning
the possibility that the number of values of x in <J>x might not be even
denumerable.... (1918, 236)
He made no attempt, however, to justify his assertion, which amounted
to a kind of compactness theorem.
A second logician who introduced an infinitary logic in the context of
Principia Mathematica was Frank Ramsey. In 1925 Ramsey presented a
critique of Principia, arguing that the Axiom of Reducibility should be
abandoned and proposing instead the simple theory of types. He enter-
tained the possibility that a truth function may have infinitely many
arguments (1925, 367), and he cited Wittgenstein as having recognized that
such truth functions are legitimate (1925, 343). Further, Ramsey argued
that "owing to our inability to write propositions of infinite length, which
is logically a mere accident, (<{>).<J>a cannot, like p.q, be elementarily ex-
pressed, but must be expressed as the logical product of a set of which
it is also a member" (1925, 368-69). It appears that Ramsey was not con-
cerned with infinitary formulas per se but only with using them in his
heuristic argument for the simple theory of types. Yet his proposal was
sufficiently serious that Godel later cited it when arguing against such in-
finitary formulas (Godel 1944, 144-46).
8. Hilbert: Later Foundational Research
Hilbert did not abandon foundational questions after his 1904 lecture
at the Heidelberg congress, though he published nothing further on them
for more than a decade. Rather, he gave lecture courses at Gottingen on
such questions repeatedly—in 1905, 1908, 1910, and 1913. It was the course
given in the winter semester of 1917-18, "Principles of Mathematics and
Logic," that first exhibited his mature conception of logic.12
That course began shortly after Hilbert delivered a lecture, "Axiomatic
Thinking," at Zurich on 11 September 1917. The Zurich lecture stressed
the role of the axiomatic method in various branches of mathematics and
physics. Returning to an earlier theme, he noted how the consistency of
several axiomatic systems (such as that for geometry) had been reduced
to a more specialized axiom system (such as that for H, reduced in turn
to the axioms for 1M and those for set theory). Hilbert concluded by stating
114 Gregory H. Moore
Now if we make use of the calculus of logic in this sense, then we will
be compelled to extend in a certain direction the rules governing the
formal operations. In particular, while we previously separated prop-
ositions and [prepositional] functions completely from objects and, ac-
cordingly, distinguished the signs for indefinite propositions and func-
tions rigorously from the variables, which take arguments, now we per-
mit propositions and functions to be taken as logical variables in a way
similar to that for proper objects, and we permit signs for indefinite
propositions and functions to appear as arguments in symbolic expres-
sions. (Hilbert 1917, 188)
Here Hilbert argued, in effect, for a logic at least as strong as second-
order logic. But his views on this matter did not appear in print until his
book Principles of Mathematical Logic, written jointly with Ackermann,
was published in 1928. In that book, which consisted largely of a revision
of Hilbert's 1917 course, he expressed himself even more strongly:
THE EMERGENCE OF FIRST-ORDER LOGIC 775
The intended meaning was that if the proposition A (x) holds when x is
r'A, then A (x) holds for an arbitrary x, say a. He thus defined (Vx)A(x)
as A(rA) and similarly for (3*M(x) (1923, 157).
Soon Hilbert modified the Transfinite Axiom, changing it into the
e-axiom:
Instead, he asserted that the same purpose was served by his treatment
of function variables (1928, 77). He still insisted, in (Hilbert and Acker-
mann 1928, 114-15), that the theory of types was the appropriate logical
vehicle for studying the theory of real numbers. But, he added, logic could
be founded so as to be free of the difficulties posed by the Axiom of
Reducibility, as he had done in his various papers. Thus Hilbert opted
for a version of the simple theory of types (in effect, o>-order logic).
Hilbert's co-workers in proof theory used essentially the same system
of logic as he did. Von Neumann (1927) gave a proof of the consistency
of a weak form of number theory, working mainly in a first-order sub-
system. However, he discussed Ackermann's work involving both a second-
order e-axiom and quantification over number-theoretic functions (von
Neumann 1927, 41-46). During the same period, Bernays discussed in some
detail "the extended formalism of 'second order'," mentioning Lowen-
heim's (1915) decision procedure for monadic second-order logic and
remarking how questions of first-order validity could be expressed by a
second-order formula (Bernays and Schonfinkel 1928, 347-48).16
In (1929), Hilbert looked back with pride and forward with hope at
what had been accomplished in proof theory. He thought that Ackermann
and von Neumann had established the consistency of the e-axiom restricted
to natural numbers, not realizing that this would soon be an empty vic-
tory. Hilbert posed four problems as important to his program. The first
of these was essentially second-order: to prove the consistency of the
e-axiom acting on number-theoretic functions (1929, 4). The second was
the same problem for higher-order functions, whereas the third and fourth
concerned completeness. The third, noting that the Peano Postulates are
categorical, asked for a proof that if a number-theoretic sentence is shown
consistent with number theory, then its negation cannot be shown consis-
tent with number theory. The fourth was more complex, asking for a
demonstration that, on the one hand, the axioms of number theory are
deductively complete and that, on the other, first-order logic with identi-
ty is complete (1929, 8).
Similarly, in (Hilbert and Ackermann 1928, 69) there was posed the
problem of establishing the completeness of first-order logic (without iden-
tity). The following year Godel solved this problem for first-order logic,
with and without identity (1929). His abstract (1930b) of this result spoke
of the "restricted functional calculus" (first-order logic without identity)
as a subsystem of logic, since no bound function variables were permitted.
120 Gregory H. Moore
where / was a function variable, and he regarded this formula (for the
given domain M) as expandable to an infinitary formula with a first-order
existential quantifier for each individual of M (1915, section 2). His next
theorem stated that, since Schroder's logic can express that a domain is
finite or denumerable, then Lowenheim's Theorem cannot be extended
to Schroder's (second-order) logic.
Some historians of mathematics have regarded Lowenheim's argument
for his theorem as odd and unnatural.18 But his argument appears so on-
ly because they have considered it within first-order logic. Although
Lowenheim's Theorem holds for first-order logic (as Skolem was to show),
this was not the logic in which Lowenheim worked.
10. Skolem: First-Order Logic as All of Logic
In 1913, after writing a thesis (1913a) on Schroder's algebra of logic,
Skolem received his undergraduate degree in mathematics. He soon wrote
several papers, beginning with (1913b), on Schroder's calculus of classes.
During the winter of 1915-16, Skolem visited Gottingen, where he discussed
set theory with Felix Bernstein. By that time Skolem was already ac-
quainted with Lowenheim's Theorem and had seen how to extend it to
a countable set of formulas. Furthermore, Skolem had realized that this
extended version of the theorem could be applied to set theory. Thus he
THE EMERGENCE OF FIRST-ORDER LOGIC 123
found what was later called Skolem's Paradox: Zermelo's system of set
theory has a countable model (within first-order logic) even though this
system implies the existence of uncountable sets (Skolem 1923, 232, 219).
Nevertheless, Skolem did not lecture on this result until the Fifth Scan-
danavian Congress of Mathematicians in July 1922. When it appeared in
print the following year, he stated that he had not published it earlier
because he had been occupied with other problems and because
I believed that it was so clear that the axiomatization of set theory would
not be satisfactory as an ultimate foundation for mathematics that, by
and large, mathematicians would not bother themselves with it very
much. To my astonishment I have seen recently that many mathemati-
cians regard these axioms for set theory as the ideal foundation for
mathematics. For this reason it seemed to me that the time had come
to publish a critique. (Skolem 1923, 232)
It was precisely in order to establish the relativity of set-theoretic no-
tions that Skolem proposed that set theory be formulated within first-order
logic. At first glance, given the historical context, this was a strange sug-
gestion. Set theory appeared to require quantifiers not only over indiv-
iduals, as in first-order logic, but also quantifiers over sets of individuals,
over sets of sets of individuals, and so on. Skolem's radical proposal was
that the membership relation e be treated not as a part of logic (as Peano
and Russell had done) but like any other relation on a domain. Such rela-
tions could be given a variety of interpretations in various domains, and
so should e. In this way the membership relation began to lose its privileged
position within logic.
Skolem made Lowenheim's article (1915) the starting point for several
of his papers. This process began when Skolem used the terms Zdhlaus-
druck and Zdhlgleichung in an article (1919), completed in 1917, on
Schroder's calculus of classes. But it was only in (1920) that Skolem began
to discuss Lowenheim's Theorem, a subject to which he returned many
times over the next forty years. Skolem first supplied a new proof for this
theorem (relying on Skolem functions) in order to avoid the occurrence
in the proof of second-order propositions (in the form of subsubscripts
on Lowenheim's relational expressions [Skolem 1920, 1]). Extending this
result, Skolem obtained what became known as the Lowenheim-Skolem
Theorem (a countable and satisfiable set of first-order propositions has
a countable model). Surprisingly, he did not even state the Lowenheim-
Skolem Theorem for first-order logic as a separate theorem but immediate-
124 Gregory H. Moore
The axiomatic restriction of the notion of set [to first-order logic] does
not prevent one from obtaining all the usual theorems... of Cantorian
set theory.... Nevertheless, one must observe that this way of making
the notion of set (or that of predicate) precise has a consequence of
another kind: the interpretation of the system is no longer necessarily
unique.... It is to be observed that the impossibility of characterizing
the finite with respect to the infinite comes from the restrictiveness of
the [first-order] formalism. The impossibility of characterizing the
denumerable with respect to the nondenumerable in a sense that is in
some way unconditional—does this reveal, one might wonder, a cer-
tain inadequacy of the method under discussion here [first-order logic]
for making axiomatizations precise? (Bernays in [Gonseth 1941], 49-50)
Skolem objected vigorously to Bernays's suggestion and insisted that a
first-order axiomatization is surely the most appropriate.
In 1958, at a colloquium held in Paris, Skolem reiterated his views on
the relativity of fundamental mathematical notions and criticized Tarski's
contributions:21
It is self-evident that the dubious character of the notion of set renders
other notions dubious as well. For example, the semantic definition
of mathematical truth proposed by A. Tarski and other logicians presup-
poses the general notion of set. (Skolem 1958, 13)
In the discussion that followed Skolem's lecture, Tarski responded to this
criticism:
[I] object to the desire shown by Mr. Skolem to reduce every theory
to a denumerable model.... Because of a well-known generalization
of the Lowenheim-Skolem Theorem, every formal system that has an
infinite model has a model whose power is any transfinite cardinal given
in advance. From this, one can just as well argue for excluding
denumerable models from consideration in favor of nondenumerable
models. (Tarski in [Skolem 1958], 17)
11. Conclusion
As we have seen, the logics considered from 1879 to 1923—such as those
of Frege, Peirce, Schroder, Lowenheim, Skolem, Peano, and Russell—
were generally richer than first-order logic. This richness took one of two
forms: the use of infinitely long expressions (by Peirce, Schroder, Hilbert,
Lowenheim, and Skolem) and the use of a logic at least as rich as second-
order logic (by Frege, Peirce, Schroder, Lowenheim, Peano, Russell, and
128 Gregory H. Moore
and the definition of truth, rejected the attempt by Skolem to restrict logic
to countable first-order languages. In time, uncountable first-order
languages and uncountable models became a standard part of the reper-
toire of first-order logic. Thus set theory entered logic through the back
door, both syntactically and semantically, though it failed to enter through
the front door of second-order logic.
Notes
1. Peano acknowledged (1891, 93) that his postulates for the natural numbers came from
(Dedekind 1888).
2. Peirce's use of second-order logic was first pointed out by Martin (1965).
3. Van Heijenoort (1967, 3; 1986, 44) seems to imply that Frege did separate, or ought
to have separated, first-order logic from the rest of logic. But for Frege to have done so
would have been contrary to his entire approach to logic. Here van Heijenoort viewed the
matter unhistorically, through the later perspectives of Skolem and Quine.
4. There is a widespread misconception, due largely to Russell (1919, 25n), that Frege's
Begriffsschrift was unknown before Russell publicized it. In fact, Frege's book quickly received
at least six reviews in major mathematical and philosophical journals by researchers such
as Schroder in Germany and John Venn in England. These reviews were largely favorable,
though they criticized various features of Frege's approach. The Begriffsschrift failed to
persuade other logicians to adopt Frege's approach to logic because most of them (Schro'der
and Venn, for example) were already working in the Boolean tradition. (See [Bynum 1972,
209-35] for these reviews, and see [Nidditch 1963] on similar claims by Russell concerning
Frege's work in general.)
5. Frege pointed this out to Hilbert in a letter of 6 January 1900 (Frege 1980a, 46, 91)
and discussed the matter in print in (1903b, 370-71).
6. The suggestion that Sommer may have prompted Hilbert to introduce an axiom of
continuity is due to Jongsma (1975, 5-6).
7. Forder, in a textbook on the foundations of geometry (1927, 6), defined the term "com-
plete" to mean what Veblen called "categorical" and argued that a categorical set of ax-
ioms must be deductively complete. Here Forder presupposed that if a set of axioms is con-
sistent, then it is satisfiable. This, as Gd'del was to establish in (1930a) and (1931), is true
for first-order logic but false for second-order logic. On categoricity, see (Corcoran 1980,
1981).
8. (Frege 1980a, 108). In an 1896 article Frege wrote: "I shall now inquire more closely
into the essential nature of Peano's conceptual notation. It is presented as a descendant of
Boole's logical calculus but, it may be said, as one different from the others.... By and
large, I regard the divergences from Boole as improvements" (Frege 1896; translation in
1984, 242). In (1897), Peano introduced a separate symbol for the existential quantifier.
9. (Frege 1980a, 78). Nevertheless, Church (1976, 409) objected to the claim that Frege's
system of 1893 is an anticipation of the simple theory of types. The basis of Church's objec-
tion is that for "Frege a function is not properly an (abstract) object at all, but is a sort
of incompleted abstraction." The weaker claim made in the present paper is that Frege's
system helped lead Russell to the theory of types when he dropped Frege's assumption that
classes are objects of level 0 and allowed them to be objects of arbitrary finite level.
10. Russell in (Frege 1980a, 147; Russell 1903, 528). See (Bell 1984) for a detailed analysis
of the Frege-Russell letters.
11. Three of these approaches are found in (Russell 1906): the zigzag theory, the theory
of limitation of size, and the no-classes theory. In a note appended to this paper in February
1906, he opted for the no-classes theory. Three months later, in another paper read to the
London Mathematical Society, the no-classes theory took a more concrete shape as the
130 Gregory H. Moore
substitution^ theory. Yet in October 1906, when the Society accepted the paper for publica-
tion, he withdrew it. (It was eventually published as [Russell 1973].) The version of the theory
of types given in the Principles was very close to his later no-classes theory. Indeed, he wrote
that "technically, the theory of types suggested in Appendix B [1903] differs little from the
no-classes theory. The only thing that induced me at that time to retain classes was the technical
difficulty of stating the propositions of elementary arithmetic without them" (1973, 193).
12. Hilbert's courses were as follows: "Logische Prinzipien des mathematischen Denkens"
(summer semester, 1905), "Prinzipien der Mathematik" (summer semester, 1908), "Elemente
und Prinzipienfragen der Mathematik" (summer semester, 1910), "Einige Abschnitte aus
der Vorlesung tiber die Grundlagen der Mathematik und Physik" (summer semester, 1913),
and "Prinzipien der Mathematik und Logik" (winter semester, 1917). A copy of the 1913
lectures can be found in Hilbert's Nachlass in the Handschriftenabteilung of the Nieder-
sachsische Staats- und Universitatsbibliotek in GOttingen: the others are kept in the "Gift-
schrank" at the Mathematische Institut in GOttingen. Likewise, all other lecture courses given
by Hilbert and mentioned in this paper can be found in the "Giftschrank."
13. See (Hilbert 1917, 190) and (Hilbert and Ackermann 1928, 83). The editors of Hilbert's
collected works were careful to distinguish the Principle of Mathematical Induction in (Hilbert
1922) from the first-order axiom schema of mathematical induction; see (Hilbert 1935,176n).
Herbrand also realized that in first-order logic this principle becomes an axiom schema (1929;
1930, chap. 4.8).
14. The course was entitled "Logische Grundlagen der Mathematik."
15. "Logische Grundlagen der Mathematik," a partial copy of which is kept in the univer-
sity archives at GOttingen. On the history of the Axiom of Choice, see (Moore 1982).
16. Bernays also wrote about second-order logic briefly in his (1928).
17.On the early history oif the w-rule see (Feferman 1986)
18. See, for example, (van Heijenoort 1967, 230; Vaught 1974, 156; Wang 1970, 27).
19. The recognition that Skolem in (1920) was primarily working in Lw is due to
Vaught (1974, 166). ''
20. For an analysis of Zermelo's views on logic, see (Moore 1980, 120-36).
21. During the same period Skolem (1961, 218) supported the interpretation of the theory
of types as a many-sorted theory within first-order logic. Such an interpretation was given
by Gilmore (1957), who showed that a many-sorted theory of types in first-order logic has
the same valid sentences as the simple theory of types (whose semantics was to be based
on Henkin's notion of general model rather than on the usual notion of higher-order model).
22. See (Quine 1970, 64-70). For a rebuttal of some of Quine's claims, see (Boolos 1975).
23. For the impressive body of recent research on stronger logics, see (Barwise and Fefer-
man 1985).
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THE EMERGENCE OF FIRST-ORDER LOGIC 737
Added in Proof:
In a recent letter to me, Ulrich Majer has argued that Hermann Weyl was the first to
formulate first-order logic, specifically in his book Das Kontinuum (1918). This is too strong
a claim. I have already discussed Weyl's role in print (Moore 1980, 110-111; 1982, 260-61),
but some further comments are called for here.
It is clear that Weyl (1918, 20-21) lets his quantifiers range only over objects (in his
Fregean terminology) rather than concepts, and to this extent what he uses is first-order
logic. But certain reservations must be made. For Weyl (1918, 19) takes the natural numbers
as given, and has in mind something closer to co-logic. Moreover, he rejects the unrestricted
application of the Principle of the Excluded Middle in analysis (1918, 12), and hence he
surely is not proposing classical first-order logic. Finally, one wonders about the interac-
tions between Hilbert and Weyl during the crucial year 1917. What conversations about foun-
dations took place between them in September 1917 when Hilbert lectured at Zurich and
was preparing his 1917 course, while Weyl was finishing Das Kontinuum on a closely related
subject?
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II. REINTERPRETATIONS IN
THE HISTORY OF MATHEMATICS
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Harold Edwards
139
140 Harold Edwards
If the use of transfinite logic diminishes its power to convince, and if,
as the intuitionists maintain, virtually all of classical mathematics can be
given an intuitionist foundation, then the use of transfinite logic in
mathematics is both unwarranted and undesirable.
Kronecker believed that "someday people will succeed in 'arithmetiz-
ing' all of mathematics, that is, in founding it on the single foundation
of the number-concept in its narrowest sense."2 He said that this was his
goal and that if he did not succeed then surely others who came after him
would. This view inevitably put him in opposition not only to Cantor,
whose "sets" could only appear to Kronecker as figments, but also to
his colleague Weierstrass, who likewise had undertaken the "arithmetiza-
tion of analysis." We have seen in our own time in the example of Errett
Bishop how an outstanding mathematician can arrive at views of
mathematics that place him, like it or not, in complete opposition to his
colleagues, however much he might respect them. Quite simply, there is
no way for a Bishop or a Kronecker to say politely, "I'm sorry, but your
proofs do not prove what you think they do, and for you to convince me
that your mathematics is worth my time you have to rethink in a fun-
damental way all you have done and show me that you have understood
and can answer my objections. Meanwhile, I will pursue what seems to
me a much more worthwhile project, that of establishing all of mathematics
on a firm foundation, something I am confident I can do."
Kronecker, unlike Bishop, published nothing on his constructive pro-
gram. He was engrossed in the last years of his life with his work on the
theory of algebraic numbers and algebraic functions, and his intention
to give an intuitionist development of the foundations of mathematics was
never, as far as we know, seriously begun. Nonetheless, his thinking on
foundational questions is evident in much of his work on mathematics
proper, and, in my opinion, it is worthy of the highest respect. Therefore,
I think it would give a truer picture of what happened between him and
Cantor if the story were told in something like the following way.
The Weierstrass school, of which Cantor was a member, found that,
in developing the theory of functions and in following up Cauchy's ef-
forts to put the calculus on a firm foundation, they needed to use
arguments dealing with infinity such as those that had been introduced
by Bolzano earlier in the nineteenth century. Cantor went even further,
completely disregarding the taboo against competed infinities and deal-
ing with infinite collections that he called Menge or sets. Kronecker believed
142 Harold Edwards
that none of these ways of dealing with infinity was acceptable or indeed
necessary. He had a grand conception that all of mathematics would be
based on the intuition of the natural numbers, but he never carried this
conception to fruition. Naturally, Kronecker's denial of the validity of
Weierstrass's arguments deeply wounded Weierstrass, and their relations
went from great friendship in the early years to almost total alienation
in Kronecker's last years, even though Kronecker insisted, perhaps insen-
sitively, that a disagreement over mathematical questions should not af-
fect their personal relations. Cantor's reaction to Kronecker's opposition
was even stronger, in the first place because he was in a much weaker posi-
tion vis-a-vis Kronecker because of his youth and his position at a pro-
vincial university, and in the second place because Cantor's personality
contained a strain of paranoia that deeply disturbed at one time or another
his relations with many contemporaries other than Kronecker, including
such ostensible allies as Weierstrass, H. A. Schwarz, and G. Mittag-Leffler.
Still, Cantor's ideas were taken up by Dedekind (who had in fact an-
ticipated many of them) and Hilbert, and they became the basis of a new
mode of mathematics that proved very fruitful and has dominated
mathematics ever since, with only an occasional Brouwer or Weyl to op-
pose it.
I believe that the pendulum is beginning to swing back Kronecker's way,
not least because of the appearance of computers on the scene, which has
fostered a great upsurge of algorithmic thinking. Mathematicians are more
and more interested in making computational sense of their abstractions.
If I am right in thinking that Kronecker has been undervalued and
caricatured by historians because they have been following the lead of the
philosophers of mathematics and of mathematicians themselves, Kro-
necker's place in history may be about to improve considerably.
I will do what I can to bring this about. Surely one of the principal
reasons Kronecker is so little studied today (except, it seems to me, by
the best mathematicians) is that his works are so very difficult to read.
Jordan very aptly called them in 1870 "1'envie et le desespoir des geome-
tres."3 This shows that the difficulty we experience today in reading
Kronecker's works is not merely the difficulty of reading an old text writ-
ten in an outdated terminology. Kronecker's difficult style was difficult
for his contemporaries, too. I have been working at it for many years,
especially his Kummer Festschrift entitled "Grundzuege einer arith-
metischen Theorie der algebraischen Groessen."4 I feel my efforts have
KRONECKER'S PLACE IN HISTORY 143
been rewarded and that I will soon be able to publish papers that will con-
vey something of what Kronecker was doing and that will, I hope, awaken
the interest of others in studying Kronecker.
Let me conclude by giving a few indications of what it is that I find
so valuable in Kronecker's works and how it relates to the history and
philosophy of mathematics. It is said of Kronecker, as it is of Brouwer,
that he succeeded in his mathematics by ignoring his intuitionist principles,
but I think this allegation is completely untrue in both cases. Kronecker's
principles permeate his mathematics—the problems he studies, the goals
he sets for himself, the way he structures his theories. One of the parts
of his Kummer Festschrift that I have studied most is his theory of what
he calls "divisors," which is a first cousin of Dedekind's theory of
"ideals." Two ways in which Kronecker's theory differs from Dedekind's
show the difference in philosophy between the two men.
First, Dedekind regarded as the principal task of the theory the defini-
tion of "ideals" in such a way that the theorem on unique factorization
into primes, which is false for algebraic numbers, becomes true for ideals.
Kronecker took an altogether different view. He noted that the notion
of "prime" was relative to the field of numbers under consideration and
that if the field was extended things that had been prime might no longer
be. For this reason, the theorem on unique factorization into primes prop-
erly belongs to a later part of the theory, after the basic concepts have
been defined in a way that is independent of the field under considera-
tion. Those of you who have studied algebraic number theory will un-
doubtedly have studied it using Dedekind's ideals and will remember that
when you go from one field to another you have to manipulate the ideals—
intersecting them with the lower field if the new field is smaller and tak-
ing the ideal in the larger field they generate if the new field is larger. In
Kronecker's theory there is none of this. Divisors are defined and handled
in such a way that nothing changes if the field is extended or if it is
contracted.
Second, Kronecker defined his divisors, in essence, by telling how to
compute with them. In Dedekind's terms, this amounts to giving an
algorithm for determining, given a set of generators of an ideal, whether
a given element of the field is or is not in the ideal. Nothing of the kind
enters in Dedekind's theory because for Dedekind the definition was com-
plete and satisfactory once the ideal was defined as a set, albeit an in-
finite set with no criterion for membership. For Kronecker, such a defini-
144 Harold Edwards
Notes
1. G. Cantor, Gesammelte Abhandlugen (Berlin: Lokay, 1932), p. 182.
2. L. Kronecker, Werke, vol. 3 (Leipzig: Teubner, 1899), p. 253.
3. C. Jordan, Introduction to "Traite des substitutions et des equations algebriques,"
Paris, 1870.
4. Kronecker's Werke was published in five volumes by Teubner, Leipzig, between 1895
and 1930. This paper is in vol. 2.
Garrett Birkhoff and M. K. Bennett
1. Introduction
Felix Klein's "Erlanger Programm" (E.P.), listed in our references as
(Klein 1872), is generally accepted as a major landmark in the mathematics
of the nineteenth century. In his obituary biography Courant (1925) termed
it "perhaps the most influential and widely read paper in the second half
of the nineteenth century." Coolidge (1940, 293) said that it "probably
influenced geometrical thinking more than any other work since the time
of Euclid, with the exception of Gauss and Riemann."
In a thoughtful recent article, Thomas Hawkins (1984) has challenged
these assessments, pointing out that from 1872 to 1890 the E.P. had a
very limited circulation; that it was "Lie, not Klein" who developed the
theory of continuous groups; that "there is no evidence.. .that Poincare
ever studied the Programm;" that Killing's classification of Lie algebras
(later "perfected by Cartan") bears little relation to the E.P.; and that
Study, "the foremost contributor t o . . . geometry in the sense of the
Erlanger Programm,.. .had a strained and distant relationship with
Klein."
Our paper should be viewed as a companion piece to the study by
Hawkins. In our view, Klein's E.P. did have a major influence on later
research, including especially his own. Moreover, Klein was the chief heir
of five outstanding Germanic geometers, all of whom died within the
decade preceding the E.P.: Mobius (1790-1868), Steiner (1796-1869), von
Staudt (1798-1867), Plucker (1801-68), and Clebsch (1833-72).
Klein's close friendship with Lie at the time of the E.P. played an im-
portant role in the careers of both men. There is much truth in the fre-
quently expressed idea that Klein's studies of 'discontinuous' groups were
in some sense complementary to Lie's theory of 'continuous' groups
(Coolidge 1940, 304). Less widely recognized is the fact that in the E.P.
and related papers, Klein called attention to basic global aspects of
145
146 Garrett Birkhoff and M. K. Bennett
geometry, whereas Lie's theorems were purely local. After reviewing these
and other aspects of Klein's relationship to Lie, we will trace the influence
of his ideas on Study, Hilbert, Killing, E. Cartan, and Hermann Weyl.
In discussing these developments, we have tried to follow a roughly
chronological order. We hope that this will bring out the well-known evolu-
tion of Klein's scientific personality, from that of a brilliant, creative young
geometer to that of a farsighted organizer and builder of institutions in
his middle years, and finally to that of a mellow elder statesman or
"doyen" and retrospective historian of ideas.1
2. Klein's Teachers
Klein's E.P. surely owes much to his two major teachers: Pliicker and
Clebsch. Already as a youth of 17, Klein became an assistant in Plticker's
physics laboratory in Bonn. Though primarily a physicist, Pliicker was
also a very original geometer. Forty years earlier, he had written a brilliant
monograph on analytic projective geometry (Analytisch-geometrische Ent-
\vickelungen, 1828, 1832), which established the use of homogeneous coor-
dinates and the full meaning of duality.
Still more original (and more influential for Klein) was Pliicker's
"geometry of lines," first proposed in 1846. Pliicker proposed taking the
self-dual four-dimensional manifold of all lines in R3 as the set of basic
"elements" of geometry. In it, the sets of all "points" and of all "planes"
can be defined as what today would be called three-dimensional algebraic
varieties. Klein's Ph.D. thesis (directed by Pliicker) and several of his early
papers dealt with this idea.
Shortly before Klein finished his thesis when still only 19, Pliicker died.
Clebsch, who had just gone to Gottingen from Giessen, invited Klein to
join him there soon after. Barely 35 himself, Clebsch became Klein's sec-
ond great teacher. After making significant contributions to the mechanics
of continua (his book on elastic bodies was translated into French and
edited by St. Venant), Clebsch had introduced together with his Giessen
colleague Gordan (1831-1912) the invariant theory of the British
mathematicians Boole (1815-64), Cayley (1821-95), and Sylvester (1814-97).
Clebsch assigned to Klein the task of completing and editing the sec-
ond half of Pliicker's work on the geometry of lines (Neue Geometrie des
Raumes gegrundet auf die Betrachtung der geraden Linie als
Raumelemente). As we shall see, Klein also absorbed from Clebsch both
the concept of geometric invariant and Clebsch's interest in the "geometric
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 147
I made the acquaintance of the Norwegian, Sophus Lie. Our work had
led us from different points of view finally to the same questions, or
at least to kindred ones. Thus it came about that we met every day and
kept up an animated exchange of ideas. Our intimacy was all the closer
because, at first, we found very little interest in our geometrical con-
cerns in the immediate neighborhood.
3. Five Dazzling Years
The next five years were ones of incredible achievement for the young
Klein. During this time he wrote 35 papers and supervised seven Ph.D.
theses. It was while carrying on these activities that he wrote his E.P.,
and so it seems appropriate to recall some of his more important contem-
porary contacts.
About his continuing interactions with Stolz, Klein states:
In the summer of 1871,1 was again with Stolz.... He familiarized me
with the work of Lobachewsky and Bolyai, as well as with that of von
Staudt. After endless debates with him, I finally overcame his resistance
to my idea that non-Euclidean geometry was part of projective
geometry, and published a note about it in.. .the Math. Annalen (1871).
(EdM, vol. 1, 152)
After a digression on the significance of Gaussian curvature, Klein next
recalls the background of his second paper on non-Euclidean geometry
(Klein 1873). In it, Klein
investigated the foundations of von Staudt's [geometric] system, and
had a first contact with modern axiomatics.... However, even this ex-
tended presentation did not lead to a general clarification.... Cayley
himself mistrusted my reasoning, believing that a "vicious circle" was
buried in it. (EdM, vol. 1, 153)
The connections of Clebsch with Jordan, whose monumental Traite
des substitutions had just appeared, surely encouraged Klein to study
groups and to go to Paris in 1870, where his new friend Lie rejoined him;
the two even had adjacent rooms. In Paris, the two friends again talked
daily, and also had frequent discussions with Gaston Darboux (1842-1917).
There they extended an old theorem of Chasles, which states that
orientation-preserving rigid transformations of space are, in general, 'screw
motions' of translation along an axis and rotation about it. This theorem
had been previously extended by Olinde Rodrigues (see Gray 1980) and
Jordan (1869). Klein and Lie wrote two joint notes in Paris, the publica-
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 149
fying geometric theories and theorems. His few readers and listeners must
have realized that they were being exposed to new and fundamental
perspectives, even if they barely understood them.
5. Klein and Geometry
In 1872, the E.P. was 20 years ahead of its time; it would take at least
that long for the new perspectives of Klein and Lie to gain general accep-
tance. (However, their ideas would take deep root; in fifty years it would
become commonplace to refer also to "metric," "projective," "affine,"
and "conformal" differential geometry; see §13.)
In the meantime, Klein's rapid rise to leadership in Germanic geometry
was based on his other writings. His preeminence rested on the impres-
sion he made on contemporaries, and not on what he might write for
posterity. The value of his distinction between the familiar Riemann sphere
(often associated with conformal geometry) and the elliptic plane, con-
sisting of the sphere with opposite points identified, was immediately
appreciated.
So was Klein's use of the name "elliptic geometry" for a manifold of
constant positive curvature, as distinguished from a "hyperbolic geometry"
for one having constant negative curvature, and "parabolic geometry"
for Euclidean geometry and its cylindrical and toroidal "space forms"
(Kline 1972, 913). (Not long after, du Bois-Reymond made an analogous
classification of differential equations into those of "elliptic," "hyper-
bolic," and "parabolic" type.)
Klein's observations stimulated the British mathematician W. K. Clif-
ford (1873) to call attention to the philosophical difference between local
and global homogeneity, which had been overlooked by Riemann and
Helmholtz. Clifford also called attention (as did Klein) to the connection
between Pliicker's line geometry and 'screws' (the Theorem of Chasles).
However, he died before he could develop these ideas very far, and Klein
would not return to them until 1890 (see §§7 and 8).
Klein also showed the logical incompleteness of von Staudt's path-
breaking introduction of coordinates into axiomatically defined "projec-
tive geometries," an observation that stimulated Liiroth and others to
clarify the assumptions underlying von Staudt's "algebra of throws."5
Although most mathematicians today would consider the E.P. as
primarily a contribution to the foundations of geometry, Klein did not
regard it as such, perhaps because in 1872 the concept of a continuous
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 753
group was so novel, and even the theory of invariants still a research fron-
tier. His prefatory remarks in his (GMA) about his contributions to the
foundations of geometry concentrate on the topics that we have discussed
above. By his own reckoning (GMA, vol. 1), Klein published ten papers
on the foundations of geometry, fourteen on line geometry (an interest
he had inherited from Pliicker), and nine on the E.P. (of which the first
was with Lie). He also lists (GMA, vol. 2) sixteen papers on intuitive
(Anschauliche) geometry; in addition, he wrote several books on geometry.
In particular, his Einleitung in die Hohere Geometric of 1893 gave a general
and quite comprehensive picture of the geometry of the day, and the sec-
ond volume of his Elementarmathematik vom Hoheren Standpunkt aus
(1908) was also devoted to geometry. But of all his geometrical contribu-
tions over the years, Klein himself apparently regarded the E.P. as his
"most notable achievement" (Young 1928, v).
6. Klein and Lie
The friendship formed by Klein and Lie in the unwelcoming atmosphere
of Berlin in 1869, renewed in Paris in 1870 and again in Erlangen in 1872,
proved invaluable for both men and for mathematics. We have already
observed that much of the inspiration for the E.P. stemmed from their
discussions during these years, and we shall now describe some of the less
immediate and more worldly benefits derived by Lie from this friendship.6
Whereas Klein had eloquently expressed, within months of conceiving
it, his idea that different branches of geometry were associated with in-
variance under different groups of transformations of underlying geometric
manifolds, Lie's deeper ideas would mature much more slowly. He would
spend the rest of his life in developing the intuitive concept of a 'continuous
group of transformations' into a powerful general theory and in applying
this concept to geometry and to partial differential equations.
Max Noether, in his obituary article (1900), described Lie's and Klein's
1869-70 sojourn in Berlin and the connections between the E.P. and Lie's
later work:
In the E.P. [we find expressed] for the first time the central role of
the [appropriate] transformation group for all geometrical investiga-
tions. . .and that, with invariant properties, there is always associated
such a group.... Lie, who had worked with the most varied groups,
but to whom the meaning of classification had remained foreign, found
the idea congenial from then on. (Noether 1900, 23-23)
154 Garrett Birkhoff and M. K. Bennett
Noether called attention in a footnote to alterations made by Lie in several
of his 1872 articles, apparently as a result of Klein's new ideas.
During the decade 1872-82, Lie worked in isolation in Christiania (now
Oslo), encouraged almost exclusively by Klein and Adolf Mayer (1839-
1908). It was then that he published the striking fact that every finite con-
tinuous group ("Lie group" in today's terminology) acting on the line is
locally equivalent (ahnlich) to either the translation group of all functions
x •->• x + b, the affine group of all x •-»• ax + b, or the projective group of all
x»(ax+ b}/(cx + d), ad ^ be (LGA, vol. 5, 1-8; Gott. Nachr. 22 [1874]:
529-42). (For an annotated summary, see [Birkhoff 1973, 299-305]. Ac-
tually, the theorem is only true locally.)
Lie's thoughts at this time are revealed in a letter of 1873 to Mayer,
which states: "I have obtained most interesting results and I expect very
many more. They concern an idea whose origin may be found in my earlier
works with Klein: namely, to apply the concepts of the theory of substitu-
tions to differential equations" (LGA, vol. 5, 584). Four years later, Lie
determined locally (almost) all finite continuous groups acting transitive-
ly on a two-dimensional manifold, the next step toward determining all
the homogeneous manifolds (or "spaces") envisioned in the E.P.7
This was the third in a series of five definitive papers, published in Chris-
tiania in the years 1876-79, in which Lie laid the foundations of his theory
of continuous groups. The introduction to the first of these states in part
(LGA, vol. 5, 9):
I plan to publish a series of articles, of which the present one is the
first, on a new theory that I will call the theory of transformation
groups. The investigations just mentioned have, as the reader will notice,
many points of contact with several mathematical disciplines, especially
with the theory of substitutions,1 with geometry and modern manifold
theory,2 and finally also with the theory of differential equations.3
These points of contact establish connections between these former
separate fields.... I must prepare later articles to present the impor-
tance and scope of the new theory.
^ee Camille Jordan's Traite des substitutions (Paris, 1870). Compare
also Jordan's investigations of groups of motions.
2
See various geometric works by Klein and myself, especially Klein's
[E.P.], which hitherto has perhaps not been studied sufficiently by
mathematicians.
3
See my investigations on differential equations.
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 755
Unfortunately, Lie was not happy in Leipzig, and there he became ex-
cessively jealous of possible rivals. Thus the introduction of the third
volume of Lie-Engel goes out of its way to assert that Lie was "not Klein's
student," but that "rather, the contrary was the case."10 Likewise, in a
paper reproduced in (LGA, vol. 2, 472-79), he takes pains to identify gaps
in the reasoning of de Tilly, Klein (M.A. 37: 364), Lindemann, Fr. Schur
(who had, on the contrary, actually rigorized Lie's somewhat cavalier dif-
ferentiability assumptions), Helmholtz, and Killing!
A turning point seems to have come when Lie was awarded the first
Lobachewsky Prize in 1893 for solving the Riemann-Helmholtz problem
(a celebrated problem, which, incidentally, Klein had suggested he work
on). Namely, Lie had shown that any ^-dimensional Riemannian manifold
admitting an n(n + l)/2-parameter group of rigid motions (the "free
mobility" condition of Helmholtz) is locally isometric to either Euclidean
«-space, the w-sphere, or the ^-dimensional "hyperbolic" geometry of
Lobachewsky-Bolyai. This classic local result of Lie stands in sharp con-
trast with Klein's continuing concern with the global Clifford-Klein pro-
blem, to which we will return in §8.
Appropriately, Klein was invited to write a suitable appreciation of Lie's
solution for the occasion of the prize presentation, which he did with his
usual imaginative, insightful style. His narrative contained, however, one
complaint: the presence of an unmotivated, and to Klein unnatural,
assumption of differentiability in the foundations of geometry.
Klein's complaint was given a positive interpretation by Hilbert. As
the fifth in his famous list of unsolved problems proposed at the 1900
International Mathematical Congress, Hilbert proposed proving that any
continuous (locally Euclidean) group was in fact an analytical group with
respect to suitable parameters. Whether this problem should be attributed
to Hilbert, Klein, or Lie, its successful solution took another 50 years,
and Klein's E.P. was at least one of its indirect sources (see [Birkhoff and
Bennett, forthcoming]).
tingen, the work place of Gauss and Riemann, into a preeminent world
center of mathematics.17 One of his first activities was to organize an in-
ternational commission to fund a monument to Gauss and the physicist
W. Weber, who had collaborated in constructing an early telegraph. (It
was to honor the dedication of this monument that Hilbert, at Klein's
invitation, wrote his famous Grundlagen der Geometric [see §12].)
In 1895 Klein invited Hilbert to Gdttingen, and Hilbert accepted with
alacrity. Although Klein continued to give masterful advanced expository
lectures, Hilbert was soon attracting the lion's share of doctoral candidates
(see §12). Indeed, by 1900 Klein had become primarily a policy-maker and
elder statesman, although barely 50. In this role, Klein obtained govern-
mental and industrial support from an Institute of Applied Mathematics,
with Prandtl and Runge as early faculty members. Sommerfeld, at one
time his assistant and later coauthor with Klein of Die Kreisel ("The Top"),
was another link of Klein with applied mathematics and physics.18
Klein and Education. Already in his Antrittsrede (see §4), Klein had
expressed his concern about separation into humanistic and scientific
education, stating that: "Mathematics and those fields connected with it
are relegated to the natural sciences, and rightly so.... On the other hand,
its conceptual content belongs to neither of the two categories" (Rowe
1985, 135). Klein's later involvement with German educational policy-
making is described in (Pyenson 1983); in fact, Klein is the main subject
of two of its chapters.
Klein was a universalist who believed strongly in integrating pure with
applied mathematics, in the importance of both logic and intuition in
geometry, and in the importance of having high-school teachers who
understood and appreciated higher mathematics. Especially widely read
by high-school teachers were his 1895 lectures on Famous Problems in
Elementary Geometry, written for this purpose and translated into English,
French, and Italian. (See R. C. Archibald, American Mathematical Month-
ly 21 [1914]: 247-59, where various slips were carefully corrected.)
In 1908, Klein became the president of the International Mathematical
Teaching Commission. In this capacity he worked closely with the
American David Eugene Smith and the Swiss Henri Fehr to improve
mathematics education throughout the western world. As with other
cooperative enterprises, this one was ended by World War I.
755 Garrett Birkhoff and M. K. Bennett
have apparently forsaken his extreme formalism and written (with S. Cohn-
Vossen) a book entitled Anschauliche Geometric ("Intuitive Geometry").
(For a fuller account of the Grundlagen der Geometric, its background
and influence, see [Birkhoff and Bennett, forthcoming].)
Hilbert's social and scientific personalities were very different from
those of the dignified and highly intuitive Klein, and there is little doubt
about Hilbert's restiveness as regards Klein's regal manner. Thus in a let-
ter to his future wife, Courant wrote in 1907-8 that "Hilbert now rebels
everywhere against Klein's assumed dictatorship" (Reid 1976, 19).
Likewise, Ostrowski (coeditor of vol. 1 of Klein's GMA) wrote one of
us in 1980: "As to Hilbert I do not think that you will find any reference
to the Erlanger Programm. As a matter of fact, Hilbert did not think very
much of it."
13. Study and Elie Cartan
Three more major mathematicians whose work reflects the influence
of the E.P. are Eduard Study (1862-1930), Elie Cartan (1869-1951), and
Hermann Weyl (1885-1955). Although they had related interests, they had
very different backgrounds and tastes. We shall discuss next the influence
of the E.P. on Study and Cartan, taking up its influence on Weyl in §14.
Eduard Study. As a geometer, Study was more influential than either
Killing or Engel. Blaschke dedicated the first volume (1921) of his famous
Vorlesungen tiber Differentialgeometrie to Study, and Study's Ph.D.
students included not only Fine but also J. L. Coolidge, the author of
several widely read books, whose History of Geometrical Methods
(Coolidge 1940) is a standard reference. From the chapter "Higher Space
Elements" in this treatise, we quote the following passage:
The connecting thread in [this chapter] is the idea of treating directly
as a space element some figure previously treated as a locus. The idea
of doing this was dominant in geometrical circles, especially in the
schools of Klein and Study, at the end of the nineteenth century....
It shades off imperceptibly into the theory of geometrical transforma-
tions. (Coolidge 1940, book 2, chap. 6)
This idea, obviously generalizing Pliicker's "line geometry," can be used
in the spirit of the E.P. to construct many "global" representations of
continuous groups as transformation groups of manifolds.
Study began his career at the University of Munich, where he wrote
his doctoral thesis Ueber die Massbestimmungen Extensiver Grossen in
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 169
relativity" theory fits much less neatly into Klein's classification scheme
than his "special theory." See (Bell 1940, chap. 20, esp. 443-49), where
a quotation from Veblen's chapter 3 is contrasted with one made by Veblen
ten years later, and one made in 1939 by J. H. C. Whitehead.22
However, the properties of Einstein's curved space-time are expressed
by local differential invariants, which can be classified as 'conformaP,
'metric', 'affine', 'projective', etc., very much in the spirit of the E.P.
J. A. Schouten (1926) spelled out this connection. Cartan's invited ad-
dress on "Lie Groups and Geometry" at the Oslo congress (1936) and
J. H. C. Whitehead's biography of Cartan ("Obituary Notices of the
Fellows of the Royal Society" 8, 1952) give perhaps the most authoritative
opinions on the subject.
In the 1920s and 1930s, Cartan also showed his consummate and
creative command of old and new mathematics by his major contribu-
tions to the "globalization" of Lie's purely local theory of Lie groups.
The resulting global theory has made both Klein's and Lie's expositions
technically obsolete.
Between 1927 and 1935, Cartan published what Chern and Che valley
call "his most important work in Riemannian geometry.. .the theory of
symmetric spaces" (Bulletin of the American Mathematical Society 58
[1952]: 244). These are Riemannian manifolds in which ds2 is invariant
under reflection in any point. In retrospect, Cartan's tortuous path (via
the parallelism of Levi-Civita) to the recognition of this simple idea seems
amazing.23 Even more amazing is the failure of Klein, after applying reflec-
tions in lines in the hyperbolic plane in so many ways, to identify "sym-
metric Riemann spaces" at all!
14. Klein and Hermann Weyl
Hermann Weyl began his career in the Gottingen that Klein had built
up, and Hilbert was his thesis advisor. Hence he was in some sense an
academic great-grandson of Klein, whom he must have known. Weyl seems
never to have been overawed by Hilbert; thus his 1908 thesis stated une-
quivocally (M.A. 66 [1909]: 273): "As will be shown below, the applica-
bility of Hilbert's method is by no means limited to the continuous kernels
treated by Hilbert... but also leads to interesting consequences in certain
more general cases." Beginning in 1917, Weyl outdistanced Hilbert (again)
in mathematicizing Einstein's then new general theory of relativity. In
papers and in his famous book Raum. Zeit. Materie (later translated into
772 Garrett Birkhoff and M. K. Bennett
English as Space, Time, Matter), he introduced local generalizations of
affine, projective, and conformal geometry that are related to their global
counterparts as Riemannian geometry is related to Euclidean, as Schouten
(1926) was later to explain. Still under 40, he then attacked Hilbert's "for-
malist" logic of mathematics in the early 1920s supporting the conflict-
ing "intuitionist" logic of L. E. J. Brouwer.
Weyl showed more respect for Klein, to whom he dedicated the first
(1912) edition of his Die Idee der Reimannschen Flache.2* This was
because, as he stated in the preface to its 1955 edition, "Klein had been
the first to develop the freer conception of a Riemann surface,... thereby
he endowed Riemann's basic ideas with their full power."
Then, in the middle 1920s, Weyl was the spark plug of the famous Peter-
Weyl theory of group representations. He used a very concrete theory of
group-invariant measure on compact Lie groups, the existence of which
permits one to extend to compact Lie groups the result of E. H. Moore
'and Maschke: that every group of linear transformations having a finite
group-invariant measure is equivalent to a group of orthogonal transfor-
mations.25 In his Gruppentheorie und Quantenmechanik (translated into
English by H. P. Robertson), Weyl later applied the analogous result for
the orthogonal group to the then new quantum mechanics.26
By an irony of fate, very little of Weyl's deep and influential research
work was done at Gottingen, the source of much of his inspiration. It
was only three years after he finally accepted a professorship there that
Hitler seized power in Germany. In the next year, Weyl emigrated to the
new Institute of Advanced Study in Princeton, where he spent his last twen-
ty years, creative and versatile to the end. His later books, The Classical
Groups (1939) and Symmetry (1952), show his spiritual affinity with Klein,
and it seems fitting to conclude our review with two quotations from the
former. First, Weyl remarks:
This is not the place for repeating the string of elementary definitions
and propositions concerning groups which fill the first pages of every
treatise on group theory. Following Klein's "Erlanger Program" (1872),
we prefer to describe in general terms the significance of groups for
the idea of relativity, in particular in geometry. (1939, 14)
Later, he makes a more specific evaluation:
The dictatorial regime of the projective idea in geometry was first
broken by the German astronomer and geometer Mobius, but the
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 773
Notes
1. In discussing this last phase of Klein's career, Rowe (1985, 278) has referred to him
as the "doyen of German mathematics for nearly three decades."
2. Weierstrass himself always viewed geometry metrically.
3. For the significance of W-curves, Klein refers his reader to ##13-20 and #34 of the
article of Scheffers (BMW, vol. 3, D4).
4. In the second of these (Klein 1873), submitted in June 1872, Klein describes briefly
how "the different methods of geometry can be characterized by an associated transforma-
tion group." This paper and the E.P. were the first publications by Klein or Lie that used
the phrase transformation group. The word invariant is conspicuous by its absence, although
invariant theory had been studied in Germany (without explicit mention of the word group)
for at least a decade.
5. See (Birkhoff and Bennett, forthcoming) for references to this work of Klein and Lu'roth,
as well as its influence on later theories.
6. A glance at the index of names in the relevant volumes of Lie's Collected Papers (LGA)
makes Lie's scientific indebtedness to Klein obvious.
7. Lie's local theory (LGA, vol. 5) was finally extended into a rigorous global theory
by G. D. Mostow (Annals of Mathematics 52 [1950]: 606-36). There Mostow showed that
the two-dimensional manifolds are the plane, cylinder, torus, sphere, projective homogeneous
plane, MObius strip, and Klein bottle.
8. Picard had used the simplicity of the (Lie) projective group on two variables to prove
the impossibility of solving u" + p(x)u' + q(x)u = 0 by quadratures—a theorem that Lie
wished he had discovered himself.
9. See also (Hawkins 1982, §3).
10. For a very human account of Lie's jealousy of Klein and their final reconciliation,
which reproduces a moving letter from Frau Klein, see (Young 1928, xviii-xix).
11. Wedekind's summary mentions his use of a result in the E.P.; Klein entitled his paper
"Binary Forms Invariant under Linear Transformations."
12. By this, Klein meant that the subgroup of the group of the icosahedron leaving each
point invariant is of order 2, as contrasted with 5 for the vertices of the icosahedron and
3 for the dodecahedron.
13. As late as 1886, Brioschi (M.A. 26 [1886]: 108) would refer to the "icosahedral"
hyper geometric equation:
x(x- l)v" + (7x-4v')/6 + llv/3 2 4 2 5 2 = 0.
14. See, for example, L. R. Ford, Automorphic Functions (New York: McGraw-Hill,
1929).
15. Another close friend of Fine was Woodrow Wilson, president of Princeton Univer-
sity, and later of the United States. Further information on H. B. Fine can be found in
William Aspray's paper in this volume.
16. "For a quarter of a century no one could think of the American Mathematical Society
apart from the personality of Professor Cole" (Archibald 1938, 101).
174 Garrett Birkhoff and M. K. Bennett
17. Many references to Klein's role in Gottingen may be found in (Reid 1970, 1976);
see also chapter 11 of (Klein 1893b).
18. Klein had shown his belief in the importance of applied mathematics in 1875, by
leaving a full professorship at Erlagen for a position at the Technische Hochschule in Munich
and referring to it as a great advance ("einem grossen Sprung") (Young 1928, vii). There
he held a seminar on pure and applied mathematics with von Linde (Pyenson 1979, 55-61).
19. For Klein's role in suggesting Hilbert's Fifth Problem, see (Birkhoff and Bennett,
forthcoming).
20. See (Ziegler 1985) and (BMW, vol. 3, parts 1, 2).
21. Principle (a) clearly refers to the method used by Hilbert in his Grundlagen der
Geometrie.
22. Actually, Whitehead did not lose interest in the E.P. as quickly as Bell suggests; see
(Annals of Mathematics 33 [1932]: 681-87).
23. For Cartan's mature exposition of the theory of symmetric spaces, see Proceedings,
International Congress of Mathematicians, Zurich, 1932, 152-61.
24. A few years earlier, Paul Koebe (also in Gottingen) had finally solved rigorously
the "uniformization problem" that had eluded both Klein and Poincard. Koebe's
"Primenden," one of his major technical tools, seem related to the ideas of Klein's "Grenz-
kreistheorem" (see §9).
25. The Peter-Weyl theory surely helped to inspire Haar's 1933 theory of invariant
Lebesgue measure on compact topological groups. It was von Neumann's subsequent ap-
plication of Haar measure to solve the Klein-Hilbert Fifth Problem for compact groups,
and Pontrjagin's parallel solution for Abelian groups, that paved the way for its complete
solution.
26. One of us adapted Weyl's title to the chapter on group theory and fluid mechanics
in his Hydrodynamics (Birkhoff, Princeton University Press, 1950). In this the notion of
'self-similar solution' (exploited earlier by Sedov and others) was generalized to arbitrary
groups. The ultimate inspiration for this chapter was Klein's group-theoretic interpretation
of special relativity in his (EdM), as an extension of the E.P.
References
[EdM] Klein, Felix. 1926-27. Entwicklung der Mathematik in 19ten Jahrhundert. Berlin:
Springer. Reprinted New York: Chelsea, 1950.
[EMW] Enzyklopadie der Mathematischen Wissenschaften, esp. vol. 3.1.
[ESM] Encyclopedic des Sciences Mathematiques, esp. vol. 3.
[GMA] Klein, Felix. 1921-23. Gesammelte Mathematische Abhandlungen. 3 vols. Berlin:
Springer.
[LGA] Lie, Sophus. 1922-37. Gesammelte Abhandlungen. 6 vols. Ed. F. Engel and P.
Heegard. Berlin: Springer.
Archibald, R. C. 1938. A semi-centennial History of the American Mathematical Society;
1888-1938. Providence, R.I.: American Mathematical Society, vol. 1.
Bell, E. T. 1940. The Development of Mathematics. New York: McGraw-Hill.
Birkhoff, G. 1973. Source Book in Classical Analysis. Cambridge, Mass.: Harvard Univer-
sity Press.
Birkhoff, G., and Bennett, M. K. Hilbert's "Grundlagen der Geometrie." Rendiconti del
Circolo Mathematico di Palermo. Forthcoming.
Cartan, E. 1952. Oeuvres completes. 12 vols. Paris: Gauthier-Villars.
Clebsch, A., and Lindemann, F. 1876-91. Vorlesungen uber Geometrie, bearb. und hrsg.
von F. Lindemann. Leipzig: Teubner.
Clifford, W. K. 1873. Preliminary Sketch of Biquaternions. Proceedings of the London
Mathematical Society 4: 381-95.
Coolidge, J. L. 1940. A History of Geometrical Methods. Oxford: Clarendon Press.
Courant, R. 1925. Felix Klein. Jahresbericht der Deutschen Mathematiker Vereiningung 34:
197-213.
FELIX KLEIN AND HIS "ERLANGER PROGRAMM" 775
Engel, F. 1900. Sophus Lie. Jahresbericht der Deutschen Mathematiker Vereiningung 8: 30-46.
Engel, F. and Faber, K. 1932. Die Liesche Theorie der Partiellen Differentialgleichungen
Erster Ordnung. Leipzig: Teubner.
Gray, Jeremy J. 1980. Olinde Rodrigues' Paper of 1840 on Transformation Groups. Ar-
chive for History of Exact Sciences 21: 375-85.
Hawkins, Thomas. 1981. Non-Euclidean Geometry and Weierstrassian Mathematics....
Historia Mathematica 1: 289-342.
. 1981. Wilhelm Killing and the Structure of Lie Algebras. Archive for History of Ex-
act Sciences 26: 127-92.
-. 1984. The Erlanger Programm of Felix Klein.... Historia Mathematica 11: 442-70.
Hilbert, David. 1899. Grundlagen der Geometrie. 1st ed. Liepzig: Teubner.
Hopf, Heinz. 1926. Zum Clifford-Kleinschen Raumproblem. Mathematische Annalen 95:
313-39.
Jordan, Camille. 1869. Mdmoire sur les groupes de mouvements. Annali di Matematica (2)
2: 167-213 and 322-451.
. 1870. Traite des substitutions. Paris: Gauthier-Villars.
Killing, Wilhelm. 1886. Zur Theorie der Lie'schen Transformationsgruppen. Braunschweig.
. 1891. Uber die Clifford-Kleinsche Raumformen. Mathematische Annalen 39: 257-78.
Klein, Felix. 1872. Vergleichende Bertrachtungen uber neuere geometrische Forschungen
Erlangen. Reprinted with additional footnotes in Mathematische Annalen 43 (1893): 63-100
and in GMA, vol. 1, 460-97.
. 1873. Uber die sogenannte nicht-Euklidische Geometrie. Mathematische Annalen 6:
112-45.
. 1884. Vorlesungen uber das Ikosaeder.... Leipzig: Teubner. (English translation
by G. G. Morrice, 1888; cf. also F. N. Cole's review in American Journal of Mathematics
9 [1886]: 45-61.
. 1892. Einleitung in die geometrischen Funktionentheorie. Lectures given in 1880-81,
Leipzig, and reproduced in Gottingen a decade later.
. 1893a. Einleitung in die Hohere Geometrie. 2 vols. Lithographed lecture notes by
Fr. Schilling.
. 1893b. The Evanston Colloquim "Lectures on Mathematics." Republished by the
American Mathematical Society in 1911.
. 1895. Vortrage uber ausgewahlte Fragen der Elementargeometrie. Gottinge
Translated as Famous Problems of Elementary Geometry by W. W. Beman and D. E.
Smith in 1900; 2d ed. reprinted by Stechert, 1936.
-. 1908. Elementarmathematik vom Hoheren Standpunkt aus. Vol. 2. Leipzig: Teubner.
Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. New York:
Oxford University Press.
Lie, Sophus. 1880. Theorie der Transformationsgruppen I. Mathematische Annalen 16:
441-528.
Lie, S., and Engel, F., 1888-93. Theorie der Transformationsgruppen. 3 vols. Leipzig:
Teubner. Reprinted in 1930-36.
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Noether, Max. 1900. Sophus Lie. Mathematische Annalen 53: 1-41.
Pliicker, Julius. 1868-69. Neue Geometrie des Raumes..., hrsg. von Felix Klein. Leipzig:
Teubner.
Poincare, Henri. 1907-8. Science et Methode. English translation published by the Science
Press in 1921 with the title Foundations of Science.
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Pyenson, L. 1979. Mathematics, Education, and the Gottingen Approach to Physical Reality,
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. 1983. Neohumanism and the Persistence of Pure Mathematics in Wilhelmian Ger-
many. Philadelphia: American Philosophical Society.
Reid, Constance. 1970. Hilbert. Berlin: Springer.
. 1976. Courant in Gottingen and New York. Berlin: Springer.
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Historia Mathematica 10: 448-57.
. 1984. Felix Klein's Erlanger Antrittsrede.... Historia Mathematica 12: 123-41.
. 1985. Essay Review. Historia Mathematica 12: 278-91.
Schouten, J. A. 1926. Erlanger Programm and Uebertragungslehre. Rendiconti del Circolo
Mathematico di Palermo 50: 142-69.
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Veblen, O., and Young, J. W. 1917. Projective Geometry. Vol. 2. (Actually written by
Veblen.) Boston: Ginn.
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tion of the 2d ed. by Gerald Mac Lane, Addison-Wesley, 1955.)
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. 1944. David Hilbert and His Mathematical Work. Bulletin of the American
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Joseph W Dauben
777
775 Joseph W. Dauben
not noticed it is being willfully blind. The crisis is due to our neglect
of philosophical issues... .2
Bishop, too, relates his crisis in part to the subject of the infinite and in-
finitesimals. Arguing that formalists mistakenly concentrate on the "truth"
rather than the "meaning" of a mathematical statement, he criticizes
Abraham Robinson's nonstandard analysis as "formal finesse," adding
that "it is difficult to believe that debasement of meaning could be car-
ried so far."3 Not all mathematicians, however, are prepared to agree that
there is a crisis in modern mathematics, or that Robinson's work con-
stitutes any debasement of meaning at all.
Kurt Godel, for example, believed that Robinson more than anyone
else had succeeded in bringing mathematics and logic together, and he
praised Robinson's creation of nonstandard analysis for enlisting the
techniques of modern logic to provide rigorous foundations for the calculus
using actual infinitesimals. The new theory was first given wide publicity
in 1961 when Robinson outlined the basic idea of his "nonstandard"
analysis in a paper presented at a joint meeting of the American
Mathematical Society and the Mathematical Association of America.4
Subsequently, impressive applications of Robinson's approach to infin-
itesimals have confirmed his hopes that nonstandard analysis could enrich
"standard" mathematics in important ways.
As for his success in defining infinitesimals in a rigorously mathematical
way, Robinson saw his work not only in the tradition of others like Leib-
niz and Cauchy before him, but even as vindicating and justifying their
views. The relation of their work, however, to Robinson's own research
is equally significant, as Robinson himself realized, and this for reasons
that are of particular interest to the historian of mathematics. Before
returning to the question of a "new" crisis in mathematics due to Robin-
son's work, it is important to say something, briefly, about the history
of infinitesimals, a history that Robinson took with the utmost seriousness.
This is not the place to rehearse the long history of infinitesimals in
mathematics. There is one historical figure, however, who especially
interested Robinson—namely, Cauchy—and in what follows Cauchy pro-
vides a focus for considering the historiographic significance of Robin-
son's own work. In fact, following Robinson's lead, others like J. P.
Cleave, Charles Edwards, Detlef Laugwitz, and W. A. J. Luxemburg have
used nonstandard analysis to rehabilitate or "vindicate" earlier infin-
itesimalists.5 Leibniz, Euler, and Cauchy are among the more promi-
ABRAHAM ROBINSON AND NONSTANDARD ANALYSIS 179
tial as a finite line segment rather than the infinitely small quantity that
was used in practice.16
This confusion between theoretical considerations and practical applica-
tions carried over to Leibniz's metaphysics of the infinite, for he was never
committed to any one view but made conflicting pronouncements.
Philosophically, as Robinson himself has argued, Leibniz had to assume
the reality of the infinite—the infinity of his monads, for example—or
the reality of infinitesimals not as mathematical points but as substance-
or force-points—namely, Leibniz's "monads" themselves.17
That the eighteenth century was concerned not with doubts about the
potential of infinitesimals but primarily with fears about their logical con-
sistency is clear from the proposal Lagrange drew up for a prize to be
awarded by the Berlin Academy for a rigorous theory of infinitesimals.
As the prize proposal put it:
It is well known that higher mathematics continually uses infinitely large
and infinitely small quantities. Nevertheless, geometers, and even the
ancient analysts, have carefully avoided everything which approaches
the infinite; and some great modern analysts hold that the terms of the
expression infinite magnitude contradict one another.
The Academy hopes, therefore, that it can be explained how so many
true theorems have been deduced from a contradictory supposition,
and that a principle can be delineated which is sure, clear—in a word,
truly mathematical—which can appropriately be substituted for the
infinite.18
Lakatos seems to appreciate all this—and even contradicts himself on
the subject of Leibniz's theory and the significance of its perceived in-
consistency. Recalling his earlier assertion that Leibniz's theory was not
overthrown because of its inconsistency, consider the following line, just
a few pages later, where Lakatos asserts that nonstandard analysis raises
the problem of "how to appraise inconsistent theories like Leibniz's
calculus, Frege's logic, and Dirac's delta function."19
Lakatos apparently had not made up his mind as to the significance
of the inconsistency of Leibniz's theory, which raises questions about the
historical value and appropriateness of the extreme sort of rational
reconstruction that he has proposed to "vindicate" the work of earlier
generations. In fact, neither Leibniz nor Euler nor Cauchy succeeded in
giving a satisfactory foundation for an infinitesimal calculus that also
demonstrated its logical consistency. Basically, Cauchy's "epsilontics"
ABRAHAM ROBINSON AND NONSTANDARD ANALYSIS 183
in fact, does not seem to be true. There is another side to this as well,
for one may also ask whether there is any truth to the assertions made
by Robinson (and emphatically by Keisler) that "the whole point of our
infinitesimal approach to calculus is that it is easier to define and explain
limits using infinitesimals."38 Of course, this claim also deserves examina-
tion, in part because Bishop's own attempt to dismiss Keisler's methods
as being equivalent to the axiom "0 = 1" is simply nonsense.39 In fact,
there are concrete indications that despite the allegations made by Bishop
about obfuscation and the nonintuitiveness of basic ideas in nonstandard
terms, exactly the opposite is true.
Not long ago a study was undertaken to assess the validity of the claim
that "from this nonstandard approach, the definitions of the basic con-
cepts [of the calculus] become simpler and the arguments more intuitive."40
Kathleen Sullivan reported the results of her dissertation, written at the
University of Wisconsin and designed to determine the pedagogical
usefulness of nonstandard analysis in teaching calculus, in the American
Mathematical Monthly in 1976. This study, therefore, was presumably
available to Bishop when his review of Keisler's book appeared in 1977,
in which he attacked the pedagogical validity of nonstandard analysis.
What did Sullivan's study reveal? Basically, she set out to answer the
following questions:
Will the students acquire the basic calculus skills? Will they really
understand the fundamental concepts any differently? How difficult
will it be for them to make the transition into standard analysis courses
if they want to study more mathematics? Is the nonstandard approach
only suitable for gifted mathematics students?41
To answer these questions, Sullivan studied classes at five schools in
the Chicago-Milwaukee area during the years 1973-74. Four of them were
small private colleges, the fifth a public high school in a suburb of
Milwaukee. The same instructors who had taught the course previously
agreed to teach one introductory course using Keisler's book (the 1971
edition) as well as another introductory course using a standard approach
(thus serving as a control group) to the calculus. Comparison of SAT scores
showed that both the experimental (nonstandard) group and the standard
(control) group were comparable in ability before the courses began. At
the end of the course, a calculus test was given to both groups. Instruc-
tors teaching the courses were interviewed, and a questionnaire was filled
out by everyone who has used Keisler's book within the last five years.
ABRAHAM ROBINSON AND NONSTANDARD ANALYSIS 191
The single question that brought out the greatest difference between
the two groups was question 3:
Notes
1. There is a considerable literature on the subject of the supposed crisis in mathematics
associated with the Pythagoreans. See, for example, (Hasse and Scholz 1928). For a recent
survey of this debate, see (Berggren 1984; Dauben 1984; Knorr 1975).
2. (Bishop 1975, 507).
3. (Bishop 1975, 513-14).
4. Robinson first published the idea of nonstandard analysis in a paper submitted to
the Dutch Academy of Sciences (Robinson 1961).
5. (Cleave 1971; Edwards 1979; Laugwitz 1975, 1985; Luxemburg 1975).
6. (Lakatos 1978).
7. (Lakatos 1978, 43).
8. (Lakatos 1978, 44).
9. (Lakatos 1978, 49).
10. (Lakatos 1978, 50). Emphasis in original.
11. Cauchy offers his definitions of infinitely large and small numbers in several works,
first in the Cours d'analyse, subsequently in later versions without substantive changes. See
(Cauchy 1821, 19; 1823, 16; 1829, 265), as well as (Fisher 1978).
12. (Cauchy 1868).
13. (Lakatos 1978, 54).
14. For details of the successful development of the early calculus, see (Boyer 1939;
Grattan-Guinness 1970, 1980; Grabiner 1981; Youshkevitch 1959).
15. (Newton 1727, 39), where he discusses the contrary nature of indivisibles as
demonstrated by Euclid in Book X of the Elements. For additional analysis of Newton's
views on infinitesimals, see (Grabiner 1981, 32).
16. See (Leibniz 1684). For details and a critical analysis of what is involved in Leibniz's
presentation and applications of infinitesimals, see (Bos 1974-75; Engelsman 1984).
17. See (Robinson 1967, 35 [in Robinson 1979, 544]).
18. In (Lagrange 1784, 12-13; Dugac 1980, 12). For details of the Berlin Academy's com-
petition, see (Grabiner 1981, 40-43; Youshkevitch 1971, 149-68).
19. (Lakatos 1978, 59). Emphasis added.
20. See (Grattan-Guinness 1970, 55-56), where he discusses "limit-avoidance" and its
role in making the calculus rigorous.
21. (Robinson 1965b).
22. (Robinson 1965b, 184); also in (Robinson 1979, vol. 2, 87).
23. I am grateful to Stephan Korner and am happy to acknowledge his help in ongoing
discussions we have had of Robinson and his work.
24. For a recent survey of the controversies surrounding the early development of the
calculus, see (Hall 1980).
25. Borel in a letter to Hadamard, in (Borel 1928, 158).
26. (KOrner 1979, xlii). Korner notes, however, that an exception to this generalization
is to be found in Hans Vaihinger's general theory of fictions. Vaihinger tried to justify in-
finitesimals by "a method of opposite mistakes," a solution that was too imprecise, Korner
suggests, to have impressed mathematicians. See (Vaihinger 1913, 51 Iff).
ABRAHAM ROBINSON AND NONSTANDARD ANALYSIS 197
27. (Robinson 1965a, 230; Robinson 1979, 507). Nearly ten years later, Robinson re-
called the major points of "Formalism 64" as follows: "(i) that mathematical theories which,
allegedly, deal with infinite totalities do not have any detailed meaning, i.e. reference, and
(ii) that this has no bearing on the question whether or not such theories should be developed
and that, indeed, there are good reasons why we should continue to do mathematics in the
classical fashion nevertheless." Robinson added that nothing since 1964 had prompted him
to change these views and that, in fact, "well-known recent developments in set theory repre-
sent evidence favoring these views." See (Robinson 1975, 557).
28. (Robinson 1970, 45-49).
29. (Robinson 1970, 45-49).
30. (Birkhoff 1975, 504).
31. (Birkhoff 1975, 504).
32. (Bishop 1975, 507).
33. (Bishop 1975, 508).
34. For Cantor's views, see his letter to the Italian mathematician Vivanti in (Meschkowski
1965, 505). A general analysis of Cantor's interpretation of infinitesimals may be found
in (Dauben 1979, 128-32, 233-38). On the question of rigor, see (Grabiner 1974).
35. (Bishop 1975, 514).
36. It should also be noted, if only in passing, that Bishop has not bothered himself,
apparently, with a careful study of nonstandard analysis or its implications, for he offhandedly
admits that he only "gathers that it has met with some degree of success" (Bishop 1975,
514; emphasis added).
37. (Bishop 1977, 208).
38. (Keisler 1976, 298), emphasis added; quoted in (Bishop 1977, 207).
39. (Bishop 1976, 207).
40. (Sullivan 1976, 370). Note that Sullivan's study used the experimental version of
Keisler's book, issued in 1971. Bishop reviewed the first edition published five years later
by Prindle, Weber and Schmidt. See (Keisler 1971, 1976).
41. (Sullivan 1976, 371).
42. (Sullivan 1976, 373).
43. (Sullivan 1976, 383-84).
44. (Bishop 1977, 208).
45. (Sullivan 1976, 373).
46. (Sullivan 1976, 375).
47. (Robinson 1966, 5).
48. See especially (Robinson 1972a, 1972b, 1974, 1975), as well as (Dresden 1976) and
(Voros 1973).
49. (Robinson 1973, 14).
50. (Robinson 1973, 14).
51. For details, see (Dauben 1979).
52. See (Dauben 1979).
53. (Robinson 1973, 16).
54. (Luxemburg 1962, 1976; Keisler 1971).
55. (Robinson 1973, 16).
56. Robinson completed his dissertation, The Metamathematics of Algebraic Systems,
at Birkbeck College, University of London, in 1949. It was published two years later; see
(Robinson 1951).
57. (Kochen 1976, 313).
References
Berggren, J. L. 1984. History of Greek Mathematics: A Survey of Recent Research. Historia
Mathematica 11: 394-410.
Birkhoff, Garrett. 1975. Introduction to "Foundations of Mathematics." Proceedings of
198 Joseph W. Dauben
Hall, A. R. 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cam-
bridge: Cambridge University Press.
Hasse, H., and Scholz, H. 1928. Die Grundlagenkrisis der griechischen Mathematik. Kant-
Studien 33: 4-34.
Keisler, H. J. 1971. Elementary Calculus: An Approach Using Infinitesimals (Experimental
Version). Tarrytown-on-Hudson, N.Y.: Bodgen and Quigley; photo-reproduced from
typescript.
. 1976. Elementary Calculus: An Approach Using Infinitesimals. Boston: Prindle, Weber
and Schmidt.
Knorr, Wilbur R. 1975. The Evolution of the Euclidean Elements. Dordrecht: Reidel.
Kochen, S. 1976. Abraham Robinson: The Pure Mathematician. On Abraham Robinson's
Work in Mathematical Logic. Bulletin of the London Mathematical Society 8: 312-15.
Kb'rner, S. 1979. Introduction to Papers on Philosophy. In Selected Papers of Abraham
Robinson, ed. H. J. Keisler et al. Vol. 2. New Haven: Yale University Press, pp. xli-xlv.
Lagrange, J. L. 1784. Prix proposes par I'Acaddmie Royale des Sciences et Belles-Lettres
pour I'ann6e 1786. Nouveaux memoires de I'Academie Royale des Sciences et Belles-
Lettres de Berlin. Vol. 15. Berlin: G. J. Decker, pp. 12-14.
Lakatos, Imre. 1978. Cauchy and the Continuum: The Significance of Non-standard Analysis
for the History and Philosophy of Mathematics. In Mathematics, Science and
Epistemology: Philosophical Papers, ed. J. Worrall and G. Currie. Vol. 2. Cambridge:
Cambridge University Press, pp. 43-60. Reprinted in The Mathematics Intelligencer 1:
151-61, with a note, Introducing Imre Lakatos, pp. 148-51.
Laugwitz, Detlef. 1975. Zur Entwicklung der Mathematik des Infinitesimalen und Infiniten.
Jahrbuch Oberblicke Mathematik. Mannheim: Bibliographisch.es Institut, pp. 45-50.
. 1985. Cauchy and Infinitesimals. Preprint 911. Darmstadt: Technische Hochschule
Darmstadt, Fachbereich Mathematik.
Leibniz, G. W. 1684. Nova methodus pro maxima et minima.... Acta Eruditorum 3: 467-73.
Reprinted in (Leibniz 1848-63, vol. 3).
. 1848-63. Mathematische Schriften, ed. G. I. Gerhardt. Berlin: Ascher (vol. 1-2); Halle:
Schmidt (vol. 3-7).
Luxemburg, W. A. J. 1962. Lectures on A. Robinson's Theory of Infinitesimals and In-
finitely Large Numbers. Pasadena: California Institute of Technology. Rev. ed. 1964.
. 1975. Nichtstandard Zahlsysteme und die Begriindung des Leibnizschen In-
finitesimalkalkiils. Jahrbuch Oberblicke Mathematik. Mannheim: Bibliographisches In-
stitut, pp. 31-44.
Luxemburg, W. A. J. and Stroyan, K. D. 1976. Introduction to the Theory of Infinitesimals.
New York: Academic Press.
Meschkowski, Herbert. 1965. Aus den Briefbiichern Georg Cantors. Archive for History
of Exact Sciences 2: 503-19.
Newton, Isaac. 1727. Mathematical Principles of Natural Philosophy. Trans. A. Motte, rev.
ed. F. Cajori. Berkeley: University of California Press, 1934.
Robinson, Abraham. 1951. On the Metamathematics of Algebra. Amsterdam: North Holland
Publishing Company.
. 1961. Non-Standard Analysis. Proceedings of the Koninklijke Nederlandse Akademie
van Wetenschappen, ser. A, 64: 432-40.
. 1965a. Formalism 64. Proceedings of the International Congress for Logic,
Methodology and Philosophy of Science. Jerusalem, 1964. Amsterdam: North Holland
Publishing Company, pp. 228-46. Reprinted in (Robinson 1979, vol. 2, 505-23).
. 1965b. On the Theory of Normal Families. Acta Philosophica Fennica 18: 159-84.
200 Joseph W. Dauben
1. Introduction
This paper could have a slightly different title with the word How
dropped, but I could argue both sides of that question. The present title
presumes the optimistic answer, and while we all hope that this is the cor-
rect answer, the present time may not be the right one for this answer.
The history of mathematics is not an easy field, and it takes a rare per-
son to be good at it. Keynes supposedly said it took a rare person to be
a great economist: one must be a second-rate historian, mathematician,
and philosopher. For him second-rate was very good, but not great. The
same is probably true about the history of mathematics, except one may
not have to be a philosopher. Some great mathematicians have made im-
portant contributions to the history of mathematics, but very few have
spent enough time and thought on the history of mathematics to be able
to write a first-rate historical account of part of mathematics. One recent
exception is A. Weil, who has added an excellent historical account of
number theory before Gauss (Weil 1984) to the historical notes he wrote
for the Bourbaki volumes (Bourbaki 1974). His paper from the Helsinki
Congress (Weil 1980) should also be read by anyone interested in the
history of mathematics.
Since my training is in mathematics, and my reading of history has been
almost random, I have found it useful to think about what history is and
how the history of mathematics differs from cultural or political history.
Collingwood starts his book The Idea of History (1956) with the follow-
ing four questions: What is history, what is it about, how does it pro-
ceed, and what is it for? Of the many answers that could be given to these
questions, he gave general ones that others could probably agree with,
although most would think the answers were incomplete. For the first ques-
tion, he wrote that "history is a kind of research or inquiry" (1956, 9).
207
202 Richard Askey
There were two reasons I wanted to call attention to Pfaff's paper. One
is historical, and should have been of interest to historians. When n -+co
204 Richard Askey
A much more general result was given by Rogers (1895, sect. 8), and a
few special cases of (2.4) were found earlier, but the earliest special case
I know appeared in (Dixon 1891). This is almost one hundred years after
Pfaff proved (2.1), but only one year after Saalschtitz rediscovered it
(1890). These two results are often given at the same time. For example,
MATHEMATICIANS AND MATHEMATICAL HISTORIANS 205
Knuth (1968, 70, #31; 73, #62) has both of them as problems. He assigned
20 points to (2.1) and 38 points to (2.4), on a scale of 0 to 50, so it is clear
he knew that (2.4) is deeper than (2.1). This depth is also illustrated by
the approximately one hundred year difference in time when they were
discovered. If the reader thought the time difference was one year (for
the special case) or thirteen years (for the general case), these results would
seem to be of the same depth. It took me a number of years before I ap-
preciated the difference in depth.
I wanted to make one other point in this paper. Many people who write
about (2.1) and (2.4) do not really understand what these identities say.
For example, Knuth did not write either of these identities in the above
form. He used binomial coefficients rather than shifted factorials. To ex-
plain the reason behind this difference, I will have to get technical.
In elementary calculus, the favorite test for convergence of an infinite
series is the ratio test. It is easy to use and works on most power series
that are given to students at this level. For these power series, the ratio
between successive terms is a rational function of the index of the term.
For example, for a power series about x = 0, if
In other words, they are all the same identity. Binomial coefficients are
important, since they count things; but when one has a series of products
of binomial coefficients, the right thing to do is to translate the sum to
the hypergeometric series for (2.6). Translation is almost always easy (there
can be some problems that require limits when division by zero arises),
and it has been known for a long time that this is the right way to handle
sums of products of binomial coefficients. Andrews spelled this out in
detail in (1974, sect. 5), but the realization that hypergeometric series are
just series with term ratio a rational function of n is very old. Horn (1889)
used this as the definition of a hypergeometric series in two variables. R.
Narasimhan told me that he found a definition of "comfortable" series
in one of the late volumes of Euler's collected works. For Euler, a com-
fortable series is a power series whose term ratio is a rational function
of n. When I asked Narasimhan to give me a specific reference, he was
unable to find it again. I will be very pleased to pay $50 U.S. for this
reference, for it would be worth that to know that Euler's insight was also
good here. An even earlier place one might look for this insight would
be in Newton's work. In any case, by the time of Kummer's early work
(1836), some mathematicians started to look at higher hypergeometric
series and write them as
I wrote a short note (1975) mentioning many of the above facts and
sent it to HistoriaMathematica. It was sent to two referees, who disliked
it because I had not written a history paper. The editor then sent it to
two other referees, who also did not like it. One thought there were the
makings of a reasonable paper if I only did some serious historical research,
whereas the other thought the paper was silly. I had written that this
material was treated badly in books of mathematical history and was not
part of the standard curriculum. His view was essentially the following:
if this work is not part of the curriculum, then it probably is not very
important.
I would like to quote one paragraph from this technical report, chang-
ing it slightly to make it readable without including the earlier text, and
then give the surprising sequel:
The real reason for the obscurity of this material seems to be that it
plays a minor role in most problems. Often this role is essential, but
there is usually some other idea involved in the solution of a problem
which seems to be more central (it usually is) and the explicit sum which
is necessary remains a lemma. These sums are easy enough to derive
so that a mathematician who has been able to come up with other ideas
on how to solve a problem can also rediscover the required sum. But
this has not always been true. For example, Good obtained the sum
Then he says "(The sum of the series on the right must be non-negative
if a > J3, an inequality that is not obvious directly.) If |3 = 0,
See Good (1958). What he did not notice is that this series can be
translated into hypergeometric form and summed by a special case of
(2.1). The result is
Now the positivity for a > J3 is obvious. Recently some very complicated
formulas for hypergeometric series have been used by Gasper (1975)
to obtain some inequalities for integrals which have not been obtained
by any other method. Askey and Gasper (1977) used some other deep
facts about hypergeometric series to extend a result of Szego (1933).
If more complicated problems of this sort are to be solved in other areas
then mathematicians are going to have to realize that even the subject
MATHEMATICIANS AND MATHEMATICAL HISTORIANS 209
For the sake of those who do not know these conjectures, the Bieber-
bach conjecture is the following. A function/(z) is univalent if it is one
to one on its domain. Let/(z) by analytic for \z \ < 1 and normalized by
If/is univalent, Bieberbach (1916) showed that \a2\ < 2 and conjectured
that an\ < n. There is equality when/(z) = z(\-z)'2. This had been proved
for n = 3, 4, 5, 6 and for all n for some subclasses of univalent func-
tions, such as those with real coefficients an, but the general case was open
and thought to be very hard. The Milin conjecture is too technical to state
here, but it was known to imply the Bieberbach conjecture, and many
experts on univalent functions thought it was false.
The final step in the proof of these conjectures has been described by
the participants (Askey 1986; de Branges 1986; Gautschi 1986). Briefly,
it is as follows. De Branges asked Walter Gautschi for aid in seeing when
(2.7) holds; Gautschi is a colleague of de Branges's and probably the best
person in the world to approach when asking for numerical results on the
integral in (2.7). De Branges had heard that many experts on univalent
functions thought the Bieberbach conjecture was false for odd values of
n starting at 17 or 19, and Gautschi was skeptical that this approach would
work; so the two of them were very excited when the numbers Gautschi
found seemed to say that (2.7) held for n up to 30. This strongly suggested
that the Milin conjecture and the Bieberbach conjecture were true for these
210 Richard Askey
and had shown that this is true for n up to about 20, and thus that the
Milin and Bieberbach conjectures were true for these values of n.
The Bieberbach conjecture was a big problem, there was a nice new
idea, and the final step dealt with hypergeometric functions, as in my
outline quoted above. However, this time the hypergeometric function
work was harder than just one identity. There is a relatively simple proof
of (2.8), but it requires three identities, only one of which was well known.
A second one is contained in the best handbooks. The third one is over
a hundred years old and is contained in a few books, but not in the stan-
dard handbooks. Gautschi eventually called me and asked if I knew how
to prove (2.7). I looked at it that evening, changed it to (2.8), and found
this in the first place I looked (Askey and Gasper 1976). George Gasper
and I had needed this inequality to prove a conjecture I had made, and
Gasper had proved it.
De Branges's paper has now appeared (1985). After a version of de
Branges's proof was available in preprint form, many mathematicians went
through the details of his argument, gave talks on it, and some wrote their
own accounts (Aharonov 1984; Anonymous [Maynooth] 1985; Fitz Gerald
and Pommerenke 1985; Korevaar 1985; Milin 1984). However, only two
of these accounts gave a complete proof of (2.7), and the general consen-
sus was that the proof of (2.7) was magic and that it would be nice to
have a more conceptual, less computational proof. One always wants
simple proofs, but if one is willing to admit that Euler, Gauss, Kummer,
Riemann et al. knew what they were doing when they studied hyper-
geometric functions, then this proof seems very natural. To prove that
something is nonnegative, one tries to write it as a square or the sum of
squares with nonnegative coefficients. The proof Gasper found is just that.
What I am afraid of is that the last part of the quotation above will also
be prophetic and that, even with this striking use of hypergeometric func-
tions, knowledge of them will not spread to the mathematical community
at large as it should.
MATHEMATICIANS AND MATHEMATICAL HISTORIANS 211
My note was not the only paper written by a mathematician about some
early mathematics of interest today that was turned down by the editors
of Historia Mathematica. Another was written by George Andrews, and
he published it elsewhere (1982). It is also not really a history paper by
the standards of Historia Mathematica, but it deals with historical doc-
uments—in his case, two unpublished letters of L. J. Rogers; it is also
of more interest to quite a few mathematicians than many of the papers
in Historia Mathematica. I suggest it would be useful to publish an occa-
sional article like this. If there were more articles of interest to mathemati-
cians, then more mathematicians would read this journal. A few more
papers by mathematicians would also tell historians what mathematicians
consider important and would suggest topics that should be looked at in
detail by historians. After my experience, the next time I had a topic of
historical interest I wrote my comments for publication elsewhere. This
topic was the orthogonal polynomials that generalize the classical
polynomials of Hermite, Laguerre, and Jacobi. Most of these are older
than is generally known. For example, a set of polynomials that is or-
thogonal on x = 0, 1 , . . . , N with respect to the function [x+a][NN-xP]
and that are known as Hahn polynomials were really discovered by
Tchebychef (1875); see my comments to (Szego 1968) in the reprinted ver-
sion. There is a need for a historical treatment of orthogonal polynomials.
Szego (1968) wrote an outline, I added further comments, and a historical
resource without equal exists in (Shohat et al. 1940). This bibliography
is not complete, but when it was written it was probably the best bibli-
ography of a part of mathematics, and I do not know of another that
equals it in the coverage of the eighteenth and nineteenth centuries. My
comments appeared in a place where some mathematicians will see them,
but mathematical historians are unlikely to hear about them.
There should be some place where mathematicians can record historical
observations that will be read by mathematical historians. For example,
consider general history books, which mostly contain material copied by
the author from other books or papers. Errors tend to be propagated from
one book to another, some of the errors being historical and others mathe-
matical. More frequently, they are errors of ignorance, where the real point
of the work is not understood. There needs to be a place where these errors
can be corrected so that they will not appear in future books. I will illus-
trate these errors by mentioning some in M. Kline's book (1972). It is used
as an illustration because I agree with the following remark of Rota (1974):
"It is easy to find something to criticize in a treatise 1,200 pages long and
212 Richard Askey
packed with information. But whatever we say for or against it, we had
better treasure this book on our shelf, for as far as mathematical history
goes, it is the best we have." If the best can have errors like the follow-
ing, then it is clear that mathematical historians need all the help they can
get.
First, a mathematical error. Kline attributed the following formula to
Euler (Kline 1972, 489):
a few years earlier, it would have been a good illustration of how a work
on mathematical history could have aided a mathematician. I never would
have come across Fisher's paper in the Journal of Agricultural Science,
and was unlikely to have seen Allan's paper in the Proceedings of the Royal
Society of Edinburgh (in 1930). However, a comment in a biography
alerted me to the existence of these very interesting papers. Mathemati-
cians read mathematical history outside the areas where they work, so this
can be a good way of helping to break down the barriers that we have
created around our work. Historians can help in ways they do not know,
but only if they treat significant mathematics.
4. Conclusion
One of the referees of my note objected that it only contained facts
and that history is much more than just facts. That is clearly true, but
history built on incorrect facts is justly suspect. There needs to be a place
where facts can be recorded, or corrected, and this needs to be a place
that will be read by mathematical historians.
Weil (1980) set very high standards for what one would like to have
in a mathematical historian, and then he demonstrated that he was not
writing about an empty set in this century by living up to his standards.
To be realistic, it is unlikely that there will be many others who match
his description of a great mathematical historian, so we are going to have
to help each other. Together we may able to do some useful history.
Separately, some will be done, but not as much as is needed, and the quality
will often be lower than it should be.
References
Abramowitz, M., and Stegun, I. 1965. Handbook of Mathematical Functions. New York:
Dover.
Aharonov, D. 1984. The de Branges Theorem on Univalent Functions. Technion-Israel In-
stitute of Technology, Haifa.
Andrews, G. 1974. Applications of Basic Hypergeometric Series. SIAMReview 16: 441-84.
. 1982. L. J. Rogers and the Rogers-Ramanujan Identities. Mathematical Chronicle
11(2): 1-15.
Anonymous. 1985. The Proof of the Bieberbach Conjecture. Maynooth Mathematics Seminar,
Maynooth University.
Askey, R. 1975. A Note on the History of Series. Mathematical Research Center Technical
Report 1532. University of Wisconsin, Madison.
. 1986. My Reaction to de Branges's Proof of the Bieberbach Conjecture. In The Bieber-
bach Conjecture: Proceedings of the Symposium on the Occasion of the Proof, ed. A.
Baernstein II, D. Drasin, P. Duren, and A. Marden. Providence, R. I.: American
Mathematical Society, pp. 213-15.
Askey, R., and Gasper, G. 1976. Positive Jacobi Polynomial Sums. II. American Journal
of Mathematics 98: 709-37.
216 Richard Askey
. 1977. Convolution Structures for Laguerre Polynomials. Journal d'Analyse Math-
ematique 31: 48-68.
Baxter, R. J. 1982. Exactly Solved Models in Statistical Mechanics. London: Academic Press.
Bieberbach, L. 1916. Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte
Abbildung des Einheitskreises vermitteln. Sitzungsberichte der Preussischen Akademie
der Wissenschaften, physikalish-mathematische Klasse 940-55.
Bourbaki, N. 1974. Elements d'histoire des mathematiques. Paris: Hermann.
Box, J. F. 1978. R. A. Fisher, the Life of a Scientist. New York: Wiley.
Clausen, T. 1828. Ueber die Falle, wenn die Reihe von der Form y = 1 + (a •/?)/(! •/)* +
(a-a+ l)/(l-2)-(/3-/3+ l)/(yy+ \)x2 + etc. ein Quadrat von der Form z = 1 + (a'fi'
y')/(\5'e')x + etc. hat. Journal fur die reine und angewandte Mathematik 3: 89-91.
Collingwood, R. G. 1956. The Idea of History. New York: Oxford University Press.
Cox, D. A. 1984. The Arithmetic-Geometric Mean of Gauss. L'Enseignement Mathemati-
que 30: 275-330.
de Branges, L. 1985. A Proof of the Bieberbach Conjecture. Acta Mathematica 154: 137-52.
. 1986. The Story of the Verification of the Bieberbach Conjecture. In The Bieber-
bach Conjecture: Proceedings of the Symposium on the Occasion of the Proof, ed. A.
Baernstein II, D. Drasin, P. Duren, and A. Marden. Providence, R.I.: American
Mathematical Society, pp. 199-203.
Dieudonne, J. 191%. Abrege d'histoire des mathematiques, 1700-1900. 2vols. Paris: Hermann.
Dixon, A. C. 1891. On the Sum of the Cubes of the Coefficients in a Certain Expansion
by the Binomial Theorem. Messenger of Mathematics 20: 79-80.
. 1903. Summation of a Certain Series. Proceedings of the London Mathematical Society
35: 284-89.
Erd61yi, A. 1953-55. Higher Transcendental Functions. 3 vols. New York: McGraw-Hill.
FitzGerald, C., and Pommerenke, C. 1985. The de Branges Theorem on Univalent Func-
tions. Transactions of the American Mathematical Society 290: 683-90.
Gasper, G. 1975. Positive Integrals of Bessel Functions. SI AM Journal on Mathematical
Analysis 6: 868-81.
. 1977. Positive Sums of the Classical Orthogonal Polynomials. SI AM Journal on
Mathematical Analysis 8: 423-47.
Gauss, C. F. 1813. Disquisitiones generales circa seriem infinitam 1 + (a/?)/(l-y)x + [a
(o+ l)0(B+ l)]/[l'2-7(7+ l)]xx + [0(0+ I)(o+2)j808+ l)(/J+2)]/[l-2-3-y(y+ 0(7+
2)]jc3 + etc. Pars prior. Commentationes societatis regiae scientiarum Gottingensis recen-
tiores, B. Vol. 2. Reprinted in Carl Friedrich Gauss Werke. Vol. 3: Analysis. Gb'ttigen,
1866, pp. 123-62.
. 1866. Determinatio seriei nostrae per aequationem differentialem secundi ordinis.
In Carl Friedrich Gauss Werke. Vol. 3: Analysis. Go'ttingen, 1866, pp. 207-29.
Gautschi, W. 1986. Reminiscences of My Involvement in de Branges' Proof of the Bieber-
bach Conjecture. In The Bieberbach Conjecture: Proceedings of the Symposium on the
Occasion of the Proof, ed. A. Baernstein II, D. Drasin, P. Duren, and A. Marden. Prov-
idence, R.I.: American Mathematical Society, pp. 205-11.
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York: Chelsea, 1969, pp. 174-82.
MATHEMATICIANS AND MATHEMATICAL HISTORIANS 217
In this paper, I shall try to do three things: first, to briefly sketch the
eighteenth-century distinction between "abstract" and "mixed" mathe-
matics, and to contrast it with our "pure" versus "applied" distinction;
second, to illustrate the significance of this philosophical distinction for
mathematical practice, using examples drawn from the history of prob-
ability theory during the eighteenth and early nineteenth centuries; and
third, to conclude with some reflections suggested by these examples about
the preconditions for applying mathematics. Throughout, I shall be
primarily concerned with the problem of how a mathematical theory
gains—and sometimes loses—a domain of applications.
1. Mixed Mathematics
In the Metaphysics, Aristotle argues that mathematical entities cannot
be truly separated from sensible things, but only abstracted from them
(M.2.1077a 9-20; b 12-30). With some Lockean and Cartesian elabora-
tions, this view continued to dominate philosophical accounts of mathe-
matics throughout the eighteenth century. The most influential of these
accounts was d'Alembert's Preliminary Discourse (1751) to the great En-
cyclopedic. Following Locke, d'Alembert asserts that all human knowledge
derives from experience; borrowing from Descartes, he assumes analysis
to be the fundamental intellectual operation turned upon the raw materials
of sensation. Property by property, the mind strips away the tangle of
particular features that compose any sensation until it arrives at the barest
skeleton or "phantom" of the object, shaped extension. It is at this rare-
fied level that mathematics studies the objects of experience. Although
these objects have been systematically denuded of all those traits that nor-
mally accompany them in perception, they are not, d'Alembert insists,
thereby denatured. Mathematics may be the "farthest outpost to which
the contemplation of properties can lead us," but it is nonetheless still
anchored in the material universe of experience.
Once the limits of analysis have been reached at magnitude and exten-
sion, the mind reverses its path and begins to reconstitute perception, prop-
erty by property, by the reciprocal operation of synthesis, until it ultimately
arrives at its departure point, the concrete experience itself. The successive
stages along this route demarcate the subject matter of the various sciences.
For example, add impenetrability and motion to the magnitude and ex-
tension of mathematics, and the science of mechanics is created. All
sciences study the same objects but embrace their perceptual complexity
to a greater or lesser extent.1
FITTING NUMBERS TO THE WORLD 223
That mathematics could describe the external world would thus have
hardly puzzled d'Alembert: mathematics came from the world, and to the
world it must return, for d'Alembert believed that mathematical abstrac-
tions, as he says, "are useful only insofar as we do not limit ourselves
to them." Even the "abstract" mathematics of pure number and exten-
sion (i.e., of arithmetic and geometry) were simply the endpoints of a con-
tinuum along which mathematics was "mixed" with sensible properties
in varying proportions. Of course, this schema brought its own quandaries:
if mathematics was joined to the natural sciences by a continuous intellec-
tual process, why did only mathematical results enjoy certainty, and ex-
actly where did the boundary between mathematics and these sciences lie?
D'Alembert was obliged to resort to Cartesian intuitions of clarity and
simplicity to answer the first question, and he tendentiously drew the
boundary to include rational mechanics within mathematics—even his
devoted admirer Montucla could not follow him in that.2 But d'Alembert's
account of "abstract" and "mixed" mathematics did explain how, for
example, geometric optics or celestial mechanics was possible, and it was
echoed by many other in the next fifty years.
So far, so Aristotelian. We recognize the kinship of Aristotle's bronze
isosceles triangle and d'Alembert's impenetrable moving body: both ex-
amples "mix" physical with abstract properties. But when we turn to
d'Alembert's table depicting the actual contents of mixed mathematics,
we see how much enlarged beyond the Aristotelian canon it had become
by 1750 in the mind of one of Europe's leading mathematicians.3 "Mixed"
or "Physico-Mathematics" dwarfs "Pure Mathematics," even though the
latter category has been swelled by the integral and differential calculus,
and each of the headings of the Aristotelian "mixed" canon—mechanics,
astronomy, optics, harmonics—has greatly expanded its range of subhead-
ings. (Harmonics, for example, now comprises only one part of "Acous-
tics," and optics only one part of a broader division including dioptics,
perspective, and catoptrics.) Moreover, the division of space in d'Alem-
bert's table corresponds roughly to the division of labor among eighteenth-
century mathematicians: Bossut's 1810 survey of mathematics from the
origins of the calculus to the year 1800 devotes 177 pages to work in
abstract mathematics and 319 pages to that in mixed mathematics.4
Not only did mixed mathematics preponderate; in the opinion of
eighteenth-century mathematicians, it bid fair to expand still further. Mon-
tucla, in the 1758 edition of his celebrated Histoire des mathematiques,
reflected on the open-ended nature of mixed mathematics, which grew
224 Lorraine J. Daston
tuating factors: if a judge erred, it was for a specific reason, not because
he had "put his hand in an urn" and made an unlucky draw.13 Yet for
the classical probabilists, these assumptions did not seem so outrageous.
Condorcet and Laplace admitted that the conditions of independence,
equality, and constancy for individual probabilities were simplifications,
but they argued that all mixed mathematics involved idealizations (witness
Newton's laws of dynamics), and further maintained that such approx-
imations "founded on the data, indicated by good sense" were preferable
to nonmathematical specious reasoning.
The eighteenth- and early nineteenth-century probabilists could be con-
fident in assumptions that their successors found absurd because they
subscribed to psychological theories that described mental operations in
terms congenial to their mathematics. According to Locke, Hartley, Hume,
and others, the sound mind reasoned by the implicit computation and
comparison of probabilities.14 Or as Montucla put it apropos of crack
gamblers:
The gambler's mind ["Pesprit du jeu"], that mind that seems to cap-
ture fortune, is nothing more than an innate or acquired talent for see-
ing at a glance all the chance combinations that could lead to gain or
loss; human prudence is ultimately nothing other than the art of ap-
preciating the probabilities of events, in order to act accordingly.15
The association of ideas in principle mirrored the regularity and frequen-
cy of events culled from experience: in an unbiased mind, associations
of ideas corresponded to real connections between the events and objects
represented by the ideas. The very workings of the human understanding,
when undistorted by strong emotion or uncritical custom, imitated Ber-
noulli's theorem, which Hartley had claimed was "evident to attentive
Persons, in a gross general way, from the Common Methods of Reason-
ing."16 Associationist psychology also emphasized the combinatorial
operations of the mind; indeed, all intellectual novelty owed to the men-
tal combination and recombination of simple ideas by, as Condillac wrote,
"a kind of calculus." Condorcet affirmed Condillac's claim that the best
intellects were those that excelled in "uniting more ideas in memory and
in multiplying these combinations."17 If good minds worked by "a kind
of calculus," then the combinatorial calculus of probabilities could be
viewed as the mathematical expression and extension of the psychological
processes that constituted right reasoning—in particular, the right reason-
ing of judges.
230 Lorraine J. Daston
Insurance
My final example, actuarial mathematics, is by way of contrast an un-
disputed success story about applying probability theory to the world—
or at least it became one after almost a century's worth of neglected
mathematical efforts to make it so. The history of how mathematical prob-
ability eventually came to be applied to the insurance trade is an intricate
one that I can barely sketch here.21 My main point in doing so will be
to make the converse point to my claim concerning mixed mathematics
and the probability of judgments. If the probability of judgments enjoyed
a bright if brief career, it was because peculiar circumstances conjoined
first to suggest a striking analogy between legal judgments and mathe-
matical probability, and later to destroy this analogy. Conversely, if eight-
eenth-century insurers were slow to make use of a mathematical techno-
logy that was in many ways tailor-made for them, it was because they
perceived the conditions of their trade as downright disanalogous to those
imposed by the mathematicians.
Why did the practitioners of risk—particularly insurers and sellers of
annuities—fail to take advantage of mathematical techniques and collec-
tions of statistics that were devised for their purposes? There seems to
have been, as the mystery writers put it, both opportunity and motive.
Opportunity, for there existed a well-worked-out theory for pricing an-
nuities and (potentially) life insurance using the new mathematics of prob-
ability and the new data of mortality statistics from circa 1700 on.
Moreover, the mathematicians (De Moivre, Simpson, and others) had ex-
pressly translated this literature into elementary handbooks aimed at the
innumerate clerk, with all algebra converted to verbal form and most
technical material relegated to appendices, plus copious tables to obviate
the need for onerous calculation. Motive, for both the Dutch and especially
the English annuity and insurance markets were large and bustling enough
by the turn of the eighteenth century to offer the mathematically based
company a competitive edge. (This is not to say that such enterprises could
not be profitable without mathematics—indeed, they were too profit-
able—but as the experience of the first mathematically based company
was to show, even the loosest connection between calculation and premium
could cut prices while still preserving a lavish margin of profit.)
Why, then, did the practice of risk taking lag so far behind the theory
of risk taking? I shall argue, in summary form, that mathematically based
insurance could not be sold until insurers and their customers came to
FITTING NUMBERS TO THE WORLD 233
believe that phenomena like human mortality, shipwrecks, fires, and other
catastrophes were regular enough to make statistical and probabilistic ap-
proaches plausible. To make this point clear, I must briefly describe the
practice of insurance without actuarial mathematics. Certain forms of risk
taking—insurance (chiefly maritime), annuities, and gambling—were wide-
ly and successfully practiced in Europe long before the formulation of
mathematical probability. Although experience no doubt honed the ability
of the underwriter, dealer in annuities, or gambler to estimate odds, their
approach to risk could hardly be described as statistical or probabilistic,
even at an intuitive level. A sixteenth-century insurer might have found
such a statistical approach impractical, for it assumes conditions that are
stable over a long period as well as the homogeneity of categories. In-
surance manuals and legal treatises of the period emphasized that the
premium in any given case depends on a judicious weighting of the par-
ticular circumstances: the cargo, the season of the year, the route taken,
the condition of the ship, the skill of the captain, the latest "good or bad
news" concerning storms, warships, and privateers.22 Moreover, in com-
mercial centers populous enough to support whole markets of insurers,
premium prices also reacted to levels of supply and demand as well as
to the latest news about the Barbary pirates.
Thus, annuity rates and insurance premiums certainly reflected past
experience, but it was a far more nuanced experience than a simple toting
up of mortality and shipwreck statistics. It was an experience sensitive
to myriad individual circumstances and their weighted interrelationships,
not to mention market pressures: it was not simply astatistical, it was anti-
statistical. Given the highly volatile conditions of both sea traffic and health
in centuries notorious for warfare, plagues, and other unpredictable mis-
fortunes, I am not persuaded that this was an unreasonable approach.
In any case, it was the prevailing one—and it evidently turned a profit.
Although mathematicians and statisticians addressed insurance and an-
nuities problems almost from the inception of mathematical probability
in the late seventeenth century—Huygens, DeWitt, Leibniz, the Bernoullis,
Halley, and De Moivre were all interested—and although by 1750 there
existed an extensive literature in Dutch, English, Latin, and French on
the subject, the impact of mathematical probability on practice was ef-
fectively nil prior to the establishment of the Equitable Society for the
Assurance of Lives in 1762 (at the instance of a mathematician, James
Dodson).23 And even then, the dictates of mathematical theory were greatly
234 Lorraine J. Daston
prosperity and the ever more regular contours of the Equitable's own mor-
tality figures as membership increased. Year after year, William Morgan,
the first mathematically trained actuary, held importunate stockholders
at bay, warning that "extraordinary events or a season of uncommon mor-
tality" might catch the Equitable unawares,27 law of large numbers or
no. Small wonder that the nineteenth-century mathematician Augustus
De Morgan quipped: "We should write upon the door of every mutual
office but one be wary; but upon that one should be written be not too
wary and over it the Equitable Society."2*
Thus, the confidence that the phenomena of human mortality revealed,
in the words of German theologian and demographer Johann Siissmilch,
"a constant, general, great, complete, and beautiful order"29 stable enough
to rest a trade upon was slow in coming. For some phenomena, that faith
came later than for others, and the disparities are hard to explain. As ear-
ly as 1662 some writers were confident enough of the regularities to col-
lect data on mortality, but it is difficult to understand why mortality should
have been assumed to be regular and other phenomena of equal practical
interest like the incidence of fires not, in an age where both were subject
to wild fluctuations: witness the plague and Great Fire of London in
1665-66. Whatever the reasons for such confidence that whole new realms
of phenomena were indeed regular, it was a prerequisite for the transfor-
mation of insurance from an enterprise based on judgments of individual
cases to one based on probability and statistics. In contrast to the case
of the probability of judgments, here a new analogy was forged rather
than an old one sundered.
3. Conclusion
These three examples drawn from the history of the classical theory
of probability illustrate three cardinal points about applying mathematics
to phenomena: (1) whether applications succeed or fail depends on the
standards set, and in this respect eighteenth-century mixed mathematics
differed significantly from nineteenth-century applied mathematics; (2)
mathematical theories can lose as well as gain applications; and (3) the
conditions that make a set of phenomena ripe for quantification—or the
reverse—depend crucially on their perceived stability and degree of analogy
with the mathematical techniques at hand. The mixed mathematics of the
eighteenth century demanded a far closer fit between mathematics and
the phenomena than did the applied mathematics of the nineteenth
236 Lorraine J. Das ton
Notes
1. Jean d'Alembert, "Discours prdliminaire," Encyclopedic, ou Dictionnaire raisonne
des sciences, des arts et metiers (Paris, 1751-80), vol. 1, pp. v ff.
2. J. F. Montucla, Histoire des mathematiques (Paris, 1758), vol. 1, p. 5.
3. D'Alembert, "Discours," Systeme figure^ des connaissances humaines.
4. Charles Bossut, Histoire generate des mathematiques (Paris, 1802-10), vol. 2.
5. Montucla, Histoire, vol. 1, p. 6; 2d ed. (Paris, An VII-X [1798-1802]), vol. 3, p. 381.
FITTING NUMBERS TO THE WORLD 237
i
Mathematics underwent, in the nineteenth century, a transformation
so profound that it is not too much to call it a second birth of the subject—
its first birth having occurred among the ancient Greeks, say from the
sixth through the fourth century B.C. In speaking so of the first birth,
I am taking the word mathematics to refer, not merely to a body of
knowledge, or lore, such as existed for example among the Babylonians
many centuries earlier than the time I have mentioned, but rather to a
systematic discipline with clearly defined concepts and with theorems
rigorously demonstrated. It follows that the birth of mathematics can also
be regarded as the discovery of a capacity of the human mind, or of human
thought—hence its tremendous importance for philosophy: it is surely
significant that, in the semilegendary intellectual tradition of the Greeks,
Thales is named both as the earliest of the philosophers and the first prov-
er of geometric theorems.
As to the "second birth," I have to emphasize that it is of the very
same subject. One might maintain with some plausibility that in the time
of Aristotle there was no such science as that we call physics; that Plato
and Aristotle were acquainted with mathematics in our own sense of the
term is beyond serious controversy: a mathematician today, reading the
works of Archimedes, or Eudoxos's theory of ratios in Book V of Euclid,
will feel that he is reading a contemporary. Then in what consists the "sec-
ond birth"? There was, of course, an enormous expansion of the sub-
ject, and that is relevant; but that is not quite it: The expansion was itself
effected by the very same capacity of thought that the Greeks discovered;
but in the process, something new was learned about the nature of that
capacity—what it is, and what it is not. I believe that what has been
238
LOGOS, LOGIC, AND LOGISTIKg 239
III
It is far beyond both the scope of this paper and the competence of
its author to do justice, even in outline, to the complex of interrelated
investigations and mathematical discoveries (or "inventions") by which
mathematics itself was deeply transformed in the nineteenth century; I
am going to consider only a few strands in the vast fabric. A good part
of the interest will center in the theory of numbers—and in more or less
related matters of algebra and analysis—but with a little attention to
geometry too.
Let us begin with a very brief glance at the situation early in the cen-
tury, when the transformation had just commenced. A first faint prefigura-
tion of it can be seen in work of Lagrange in the 1770s on the problems
of the solution of algebraic equations by radicals and of the arithmetical
theory of binary quadratic forms. On both of these problems, Lagrange
brought to bear new methods, involving attention to transformations or
mappings and their invariants, and to classifications induced by equiva-
lence-relations. (That Lagrange was aware of the deep importance of his
methods is apparent from his remark that the behavior of functions of
the roots of an equation under permutations of those roots constitutes
"the true principles, and, so to speak, the metaphysics of the resolution
of equations of the third and fourth degree." Bourbaki1 suggests that one
can see here a first vague intuition of the modern concept of structure.}
This work of Lagrange was continued, on the number-theoretic side, by
Gauss; on the algebraic side, by Gauss, Abel, and Galois; and, despite
the failure of Galois's investigations to attract attention and gain recogni-
tion until nearly fifteen years after his death, the results attained by the
early 1830s were enough to ensure that—borrowing Lagrange's word—
the new "metaphysics" would go on to play a dominant role in algebra
and number-theory. (I should not omit to remark here that a procedure
closely related to that of classification was the introduction of new "ob-
jects" for consideration in general, and for calculation in particular. In
the Disquisitiones Arithmeticae of Gauss, explicit introduction of new ideal
"objects" is avoided in favor of the device of introducing new "quasi-
equalities" [congruences]; Galois, on the other hand, went so far as to
introduce "ideal roots" of polynomial congruences modulo a prime
number, thus initiating the algebraic theory of finite fields.2)
By the same period, the early 1830s, decisive developments had also
taken place in analysis and in geometry: in the former, the creation—
LOGOS, LOGIC, AND LOGISTIKE 241
IV
As pivotal figures for the history now to be discussed, I would name
Dirichlet, Riemann, and Dedekind—all closely linked personally to one
another, and to Gauss. Among these, Dirichlet is perhaps the poet's
poet—better appreciated by mathematicians, more especially number-
theorists, with a taste for original sources, than by any wide public. It
is not too much to characterize Dirichlet's influence, not only upon those
who had direct contact with him—among those in our story: Riemann
and Dedekind, Kummer and Kronecker—but upon a later generation of
mathematicians, as a spiritual one (the German geistig would do better).
Let me cite, in this connection, Hilbert and Minkowski. In his Gottingen
address of 1905, on the occasion of Dirichlet's centenary, Minkowski, nam-
ing a list of mathematicians who had received from Dirichlet "the strongest
impulse of their scientific aspiration," refers to Riemann: "What math-
ematician could fail to understand that the luminous path of Riemann,
this gigantic meteor in the mathematical heaven, had its starting-point in
the constellation of Dirichlet"; and he then remarks that although the
assumption to which Riemann gave the name "Dirichlet's Principles"—
we may recall that this had just a few years previously been put on a sound
basis by Hilbert—was in fact introduced not by Dirichlet but by the young
William Thomson, still "the modern period in the history of mathematics"
dates from what he calls "the other Dirichlet Principle: to conquer the
problems with a minimum of blind calculation, a maximum of clear-seeing
thoughts."3 Just four years later, in his deeply moving eulogy of Min-
kowski, Hilbert says of his friend: "He strove first of all for simplicity
and clarity of thought—in this Dirichlet and Hermite were his models."4
And for perhaps the strongest formulation of Minkowski's "other Dirich-
let Principle," consider this passage quoted by Otto Blumenthal, in his
biographical sketch of Hilbert, from a letter of Hilbert to Minkowski:
242 Howard Stein
"In our science it is always and only the reflecting mind der uberlegende
Geist], not the applied force of the formula, that is the condition of a
successful result."5
V
I am going to make a sudden jump here: Why did Dedekind write his
little monograph on continuity and irrational numbers? To be sure, that
work is in no need of an excuse; but I have long been struck by these cir-
cumstances: (1) Dedekind himself tells us in his foreword that when, some
dozen years earlier, he first found himself obliged to teach the elements
of the differential calculus, he "felt more keenly than ever before the lack
of a really scientific foundation of arithmetic." He goes on to express
his dissatisfaction with "recourse to the geometrically evident" for the
principles of the theory of limits.6 (Note that Dedekind speaks of a "foun-
dation of arithmetic" rather than of analysis.) (2) In the foreword to the
first edition of his monograph on the natural number, Dedekind invokes,
as something "self-evident," the principle that every theorem of algebra
and of the higher analysis can be expressed as a theorem about the natural
numbers: "an assertion," he says, "that I have also heard repeatedly from
the mouth of Dirichlet."7 (3) From antiquity (e.g., Aristotle, Euclid)
through the late eighteenth and early nineteenth centuries (e.g., Kant,
Gauss), a prevalent view was that there are two distinct sorts of "quanti-
ty": the discrete and the continuous, represented mathematically by the
theories of number and of continuous magnitude. Gauss, for instance,
excludes from consideration in the Disquisitiones Arithmeticae "fractions
for the most part, surds always."8 But Dirichlet, in 1837, succeeded in
proving that there are infinitely many prime numbers in any arithmetic
progression containing two relatively prime terms, by an argument that
makes essential use of continuous variables and the theory of limits. This
famous investigation was the beginning of analytic number theory; and
Dirichlet himself signalizes the importance of the new methods he has in-
troduced into arithmetic: "The method I employ seems to me above all
to merit attention by the connection it establishes between the infinitesimal
Analysis and the higher Arithmetic [I'Arithmetique transcendante];9
I have been led to investigate a large number of questions concerning
numbers [among them, those related to the number of classes of binary
quadratic forms and to the distribution of primes] from an entirely
new point of view, which attaches itself to the principles of infinitesimal
LOGOS, LOGIC, AND LOGISTIKE 243
VI
It would be tempting to spend much time on Dedekind's theory of
continuity—on its relation to Eudoxos and to geometry, and on the dif-
ficulty his contemporaries had in understanding its point. Let me just refer
to the extracts from Dedekind's correspondence with Rudolf Lipschitz,
given by Emmy Noether in vol. Ill of Dedekind's Gesammelte Mathe-
matische Werke, and particularly to the fact that Lipschitz objected to
Dedekind that the property the latter calls "completeness" or "continu-
ity"—in the terminology now standard, connectedness—is self-evident and
doesn't need to be stated—that no man can conceive of a line without
that property. Dedekind replies that this is incorrect, since he himself can
conceive of all of space and each line in it as entirely discontinuous (add-
ing that Herr Professor Cantor in Halle is evidently another man of the
same sort). He refers to §3 of his monograph, where he had said, "If space
has a real existence at all, it does not have necessarily to be continuous."13
But it is in the foreword to the first edition of the monograph on the natural
numbers that Dedekind returns to this point and indicates the particular
grounds for his claim: namely, the existence (as we would say) of models
of Euclid's geometry in which all ratios of lengths of straight segments
are algebraic numbers. For the latter concept, he refers the reader to
Dirichlet's Vorlesungen tiber Zahlentheorie, §159 of the second, §160 of
the third edition. The second edition of Dirichlet's lectures—of which,
of course, Dedekind was the editor—was published the year before Stetig-
keit und irrationale Zahlen; and the section indicated14 is part of the fa-
mous Supplement written by Dedekind in which the theory of algebraic
number fields and algebraic integers was developed for the first time.
I have the impression that the central importance of that very great
work of Dedekind for the entire subsequent development of mathematics
has not been generally appreciated. It is certainly well known—to those
who know such things—that this Supplement was of the first importance
for algebraic number theory and for what is now called "commutative
algebra." It is perhaps less well known that this is also the place in which
Galois's theory was developed for the first time in its modern form—as
a theory of field extensions and their automorphisms, rather than of
substitutions in formulas and of functions invariant under substitutions.
But that new perspective upon Galois's achievement is itself only one
manifestation of a general principle that permeates the work—one that
could be summed up in Minkowski's phrase expressing the "other Dirichlet
LOGOS, LOGIC, AND LOGISTIKE 245
ferences, so that no step occurs that does not conform to one of a small
number of modes of inference recognized as logical"; but he quite fails
to tell us how such recognition occurs—or what its content is. The trou-
ble lies in the reigning presumption that to recognize a proposition as a
"logical truth" is to identify its epistemological basis', and this is a trou-
ble because—if I may be forgiven for pontificating on the point—no cogent
theory of the "epistemological basis" of any kind of knowledge has ever
been formulated.
On the other hand, some things can be said about Dedekind's claim—
and even, I think, some epistemological things. In a general way, we should
remember, "logic" in the period in question was taken to be concerned
with the "laws of thought." I have said in my opening remarks that the
discovery of mathematics was also the discovery of a capacity of the human
mind. Dedekind tells us what, in his opinion, this capacity is. His earliest
published statement dates from 1879 (by coincidence, the year of Frege's
Begriffsschrift) and occurs in §161 of the Eleventh Supplement to
Dirichlet's Zahlentheorie, third edition:18 in the text he introduces the no-
tion of a mapping; in a footnote he remarks, and repeats the statement
in the foreword to the first edition of Was sind und was sollen die Zahlen?19
that the whole science of numbers rests upon this capacity of this mind—
the capacity to envisage mappings—without which no thinking at all is
possible. So the claim that arithmetic belongs to logic is the claim that
the principles of arithmetic are essentially involved in all thought—with-
out anything said about an epistemological basis. Moreover, the principles
involved are indeed those employed explicitly in Dedekind's algebraic-num-
ber-theoretic investigations: the formation of what he calls "systems"—
fields, rings (or "orders"), modules, ideals—and mappings.
But there is another aspect of Dedekind's view that should not be
overlooked. What is his answer to the question posed by the title of his
monograph? But first, what is the question? The version given by the
English translator of the work, The Nature and Meaning of Numbers,
is (setting aside its abandonment of the interrogative form) quite mislead-
ing: the German idiom "Was soil [etwas]?" is much broader than the
English "What does [something] mean?" What it connotes is always, in
some sense, "intention"; but with the full ambiguity of the latter term.
We can, however, see the precise sense of Dedekind's question from the
quite explicit answer he gives:
My general answer [or "principal answer": Hauptantwort] to the ques-
LOGOS, LOGIC, AND LOGISTIKE 247
tion posed in the title of this tract is: numbers are free creations of the
human mind; they serve as a means for more easily and more sharply
conceiving the diversity of things.20
The title, therefore, asks: What are numbers, and what are they for? (What
is their use, their function?)
The answer is not very satisfying. I think Dedekind makes a mistake
by assuming that numbers are "for" some one thing. But it is a venial
mistake that lies really only on the surface of his formulation. It is, in-
deed, part of the great discovery of the nineteenth century that mathe-
matical constructs may have manifold "uses," and uses that lie far from
those envisaged at their "creation." On the other hand, this formulation
of Dedekind's is so general and vague that perhaps it could be stretched
to cover absolutely any application of the concept of number.
What I think is more interesting in all this is that it is not what numbers
"are" intrinsically that concerns Dedekind. He is not concerned, like Frege,
to identify numbers as particular "objects" or "entities"; he is quite free
of the preoccupation with "ontology" that so dominated Frege, and has
so fascinated later philosophers. Dedekind's general answer to his first
question, "Numbers are free creations of the human mind," later takes
the following specific form: He defines—or, equivalently, axiomatizes—the
notion of a "simply infinite system"; and then says (in effect: this is my
own free rendering) it does not matter what numbers are; what matters
is that they constitute a simply infinite system. He adds—and here I
translate literally—"In respect of this freeing of the elements from any
further content (abstraction), one can justly call the numbers a free crea-
tion of the human mind."21 Of course it follows from this characteriza-
tion that numbers are "for" any use to which a simply infinite system
can be put; it is because this answer does follow, and because it is the
right one, that I described Dedekind's mistake about this as venial.
It should be noted that a very similar point applies to Dedekind's
analysis of real numbers. Here again the contrast with Frege is instruc-
tive. In the second volume of his Grundgesetze der Arithmetik, Frege
moves slowly toward a definition of the real numbers.22 He does not quite
reach it—that was reserved for the third volume, which never appeared;
but what it would have been is pretty clear. Frege's idea was that real
numbers are "for" representing ratios of measurable magnitudes (as, for
him, whole numbers—Anzahlen—are "for" representing sizes of sets—
in his terminology, sizes of "extensions of concepts"); and he wants the
248 Howard Stein
VIII
Dedekind's term "free creation" also deserves some attention. (The
theme has, again, some Dirichletian resonance, since Dirichlet in his work
on trigonometric series played a significant role in legitimating the notion
of an "absolutely arbitrary function," unrestricted by any necessary
reference to a formula or "rule.") It is very characteristic of Dedekind
to wish to open up the possibilities for developing concepts, and to wish
also that alternative, and new, paths be explored. We have just seen him
urging Weber to develop his own views on the natural numbers; he
repeatedly urged Kronecker to make known his way of developing alge-
braic number theory; in the foreword to his fourth (and last) edition of
Dirichlet's Vorlesgunen uber Zahlentheorie, containing his final revision
of the Eleventh Supplement, he expresses the hope that one of Kronecker's
students may prepare a complete and systematic presentation of Kroneck-
er's theory—and also recommends the attempt to simplify the founda-
tions of his own theory to younger mathematicians, who enter the field
without preconceived notions, and to whom therefore such simplification
may be easier than to himself.26 (This was in September 1893. Kronecker
had recently died; Hilbert had just begun to work on algebraic number
theory.)
If this has come to sound too much like a panegyric on Dedekind, I
can only say that that is because he does seem to be a great and true pro-
phet of the subject—a genuine philosopher, of and in mathematics.
IX
In his brief account of Dirichlet, Felix Klein mentions27 as "a particular
characteristic" of Dirichlet's number-theoretic investigations the type of
proof, which he was the first to employ, that establishes the existence of
something without furnishing any method for finding or constructing it.
(One recalls that when, some forty-five years later, Hilbert published
proofs of a similar type in the algebraic theory of invariants, they were
250 Howard Stein
because it is not purely discursive but has a definite content, given in the
a priori intuition of space and time.38 But the geometry of Riemann is
quite freed from any such specific content: Kant's view that the creative
or productive power of geometry rested upon its concrete "intuitive"
spatial content was simply mistaken, and mathematics is seen to be an
abstract and a priori "organon" of knowledge (whether one chooses to
call it a "logic," or a "dialectic," or whatever). And again, in view of
this its general capacity, there is no more reason to hamper it by restric-
tions to the effectively computable than to hamper it by restrictions to
the "spatially intuitive."
But then—so this dialectic goes—why restrict the license at all (e.g.,
to mathematical disciplines that one thinks might serve the ends of some
empirical science)? Why not complete freedom of conceptual elaboration
in mathematics? Then, within any logos so freely developed, one can pay
appropriate attention to the distinction between what is and what is not
constructive.
XI
This point of view—which is really not very far from the moderate posi-
tion attributed to Kronecker by Hensel—is the one that Hilbert so vigorous-
ly championed. Let me call attention, in briefest outline, to a few salient
points.
Notice of Hilbert's interest in the foundations of geometry dates back
to the earliest days of his work in algebraic number-theory, or even
somewhat before: it was in 1891, according to Otto Blumenthal,39 that
Hilbert, in a mathematical discussion in a Berlin railway waiting room,
made his famous statement that in a proper axiomatization of geometry
"one must always be able to say, instead of 'points, straight lines, planes',
'tables, chairs, beer mugs'." This view—that the basic terms of an ax-
iomatized system must be "meaningless"—is often misconstrued as "for-
malism." But the very same requirement was stated, fifteen years earlier,
by the "logicist" Dedekind, in his letter to Lipschitz already cited: "All
technical expressions [are to be] replaced by arbitrary newly invented
(heretofore nonsensical) words; the edifice must, if it is rightly constructed,
not collapse.'40
Hilbert uses as epigraph to his Grundlagen der Geometrie a well-known
Kantian aphorism:41 "Thus all human knowledge begins with intuitions,
proceeds to concepts, and ends with ideas." For Hilbert, axiomatization,
254 Howard Stein
"Naturerkennen und Logik" (1930);43 and he surely did not think that
physics is meaningless, or its discourse a play with "blind" symbols. His
point is, I think, this rather: that the mathematical logos has no respon-
sibility to any imposed standard of meaning: not to Kantian or Brouwerian
"intuition," not to finite or effective decidability, not to anyone's meta-
physical standards for "ontology"; its sole "formal" or "legal" respon-
sibility is to be consistent (of course, it has also what one might call a
"moral" or "aesthetic" responsibility: to be useful, or interesting, or
beautiful; but to this it cannot be constrained—poetry is not produced
through censorship).
In proceeding to his "program," however, Hilbert set as his goal the
mathematical investigation of the mathematical logos itself, with the prin-
cipal aim of establishing its consistency (or "their" consistency—for he
did not envisage a single canonical axiom-system for all of mathematics).
And here he made essential use both of Frege and of Kronecker. For
Frege's extremely careful and minute regimentation of logical language
or "concept-writing" did not, as Frege himself thought it would, serve
as a guarantee of consistency; but the techniques he used for that regimen-
tation did render the formal languages of mathematical theories, and their
formal rules of derivation, subject to mathematical study in their own right,
regarded as purely "blind" symbolic systems. Moreover, the regimenta-
tion was itself of such a kind that the play with symbols was a species
of calculation—of logistike. Without Frege, proof-theory in Hilbert's sense
would have been impossible.
Of course, the final irony of this story, and the collapse of Hilbert's
dream of establishing the consistency of the logic of the logos by means
restricted to logistike, lies in the discovery by Godel, Post, Church, and
Turing that there is a general theory of logistike, and that this theory
is nonconstructive; in particular, that neither the notion of consistency
nor that of provability is (in general) effective; and further that all suf-
ficiently rich consistent systems fall short of the Kantian "ideal"—are
incomplete.
This leaves us with a mystery—a subject of "wonder," in which, ac-
cording to Aristotle, philosophy begins. He says it ends in the contrary
state; I am inclined to believe, as I think Aristotle's master Plato did, that
philosophy does not "end," but that mysteries become better under-
stood—and deeper. The mysteries we now have about mathematics are
certainly better understood—and deeper—than those that confronted Kant
255 Howard Stein
or even Gauss. To attempt, however, to survey what they are, even in the
sketchiest way, although enticing, is a task that would require another
paper, and is certainly beyond the scope of this one.44
Notes
1. N. Bourbaki, Elements d'histoire des math£matiques, 2ded. (Paris: Hermann, 1974),
p. 100.
2. Ibid., pp 108-9.
3. Hermann Minkowski, Gesammelte Abhandlungen, vol. 2, (Leipzig, 1911; reprinted
New York: Chelsea 1967), pp. 460-61.
4. Ibid., vol. 1, p. xxix; David Hilbert, Gesammelte Abhandlungen, vol. 3, (Berlin:
Springer, 1935), p. 362.
5. Ibid., p. 394.
6. Richard Dedekind, Stetigkeit und irrationale Zahlen, Vorwort; in his Gesammelte
Mathematische Werke, vol. 3, ed. R. Fricke, E. Noether, and O. Ore (Braunschweig:
F. Vieweg, 1932), pp. 315-16; translation in Dedekind, Essays on the Theory of Numbers,
trans. W. W. Beman (Open Court, 1901; reprinted New York: Dover, 1963), p. 1.
7. Dedekind, Wassindund wassollen die Zahlen?, Vorwort zur ersten Auflage; Werke,
vol. 3, p. 338; Essays, p. 35.
8. Carl Friedrich Gauss, Disquisitiones arithmeticae, preface; quoted by Leopold
Kronecker, Vorlesungen iiber Zahlentheorie, vol. 1, ed. Kurt Hensel (Leipzig: Teubner, 1901),
p. 2.
9. Peter Gustav Lejeune Dirichlet, "Sur 1'usage des sdries infinies dans la th6orie des
nombres," in G. Lejeune Dirichlet's Werke, vol. 1, ed. L. Kronecker (Berlin, 1889; reprinted
New York: Chelsea, 1969), p. 360.
10. Dirichlet, "Recherches sur diverses applications de 1'analyse infinitesimale a la thferie
des nombres," ibid., p. 411.
11. Kronecker, Vorlesungen iiber Zahlentheorie, vol. 1, pp. 2-5 (italics added).
12. Kronecker, Werke, vol. 3, 1st half-volume, ed. K. Hensel (Leipzig: Teubner, 1899),
p.253 (emphasis in original; Kronecker quotes, in a footnote, a letter from Gauss to Bessel,
9 April 1830).
13. Dedekind, Werke, vol. 3, p. 478.
14. See Dedekind, Werke, vol. 3, pp. 223ff. (§159, in Supplement X to the second edi-
tion of Dirichlet's lectures), and ibid., pp. 2ff. (§160, in Supplement XI to the third and
fourth editions of that work).
15. Dedekind, "Sur la th6orie des Nombres entiers alg6brique," in his Werke, vol. 3,
p. 296.
16. Dedekind, Werke, vol. 3, p. 335; Essays, p. 31.
17. Gottlob Frege, Die Grundlagen der Arithmetik (Breslau: Verlag von W. Koebner,
1884; edition with parallel German and English texts, trans. J. L. Austin, reprinted Evanston,
111.: Northwestern University Press, 1968), p. 102; cf. also his Grundgesestze der Arithmetik,
vol. 1 (Jena, 1893; reprinted (two volumes in one) Hildesheim: Georg Olms Verlagsbuch-
handlung, 1962), p. vii.
18. Dedekind, Werke, vol. 3, p. 24.
19. Ibid., p. 336; Essays, p. 32.
20. Dedekind, Werke, vol. 3, p. 335; Essays, p. 31.
21. Dedekind, Was sind und was sollen die Zahlen?, §73; Werke, vol. 3, p. 360, Essays,
p. 68.
22. Frege, Grundgesetze der Arithmetik, vol. 2 (Jena, 1903; reprinted, together with vol.
1, as cited in n. 17 above), pp. 155-243; for general discussion, see §§156-65, 173, 175, 197,
and 245—pp. 154-63, 168-69, 170-72, 189-90, 243.
23. Dedekind, Werke, vol. 3, p. 325; Essays, p. 15.
24. Dedekind, Werke, vol. 3, pp. 489-90 (letter dated 24 January 1888).
LOGOS, LOGIC, AND LOGISTIKE E 257
25. Felix Klein, Vorlesungen iiber die Entwicklung der Mathematik im 19ten Jahrhundert,
vol. 1, ed. R. Courant and O. Neugebauer (reprinted New York: Chelsea 1956), p. 322.
26. Dedekind, Werke, vol. 3, p. 427. Another passage is worth quoting as an illustration
both of Dedekind's generous attitude toward alternative constructions and of the Dirichle-
tian ideal that governs his own preferences; it comes, again, from the foreword to Was sind
und was sollen die Zahlen?, but refers back to his procedure in constructing the real numbers.
Dedekind recognizes that the theories of Weierstrass and of Cantor are both entirely ade-
quate—possess complete rigor; but of his own theory, he says that "it seems to me somewhat
simpler, I might say quieter," than the other two. (See Werke, vol. 3, p. 339. Italics added—
Dedekind's word is ruhiger; the published English translation, "easier," loses the point of
the expression.)
27. Klein, Entwicklung der Mathematik, vol. 1, p. 98.
28. See Hilbert, Gesammelte Abhandlungen, vol. 3, pp. 394-95.
29. Kronecker, Vorlesungen uber Zahlentheorie, vol. 1, p. 11.
30. Kronecker had, of course, a generally workable technique for dealing with algebraic
irrationals; but when he speaks in Lecture 1 of the definitions that occur in analysis, he
remarks that "from the entire domain of this branch of mathematics, only the concept of
limit or bound has thus far remained alien to number theory" (ibid., vol. 1, pp. 4-5).
31. Ibid., p. viii.
32. Ibid., p. vi.
33. Dedekind, Werke, vol. 3, p. 156 n. What Kronecker criticizes in the second instance
here is "the various concept-formations with the help of which, in recent times, it has been
attempted from several sides (first of all by Heine) to conceive and to ground the 'irrational'
in general." That Kronecker mentions Heine—who in fact adopted, with explicit acknowledg-
ment, the method introduced by Cantor—as "first" is striking; was his antipathy to Cantor
so great that he refused to mention the latter's name at all—or, indeed, suppressed his recollec-
tion of it? The suspicion that some degree of pathological aversion is involved here is in-
creased by an example cited by Fraenkel in his biographical sketch of Cantor appended to
Zermelo's edition of Cantor's works (Georg Cantor, Abhandlungen mathematischen und
philosophischen Inhalts, ed. Ernst Zermelo [Berlin, 1932; reprinted Hildesheim: Georg Olms
Verlagsbuchhandlung, 1966], p. 455, n. 3). Cantor published in 1870 his famous theorem
on the uniqueness of trigonometric series (see his Abhandlungen, pp. 80ff.). In the follow-
ing year, he published the first stage in his extension of that result (allowing a finite number
of possibly exceptional points in the interval of periodicity); this paper also contains a
simplification of his earlier proof, which he acknowledges as due to "a gracious oral com-
munication of Herr Professor Kronecker" (ibid., p. 84). What Fraenkel reports—with an
exclamation point—is that in his posthumously published lectures on the theory of integrals,
Kronecker cites the problem of the uniqueness of trigonometric expansions as still open!
It is worth remembering that Cantor achieved his broadest generalization of his uniqueness
theorem in 1872, in the same paper (ibid.) pp. 92ff.) in which he first presented his con-
struction of the real numbers and initiated the study of the topology of point sets. As to
Dedekind, he responds, in his own quiet style, to Kronecker's footnote attack in a footnote
of his own to §1 of Was sind und was sollen die Zahlen? (see Dedekind, Werke, vol. 3, p. 345).
34. This point deserves a little further emphasis. Kronecker had aprogram for the elimina-
tion of the nonconstructive from mathematics. Such a program was of unquestionably great
interest. It continues to be so, in the double sense that a radical elimination—showing how
to render constructive a sufficient body of mathematics, and at the same time so greatly
simplifying that body, as to make it plausible that this constructive part is really all that
deserves to be studied and that the constructive point of view is the clearly most fruitful
way of studying it—would be a stupendous achievement; whereas in the absence of such
radical elimination, it remains always important to investigate how far constructive meth-
ods may be applied. But Kronecker's own published ideas for constructivization appear to
have extended only to the algebraic, as indeed the remark cited in n. 28 above explicitly
acknowledges (for when Kronecker says that "the concept of the limit... has thus far re-
mained alien to number theory," that last phrase has to be taken to mean irreducible to
258 Howard Stein
For over two decades, one of my major interests has been reading,
teaching, and writing history of mathematics. During those decades, I have
become convinced that ten claims I formerly accepted concerning math-
ematics and its development are both seriously wrong and a hindrance
to the historical study of mathematics. In analyzing these claims, I shall
attempt to establish their initial plausibility by showing that one or more
eminent scholars have endorsed each of them; in fact, all seem to be held
by many persons not fully informed about recent studies in history and
philosophy of mathematics. This paper is in one sense a case study; it has,
however, the peculiar feature that in it I serve both as dissector and frog.
In candidly recounting my changes of view, I hope to help newcomers
to history of mathematics to formulate a satisfactory historiography and
to encourage other practitioners to present their own reflections. My at-
tempt to counter these ten claims should be prefaced by two qualifica-
tions. First, in advocating their abandonment, I am not in most cases urg-
ing their inverses; to deny that all swans are white does not imply that
one believes no swans are white. Second, I realize that the evidence I ad-
vance in opposition to these claims is scarcely adequate; my arguments
are presented primarily to suggest approaches that could be taken in more
fully formulated analyses.
1. The Methodology of Mathematics Is Deduction
In a widely republished 1945 essay, Carl G. Hempel stated that the
method employed in mathematics "is the method of mathematical dem-
onstration, which consists in the logical deduction of the proposition
to be proved from other propositions, previously established." Hempel
added the qualification that mathematical systems rest ultimately on ax-
ioms and postulates, which cannot themselves be secured by deduction.1
Hempel's claim concerning the method of mathematics is widely shared;
260
TEN MISCONCEPTIONS ABOUT MATHEMATICS 261
has faded. The Age of Reason is gone."7 Much in what follows sheds
further light on the purported certainty of mathematics, but let us now
proceed to two related claims.
3. Mathematics Is Cumulative
An elegant formulation of the claim for the cumulative character of
mathematics is due to Hermann Hankel, who wrote: "In most sciences
one generation tears down what another has built and what one has
established another undoes. In Mathematics alone each generation builds
a new story to the old structure."8 Pierre Duhem made a similar claim:
"Physics does not progress as does geometry, which adds new final and
indisputable propositions to the final and indisputable propositions it
already possessed... ."9 The most frequently cited illustration of the
cumulative character of mathematics is non-Euclidean geometry. Consider
William Kingdon Clifford's statement: "What Vesalius was to Galen, what
Copernicus was to Ptolemy, that was Lobatchewsky to Euclid."10 Clif-
ford's claim cannot, however, be quite correct; whereas acceptance of
Vesalius entailed rejection of Galen, whereas adoption of Copernicus led
to abandonment of Ptolemy, Lobachevsky did not refute Euclid; rather
he revealed that another geometry is possible. Although this instance il-
lustrates the remarkable degree to which mathematics is cumulative, other
cases exhibit opposing patterns of development. As I wrote my History
of Vector Analysis,11 I realized that I was also, in effect, writing The
Decline of the Quaternion System. Massive areas of mathematics have,
for all practical purposes, been abandoned. The nineteenth-century
mathematicians who extended two millennia of research on conic section
theory have now been forgotten; invariant theory, so popular in the nine-
teenth century, fell from favor.12 Of the hundreds of proofs of the Py-
thagorean theorem, nearly all are now nothing more than curiosities.13
In short, although many previous areas, proofs, and concepts in mathe-
matics have persisted, others are now abandoned. Scattered over the land-
scape of the past of mathematics are numerous citadels, once proudly
erected, but which, although never attacked, are now left unoccupied by
active mathematicians.
4. Mathematical Statements Are Invariably Correct
The most challenging aspect of the question of the cumulative character
of mathematics concerns whether mathematical assertions are ever refuted.
264 Michael J. Crowe
The previously cited quotations from Hankel and Duhem typify the
widespread belief that Joseph Fourier expressed in 1822 by stating that
mathematics "is formed slowly, but it preserves every principle it has once
acquired.... "u Although mathematicians may lose interest in a particular
principle, proof, or problem solution, although more elegant ways of for-
mulating them may be found, nonetheless they purportedly remain. In-
fluenced by this belief, I stated in a 1975 paper that "Revolutions never
occur in mathematics."15 In making this claim, I added two important
qualifications: the first of these was the "minimal stipulation that a
necessary characteristic of a revolution is that some previously existing
entity (be it king, constitution, or theory) must be overthrown and ir-
revocably discarded"; second, I stressed the significance of the phrase "in
mathematics," urging that although "revolutions may occur in mathe-
matical nomenclature, symbolism, metamathematics, [and] methodolo-
g y . . . , " they do not occur within mathematics itself.16 In making that
claim concerning revolutions, I was influenced by the widespread belief
that mathematical statements and proofs have invariably been correct.
I was first led to question this belief by reading Imre Lakatos's brilliant
Proofs and Refutations, which contains a history of Euler's claim that
for polyhedra V-E + F = 2, where Kis the number of vertices, E the num-
ber of edges, and F the number of faces.17 Lakatos showed not only that
Euler's claim was repeatedly falsified, but also that published proofs for
it were on many occasions found to be flawed. Lakatos's history also
displayed the rich repertoire of techniques mathematicians possess for
rescuing theorems from refutations.
Whereas Lakatos had focused on a single area, Philip J. Davis took
a broader view when in 1972 he listed an array of errors in mathematics
that he had encountered.18 Philip Kitcher, in his recent Nature of Math-
ematical Knowledge, has also discussed this issue, noting numerous er-
rors, especially from the history of analysis.19 Morris Kline called atten-
tion to many faulty mathematical claims and proofs in his Mathematics:
The Loss of Certainty. For example, he noted that Ampere in 1806 proved
that every function is differentiable at every point where it is continuous,
and that Lacroix, Bertrand, and others also provided proofs until Weier-
strass dramatically demonstrated the existence of functions that are every-
where continuous but nowhere differentiable.20 In studying the history
of complex numbers, Ernest Nagel found that such mathematicians as Car-
dan, Simson, Playfair, and Frend denied their existence.21 Moreover,
TEN MISCONCEPTIONS ABOUT MATHEMATICS 255
Maurice Lecat in a 1935 book listed nearly 500 errors published by over
300 mathematicians.22 On the other hand, Rene Thorn has asserted: "There
is no case in the history of mathematics where the mistake of one man
has thrown the entire field on the wrong track.... Never has a signifi-
cant error slipped into a conclusion without almost immediately being
discovered."23 Even if Thorn's claim is correct, the quotations from Duhem
and Fourier seem difficult to reconcile with the information cited above
concerning cases in which concepts and conjectures, principles and proofs
within mathematics have been rejected.
5. The Structure of Mathematics Accurately Reflects Its History
In recent years, I have been teaching a course for humanities students
that begins with a careful reading of Book I of Euclid's Elements. That
experience has convinced me that the most crucial misconception that
students have about mathematics is that its structure accurately reflects
its history. Almost invariably, the students read this text in light of the
assumption that the deductive progression from its opening definitions,
postulates, and common notions through its forty-eight propositions ac-
curately reflects the development of Euclid's thought. Their conviction in
this regard is reinforced by the fact that most of them have earlier read Aris-
totle's Posterior Analytics, in which that great philosopher specified that
for a valid demonstration "the premises... must b e . . . better known than
and prior to the conclusion.... "24 My own conception is that the develop-
ment of Euclid's thought was drastically different. Isn't it plausible that
in composing Book I of the Elements, Euclid began not with his defini-
tions, postulates, and common notions but rather either with his extremely
powerful 45th proposition, which shows how to reduce areas bounded by
straight lines to a cluster of measurable triangular areas, or with his magnifi-
cent 47th proposition, the Pythagorean theorem, for which he forged a proof
that has been admired for centuries. Were not these two propositions the
ones he knew best and of which he was most deeply convinced? Isn't it
reasonable to assume that it was only after Euclid had decided on these
propositions as the culmination for his first book that he set out to con-
struct the deductive chains that support them? Is it probable that Euclid
began his efforts with his sometimes abstruse and arbitrary definitions—
"a point is that which has no parts"—and somehow arrived forty-seven
propositions later at a result known to the Babylonians fifteen centuries
earlier? An examination of Euclid's 45th and 47th propositions shows that
266 Michael J. Crowe
they depend upon the proposition that if two coplanar straight lines meet
at a point and make an angle with each other equal to two right angles,
then those lines are collinear. Should it be seen as a remarkable coincidence
that thirty-one propositions earlier Euclid had proved precisely this result,
but had not used it a single time in the intervening propositions? It seems
to me that accepting the claim that the history and deductive structures
of mathematical systems are identical is comparable to believing that Sac-
cheri was surprised when after proving dozens of propositions, he finally
concluded that he had established the parallel postulate.
Is not the axiomatization of a field frequently one of the last stages,
rather than the first, in its development? Recall that it took Whitehead and
Russell 362 pages of their Principia Mathematica to prove that 1 + 1=2.
Calculus texts open with a formulation of the limit concept, which took
two centuries to develop. Geometry books begin with primary notions and
definitions with which Hilbert climaxed two millennia of searching.
Second-grade students encounter sets as well as the associative and com-
mutative laws—all hard-won attainments of the nineteenth century. If
these students are gifted and diligent, they may years later be able to
comprehend some of the esoteric theorems advanced by Archimedes or
Apollonius. When Cauchy established the fundamental theorem of the
calculus, that subject was nearly two centuries old; when Gauss proved
the fundamental theorem of algebra, he climaxed more than two millen-
nia of advancement in that area.25 In teaching complex numbers, we first
justify them in terms of ordered couples of real numbers, a creation of
the 1830s. After they have magically appeared from this process, we
develop them to the point of attaining, say, Demoive's theorem, which
came a century before the Hamilton-Bolyai ordered-couple justification
of them. In presenting a theorem, first we name it and state it precisely
so as to exclude the exceptions it has encountered in the years since its
first formulation; then we prove it; and, finally, we employ it to prove
results that were probably known long before its discovery. In short, we
reverse history. Hamilton created quaternions in 1843 and simultaneous-
ly supplied a formal justification for them, this being the first case in which
a number system was discovered and justified at the same time; half a
century later Gibbs and Heaviside, viewing the quaternion method of space
analysis as unsatisfactory, proposed a simpler system derived from quater-
nions by a process now largely forgotten.
Do not misunderstand: I am not claiming that the structure of math-
TEN MISCONCEPTIONS ABOUT MATHEMATICS 267
ematics, as a whole or in its parts, is in every case the opposite of its history.
Rather I am suggesting that the view that students frequently have, im-
plicitly or explicitly, that the structure in which they encounter areas of
mathematics is an adequate approximation of its history, is seriously defec-
tive. Mathematics is often compared to a tree, ever attaining new heights.
The latter feature is certainly present, but mathematics also grows in root
and trunk; it develops as a whole. To take another metaphor, the math-
ematical research frontier is frequently found to lie not at some remote
and unexplored region, but in the very midst of the mathematical domain.
Mathematics is often compared to art; yet reflect for a moment. Homer's
Odyssey, Da Vinci's Mona Lisa, and Beethoven's Fifth Symphony are
completed works, which no later artist dare alter. Nonetheless, the latest
expert on analysis works alongside Leibniz and Newton in ordering the
area they created; a new Ph.D. in number theory joins Euclid, Fermat,
and Gauss in perfecting knowledge of the primes. Kelvin called Fourier's
Theorie analytique de la chaleur a "mathematical poem,"26 but many
authors shaped its verses. Why did some mathematicians oppose introduc-
tion of complex or transfinite numbers, charging that they conflicted with
the foundations of mathematics? Part of the reason is that, lacking a
historical sense, they failed to see that foundations are themselves open
to alteration, that not only premises but results dictate what is desirable
in mathematics.
6. Mathematical Proof in Unproblematic
Pierre Duhem in his Aim and Structure of Physical Theory reiterated
the widely held view that there is nothing problematic in mathematical
proof by stating that geometry "grows by the continual contribution of
a new theorem demonstrated once and for all and added to theorems
already demonstrated.... "27 In short, Duhem was claiming that once a
proposition has been demonstrated, it remains true for all time. Various
authors, both before and after Duhem, have taken a less absolutist view
of the nature and conclusiveness of proof. In 1739, David Hume observed:
There is n o . . . Mathematician so expert... as to place entire confidence
in any truth immediately upon his discovery of it, or regard it as any
thing, but a mere probability. Every time he runs over his proofs, his
confidence encreases; but still more by the approbation of his friends;
and is rais'd to its utmost perfection by the universal assent and ap-
plauses of the learned world."28
268 Michael J. Crowe
only intuition which can not deceive us. It may be said that today absolute
rigor is attained."37 More recently, Morris Kline remarked: "No proof
is final. New counterexamples undermine old proofs. The proofs are then
revised and mistakenly considered proven for all time. But history tells
us that this merely means that the time has not yet come for a critical
examination of the proof."38
Not only do standards of rigor intensify, they also change in nature;
whereas in 1700 geometry was viewed as providing the paradigm for such
standards, by the late nineteenth century arithmetic-algebraic considera-
tions had assumed primacy, with these eventually giving way to standards
formulated in terms of set theory. Both these points, as well as a number
of others relating to rigor, have been discussed with unusual sensitivity
by Philip Kitcher. For example, in opposition to the traditional view that
rigor should always be given primacy, Kitcher has suggested in his essay
"Mathematical Rigor—Who Needs It?" the following answer: "Some
mathematicians at some times, but by no means all mathematicians at all
times."39 What has struck me most forcefully about the position Kitcher
developed concerning rigor in that paper and in his Nature of Mathematical
Knowledge are its implications for the historiography of mathematics. I
recall being puzzled some years ago while studying the history of com-
plex numbers by the terms that practitioners of that most rational dis-
cipline, mathematics, used for these numbers. Whereas their inventor
Cardan called them "sophistic," Napier, Girard, Descartes, Huygens, and
Euler respectively branded them "nonsense," "inexplicable," "imagi-
nary," "incomprehensible," and "impossible." Even more mysteriously,
it seemed, most of these mathematicians, despite the invective implied
in thus naming these numbers, did not hesitate to use them. As Ernest
Nagel observed, "for a long time no one could defend the 'imaginary
numbers' with any plausibility, except on the logically inadequate ground
of their mathematical usefulness." He added: "Nonetheless, mathema-
ticians who refused to banish t h e m . . . were not fools... as subsequent
events showed."40
What I understand Kitcher to be suggesting is that the apparent irra-
tionality of the disregard for rigor found in the pre-1830 history of both
complex numbers and the calculus is largely a product of unhistorical,
present-centered conceptions of mathematics. In particular, if one
recognizes that need for rigor is a relative value that may be and has at
times been rationally set aside in favor of such other values as usefulness,
TEN MISCONCEPTIONS ABOUT MATHEMATICS 277
Notes
1. Carl G. Hempel, "Geometry and Empirical Science," in Readings in the Philosophy
of Science, ed. Philip P. Wiener (New York: Appleton-Century-Crofts, 1953), p. 41; reprinted
from American Mathematical Monthly 52 (1945): 7-17.
2. Carl G. Hempel, Philosophy of Natural Science (Englewood Cliffs, N.J.: Prentice-
Hall, 1966) pp. 16-17.
3. Hempel, "Geometry," pp. 40-41.
4. As quoted in On Mathematics and Mathematicians, ed. Robert Edouard Moritz (New
York: Dover, 1942), p. 295.
5. Charles S. Peirce, "The Non-Euclidean Geometry," in Collected Papers of Charles
SandersPeirce, vol. 8, ed. Arthur W. Burks (Cambridge, Mass.: Harvard University Press,
1966), p. 72.
6. Reuben Hersh, "Some Proposals for Reviving the Philosophy of Mathematics," Ad-
vances in Mathematics 31 (1979): 43.
7. Morris Kline, Mathematics: The Loss of Certainty (New York: Oxford University Press,
1980), p. 7.
8. As quoted in Moritz, Mathematics, p. 14.
9. Pierre Duhem, The Aim and Structure of Physical Theory, trans. Philip P. Wiener
276 Michael J. Crowe
1. Introduction
The principal thrust of this essay is to describe the current state of in-
teraction between mathematics and the sciences and to relate the trends
to the historical development of mathematics as an intellectual discipline
and of the sciences as they have developed since the seventeenth century.
This story is interesting in the context of the history of present-day
mathematics because it represents a shift in the preconceptions and stereo-
types of both mathematicians and scientists since World War II.
The notion that significant mathematical and scientific advances are
closely interwoven is not particularly new. The opposing notion (asso-
ciated, with whatever degree of justice, with the name of Bourbaki) was
never as fashionable, at least among working mathematicians, as in the
two decades immediately after World War II. The situation has changed
significantly during the past decade and had begun turning even earlier.
It turned not only among mathematicians, but even more significantly in
such sciences as physics. The frontier of mathematical advance was seen
again to be in forceful interaction with the basic problems and needs of
scientific advance.
This is an essay on significant trends in mathematical practice. The rela-
tions between the history and philosophy of mathematics as usually con-
ceived and mathematical practice have often been very ambiguous. In part,
this has resulted from the efforts of some historians and philosophers to
impose a framework of preconceptions upon mathematical practice that
had little to do with the latter. In part, however, it resulted from the diver-
sity of mathematical practice, to lags in its perception, and to the com-
plexity of viewpoints embedded in that practice.
Let me preface this account with two statements by great American
mathematicians of an earlier period who put the case in a sharp form.
278
MATHEMATICS AND THE SCIENCES 279
The first is John von Neumann in a 1945 essay titled "The Mathema-
tician":1
Most people, mathematicians and others, will agree that mathematics
is not an empirical science, or at least that it is practiced in a manner
which differs in several decisive respects from the techniques of the em-
pirical sciences. And, yet, its development is very closely linked with
the natural sciences. One of its main branches, geometry, actually
started as a natural, empirical science. Some of the best inspirations
of modern mathematics (I believe, the best ones) clearly originated in
the natural sciences. The methods of mathematics pervade and dominate
the "theoretical" divisions of the natural sciences. In modern empirical
sciences it has become more and more a major criterion of success
whether they have become accessible to the mathematical method or
to the near-mathematical methods of physics. Indeed, throughout the
natural sciences an unbroken chain of successive pseudomorphoses, all
of them pressing toward mathematics, and almost identified with the
idea of scientific progress, has become more and more evident. Biology
becomes increasingly pervaded by chemistry and physics, chemistry by
experimental and theoretical physics, and physics by very mathematical
forms of theoretical physics.
The second is Norbert Wiener in a 1938 essay titled "The Historical
Background of Harmonic Analysis:"2
While the historical facts in any concrete situation rarely point a clear-
cut moral, it is worth while noting that the recent fertility of harmonic
analysis has followed a refertilization of the field with physical ideas.
It is a falsification of the history of mathematics to represent pure
mathematics as a self-contained science drawing inspiration from itself
alone and morally taking in its own washing. Even the most abstract
ideas of the present time have something of a physical history. It is
quite a tenable point of view to urge this even in such fields as that
of the calculus of assemblages, whose exponents, Cantor and Zermelo,
have been deeply interested in problems of statistical mechanics. Not
even the influence of this theory on the theory of integration, and in-
directly on the theory of Fourier series, is entirely foreign to physics.
The somewhat snobbish point of view of the purely abstract mathemati-
cian would draw but little support from mathematical history. On the
other hand, whenever applied mathematics has been merely a technical
employment of methods already traditional and jejune, it has been very
poor applied mathematics. The desideratum in mathematical as well
as physical work is an attitude which is not indifferent to the extremely
280 Felix E. Browder
and because of the prevalence and intensity of myths in this domain that
prevent realistic assessment of the situation.
We are all very conscious of the role of the high-speed digital com-
puter as one of the decisive facts of the present epoch and for the
foreseeable future. We all know of the tremendous impact it has had on
the structure of processes in industrial society that depend on calculation,
communication, and control. In practice, this excludes very few domains
of human existence in modern society, whether technological, economic,
social, political, or military. The sciences and mathematics have not been
immune from this impact. Indeed, the scope and nature of scientific and
mathematical instrumentation and practice in our society have already been
radically changed by the existence of high-speed digital computation and
its continual decrease in cost during recent decades. I have deliberately
used the unusual phrase mathematical instrumentation to point up the
radically new fact that such a phenomenon now exists and is an impor-
tant component of our situation.
At the same time, although we are all conscious of the importance of
the digital computer (sometimes to the point of hysteria), and indeed are
inundated with advertising hyperbole from the most diverse quarters about
all the wonders that supercomputers will accomplish, many are much less
conscious of what is ultimately an even more important fact: the com-
puter is as much a problem as it is a tool. We must understand the nature
and limitations of this most powerful of all human tools. It is important
to know what cannot be computed and the dangers of what can be
miscomputed.
These limitations can be seen most plainly in the context of mathe-
matical and scientific practice. Perhaps the most significant use of the com-
puter in this context is as an experimental tool, sometimes even displac-
ing the laboratory experiment altogether. One translates a scientific or
mathematical problem into a simpler mathematical model and then uses
the computational power of the computer to study particular cases of the
general model. This approach has turned out to be very useful, particularly
when the conditions for experiment in the usual sense or of precise calcula-
tion become impossibly difficult. The mystique of such practices has grown
to such an extent that some speak of replacing Nature, an analog com-
puter, by a newer and better model of a digitalized nature.
The drawbacks and dangers of such practices without a background
of thorough critical analysis are equally clear. We must ask about the ade-
284 Felix E. Browder
quacy of the model, about the accuracy (not to say the meaningfulness)
of the computational process, and, last but not least, about the represent-
ative character of the particular cases that one computes. Without serious
cross-checks on these factors, we are left with yet another case of the zeroth
law of the computer: garbage in, garbage out, particularly with serious
scientific and mathematical problems that cannot be solved by computa-
tion as they stand. One replaces them by manageable problems, and the
validity of the replacement is precisely the crucial question. It is the im-
portance of this question that has led to pointed comments about the ad-
jective scientific in the currently fashionable emphasis on programs for
scientific computation on supercomputers.
These critical questions do not mean that we should neglect the com-
puter as a tool in science and mathematics. They do point up a sometimes
neglected fact—namely, that the computer is a difficult tool whose use
must be studied and refined. Computers are brute force instruments; their
effective use depends vitally on human insight and ingenuity. I intend here
to emphasize the importance of the intellectual arts and insights that are
or can be connected with the digital computer and its uses. These intellec-
tual arts have a vital relation to the mathematical enterprise. They con-
stitute a specialized and different way of applying classical mathematical
ideas and techniques with radically new purposes in mind. Their vitality,
both intellectual and practical, depends in an essential way upon a con-
tinuing contract with the central body of mathematical activity.
There is an interesting and slightly ironic aspect to the relationship be-
tween computer science and the central body of mathematics. Since the
mid-nineteenth century, mathematicians and physical scientists have tended
to see a dichotomy between mathematics that is applicable to the uses of
physical modeling and calculation and another kind that is not applicable.
The rules for this division have changed in recent years, with an ever-
increasing diversity of mathematical themes and theories falling into the
first category. Even so, the stereotype tends to persist, and some areas
of active mathematical research—like algebraic number theory or math-
ematical logic—tend to be relegated to the second category. Yet it is pre-
cisely these areas, grouped together with various forms of combinatorics
under the general label of discrete mathematics, that have turned out to
be most vital in major areas of advance in computer science. The basic
theoretical framework of computer science and the development of com-
plexity of computation rest upon the foundation of mathematical logic.
MATHEMATICS AND THE SCIENCES 285
5. Applicable Mathematics
Mathematical research in its various forms is an enterprise of great vital-
ity in the present-day world (although it is invisible to some outsiders).
Despite its fundamental autonomy, the enterprise of advanced mathemat-
ical research has interacted strongly in the last two decades with various
advances in the sciences. For the purposes of the present discussion, I
present two kinds of evidence.
The first consists of taking a conventional breakdown of the principal
active branches of contemporary mathematical research and inquiring in
general terms whether these branches have interactions of the type de-
scribed with the sciences. In the table of organization for the Internation-
al Congress of Mathematicians in Berkeley, California, in the summer
of 1986, we have such a breakdown in the division of the Congress into
nineteen sections. Of these nineteen sections, we may set aside two (his-
tory of mathematics, teaching of mathematics) and ask about the appli-
cability of the seventeen mathematical areas in this classification. Five
(probability and mathematical statistics, mathematical physics, numerical
methods and computing, mathematical aspects of computer science, ap-
plications of mathematics to nonphysical sciences) relate directly to the
sciences and technology. Eight have direct relation in contemporary prac-
tice to theory and practice in the natural sciences (geometry, topology,
algebraic geometry, complex analysis, Lie groups and representations, real
and functional analysis, partial differential equations, ordinary differen-
tial equations and dynamical systems). The remaining four (mathematical
logic and foundations, algebra, number theory, discrete mathematics and
combinatorics) have an equally vital relation to computer science. There
is no residue of mathematics that is fundamentally not applicable on this
list.
The second kind of evidence is illustrated by the study of the soliton
theory of the Korteweg-De Vries equation in the periodic case. The ap-
plications of algebraic geometry and complex analysis to the study of the
Korteweg-De Vries equation under periodic boundary conditions not only
contributed to the understanding of the physical model involved but reacted
upon the disciplines involved. New ideas and methods in both math-
ematical disciplines arose from this interaction, resulting in the solution
of classical problems in algebraic geometry and function theory. In an
even more striking case, the young Oxford mathematician Simon Donald-
288 Felix E. Browder
matter of major scientific disciplines in their own right, I doubt that this
will lead to the disappearance of professional differences between special-
ists in various disciplines in attacking these scientific problems. The dif-
ference between specialties has a positive function as well as negative
consequences. Specialists can rely upon the intellectual traditions and
resources of their scientific specialty, and this applies with the greatest
force to the mathematician. We can ask for a broader and more effective
effort at communication, however, among those concerned with common
problems, and we can cultivate an active interest in and sympathy with
the thematic concerns of other specialties than our own.
Notes
1. In The Works of the Mind, ed. Heywood and Nef (Chicago: University of Chicago
Press, 1945); reprinted in The World of Mathematics, ed. J. Newman, vol. 4 (New York:
Simon and Schuster, 1956), pp. 2053-63.
2. In Semi-Centennial Addresses of the American Mathematical Society, 1938.
Philip Kitcher •
Mathematical Naturalism
293
294 Philip Kitcher
1. Epistemological Commitments
Seeking a foundation for a part of mathematics can make exactly the
same sense as looking for a foundation for some problematic piece of scien-
tific theory. At some times in the history of mathematics, practitioners
have self-consciously set themselves the task of clarifying concepts whose
antecedent use skirted paradox or of systematizing results whose connec-
tions were previously only dimly perceived. Weierstrass's efforts with con-
cepts of convergence and Lagrange's explanation of the successes of techni-
ques for solving cubic and quartic equations are prime examples of both
forms of activity. But the grand foundational programs move beyond these
local projects of intellectual slum clearance.1
Foundationalist philosophies of mathematics bear a tacit commitment
to apriorist epistemology. If mathematics were not taken to be a priori,
then the foundational programs would have point only insofar as they
responded to some particular difficulty internal to a field of mathematics.
Subtract the apriorist commitments and there is no motivation for think-
ing that there must be some first mathematics, some special discipline from
which all the rest must be built.
Mathematical apriorism has traditionally been popular, so popular that
there has seemed little reason to articulate and defend it, because it has
been opposed to the most simplistic versions of empiricism. Apriorists
come in two varieties. Conservative apriorists claim that there is no
possibility of obtaining mathematical knowledge without the use of cer-
tain special procedures: one does not know a theorem unless one has
carried out the appropriate procedures for gaining knowledge of the ax-
ioms (enlightenment by Platonic intuition, construction in pure intuition,
stipulative fixing of the meanings of terms, or whatever) and has followed
a gapless chain of inferences leading from axioms to theorem. Frege, at
his most militant, is an example of a conservative apriorist. By contrast,
liberals do not insist on the strict impossibility of knowing a mathematical
truth without appealing to the favored procedures. Their suggestion is that
any knowledge so obtained can ultimately be generated through the use
of a prior procedures and that mathematical knowledge is ultimately im-
proved through the production of genuine proofs.
Empiricists and naturalists2 dissent from both versions of apriorism
by questioning the existence or the power of the alleged special procedures.
Insofar as we can make sense of the procedures to which apriorist epis-
temologies make their dim appeals, those procedures will not generate
MATHEMATICAL NATURALISM 295
contentless. I shall henceforth ignore this approach and leave it to the in-
terested reader to see how the argument would be developed from the
preferred point of view.)
The obvious way to distinguish mathematical knowledge from mere
true belief is to suggest that a person only knows a mathematical state-
ment when that person has evidence for the truth of the statement—
typically, though not invariably, what mathematicians count as a proof.6
But that evidence must begin somewhere, and an epistemology for
mathematics ought to tell us where. If we trace the evidence for the state-
ment back to the acknowledged axioms for some part of mathematics,
then we can ask how the person knows those axioms.
In almost all cases, there will be a straightforward answer to the ques-
tion of how the person learned the axioms. They were displayed on a
blackboard or discovered in a book, endorsed by the appropriate author-
ities, and committed to the learner's memory. But nonnaturalistic epis-
temologies of mathematics deny that the axioms are known because they
were acquired in this way. Apriorists offer us the picture of individuals
throwing away the props that they originally used to obtain their belief
in the axioms and coming to know those axioms in special ways. At this
point we should ask a series of questions. What are these special ways
of knowing? How do they function? Are they able to produce knowledge
that is independent of the processes through which the beliefs were original-
ly acquired? One line of naturalistic argument consists in examining the
possibilities and showing that the questions cannot be answered in ways
that are consistent with apriorism.7
There is a second, simpler way to argue for a naturalistic epistemology
for mathematics. Consider the special cases, the episodes in which a new
axiom or concept is introduced and accepted by the mathematical com-
munity. Naturalists regard such episodes as involving the assembly of
evidence to show that the modification of mathematics through the adop-
tion of the new axiom or concept would bring some advance in math-
ematical knowledge. Often the arguments involved will be complex—in
the way that scientific arguments in behalf of a new theoretical idea are
complex—and the pages in which the innovators argue for the merits of
their proposal will not simply consist in epistemologically superfluous
rhetoric. On the rival picture, the history of mathematics is punctuated
by events in which individuals are illuminated by new insights that bear
no particular relation to the antecedent state of the discipline.
298 Philip Kitcher
3. Varieties of Rationality
Those who offer theories of the growth of scientific knowledge must
answer two main problems. The first of these, the problem of progress,
requires us to specify the conditions under which fields of science make
progress. The second, the problem of rationality, requires us to specify
the conditions under which fields of science proceed rationally. Everyone
ought to agree that the two problems are closely connected. For a field
to proceed rationally, the transitions between states of the field at dif-
ferent times must offer those who make the transitions the best available
strategies for making progress. Rationality, as countless philosophers have
remarked, consists in adjustment of means to ends. Our typical judgments
of rationality in discussions about scientific change or the growth of
knowledge tacitly assume that the ends are epistemic. When we attribute
rationality to a past community of scientists, we consider how people in
their position would best act to achieve those ends that direct all inquiry,
and we recognize a fit between what was actually done and our envisaged
ideal.
So I start with a general thesis, one that applies equally to mathematics
and to the sciences. Interpractice transitions count as rational insofar as
they maximize the chances of attaining the ends of inquiry. The notion
of rationality with which I am concerned is an absolute one. I suppose
that there are some goals that dominate the context of inquiry, that are
not goals simply because they would serve as stepping stones to yet fur-
ther ends. The assumption is tricky. What are these goals? Who (or what)
MATHEMATICAL NATURALISM 305
ods by which they are generated. Newtonians emphasize the need for clari-
ty first, holding that secure results in problem solving will be found once
the concepts and reasonings are well understood. Leibnizians suggest that
the means of clarification will emerge once a wide variety of problems
have been tackled. Each tradition is gambling. What are the expected costs
and benefits?
Those who are uneasy, as I am, about attributing rationality to one
tradition and denying it to the other have an obvious first response to the
situation. They may propose that the alternative strategies were initially
so close in terms of their objective merits that it would have been perfect-
ly reasonable to pursue either of them. Hence there can be no condemna-
tion of either the early Newtonians or the early Leibnizians. Neither group
behaved irrationally. Yet this proposal is unsatisfactory as it stands. For
it is hard to extend it to account for the continued pursuit of the New-
tonian tradition once it had become apparent that the Continental ap-
proach stemming from Leibniz and the Bernoullis was achieving vast
numbers of solutions to problems that the Newtonians also viewed as
significant (problems that could readily be interpreted in geometric or
kinematic terms, even though the techniques used to solve them could
not). Moreover, historians who like to emphasize the role of social fac-
tors in the development of science will note, quite correctly, that the na-
tional pride of the British mathematicians and the legacy of the dispute
over priority in the elaboration of the calculus both played an important
role in the continued opposition to Continental mathematics.
I suggest that we can understand why British mathematicians dogged-
ly persisted in offering clumsy and often opaque geometric arguments if
we recognize the broader set of goals that they struggled to attain. Among
these goals was that of establishing the eminence of indigenous British
mathematics, and we can imagine that this end became especially impor-
tant after the Hanoverian succession and after Berkeley's clever challenge
to the credentials of the Newtonian calculus.26 Moreover, before we deplore
the fact that some of Newton's successors in the 1740s and 1750s may
have been moved by such nonepistemic interests as national pride, we
should also appreciate the possibility that the maintenance of a variety
of points of view (which chauvinism may sometimes achieve) can advance
the epistemic ends of the community. The goals of promoting acceptance
of truth and understanding in the total mathematical community were
ultimately achieved (in the nineteenth century) because both traditions were
MATHEMATICAL NATURALISM 309
kept alive through the eighteenth century. If, as I suspect, one of the tradi-
tions was maintained because some mathematicians were motivated by
nonepistemic interests, then perhaps we should envisage the possibility
that the deviation from individual epistemic rationality signals the presence
of an institution in the community of knowers that promotes community
epistemic rationality. A rational community of knowers will find ways
to exploit individual overall rationality in the interests of maximizing the
chances that the community will attain its epistemic ends.27
Only if we restrict ourselves to the notion of individual epistemic ra-
tionality and seek to find this everywhere in the history of mathematics
(or of science generally) does the search for rationality in that history com-
mit us to a Whiggish enterprise of distributing gold stars and black marks.
The general form of an interpractice transition is more complex than we
might have supposed, and the historiography of mathematics should reflect
the added complexity. We are to imagine that the community of math-
ematicians is initially divided into a number of homogeneous groups that
pursue different practices. In some cases, the claims made by members
of the groups will be incompatible (Newtonians versus Leibnizians,
Kronecker and his disciples against Dedekind and Cantor, militant con-
structivists against classical mathematicians). In other cases, they will just
be different. An interpractice transition may modify the particular prac-
tices, the group structure, or both. Such interpractice transitions may be
viewed from the perspective of considering whether (a) they maximize the
chances that the individuals who participate in them will achieve their in-
dividual epistemic ends, (b) they maximize the chances that those in-
dividuals will attain their total set of ends with some epistemic ends being
sacrificed to nonepistemic ends, (c) they maximize the chances that the
community will attain its epistemic ends, or (d) they maximize the chances
that the community will attain its total set of ends, with epistemic ends
being sacrificed to nonepistemic ends. Moreover, with respect to each case
we may focus on epistemic ends that are internal to mathematics or we
may look to see whether the ends of other areas of inquiry are also in-
volved, and, where (b) and (c) both obtain, we may look for institutions
within the total mathematical community that promote the attainment of
epistemic ends by the community at cost to the individual.
I have been concerned to stress the broad variety of questions that arise
for the history of mathematics once we adopt the naturalistic perspective
I have outlined and once we have extended it by differentiating notions
310 Philip Kitcher
5. An Agenda
In The Nature of Mathematical Knowledge, I focused on internal tran-
sitions and on individual epistemic rationality. My project was to show
that there have been important and unrecognized inferences in the history
of mathematics and that similar inferences underlie our knowledge of those
statements philosophers have often taken to be the foundations of our
mathematical knowledge. I now think that the position I took was too
conservative in several different ways.
First, external interpractice transitions need more emphasis. As I have
suggested above, mathematics is dependent on other sciences and on our
practical interests for the concepts that are employed in the spinning of
our mathematical stories. It is shortsighted to think that the systemati-
zation of a branch of mathematics can proceed in neglect of the ways
in which adjacent fields are responding to external demands. Hamilton
thought that generalization of claims about complex numbers would nec-
essarily be a fruitful project. Ultimately, the field in which he labored was
changed decisively through attempts to come to terms with problems in
mathematical physics. Vector algebra and analysis offered a perspective
from which the lengthy derivations of recondite properties of quaternions
look beside the point.36
Second, the history of mathematics, like the history of other areas of
science, needs to be approached from both the individual and the com-
munity perspectives. We should ask not only about the reasons people
have for changing their minds, but also about the fashion in which the
community takes advantage of our idiosyncrasies to guide us toward ends
we might otherwise have missed. As I have already noted, the pursuit of
more than one research program may often advance the community's
epistemic projects—despite the fact that it will require of some members
of the community that they act against their individual epistemic interests.
We can properly ask whether there are enough incentives in mathematics
for the encouragement of diversity.
Third, there are serious questions about the balance between the pur-
suit of epistemic ends and the pursuit of nonepistemic ends. I have been
MATHEMATICAL NATURALISM 377
Notes
1. The transition is evident in the work of Frege, who notes in the Introduction to the
Grundlagen (The Foundations of Arithmetic [Oxford: Blackwell, 1959]) that he has "felt
bound to go back rather further into the general logical foundations of our science than
perhaps most mathematicians will consider necessary" (p. x). His stated reason is that, with-
out the successful completion of the project he undertakes, mathematics has no more than
"an empirical certainty" (ibid.). Later, he suggests that the achievements of the nineteenth
century in defining the main concepts of analysis point inexorably to an analogous clarifica-
tion of the concept of natural number. To those who ask why, Frege offers the same
epistemological contrast, pointing out that mathematics prefers "proof, where proof is possi-
ble, to any confirmation by induction" (pp. 1-2). For a more detailed investigation of Frege's
MATHEMATICAL NATURALISM 321
19. Here I draw on an interpretation of Kant that I have developed in "Kant's Philosophy
of Science," in Self and Nature in Kant's Philosophy, ed. A. Wood (Ithaca, N.Y.: Cornell
University Press, 1984), pp. 185-215; and "Projecting the Order of Nature" (to appear in
Kant's Philosophy of Physical Science, ed. R. E. Butts). I should note that my interpreta-
tion is heretical in cutting away the apriorist strands in Kant's thought.
20. Of course, external transitions may create new branches of mathematics that are then
subject to internal interpractice transitions. If we credit popular anecdotes about Pascal and
Euler, then the fields of probability theory and topology may have originated in this way.
21. A large number of further distinctions may obviously be drawn here, for we may
take very different approaches to the question of how the maximization is to be done. This
is especially clear in cases where our ends admit of degrees, so that we may contrast max-
imizing the expected value with minimizing the risk of failing to obtain a certain value, and
so forth. I ignore such niceties for the purposes of present discussion.
22.1 think that it can often be shown in cases of this type that consideration of the prac-
tical interests of individuals reveals that the community optimum is more likely to result
if the individuals are motivated by nonepistemic factors. In other words, a sine qua non
for community epistemic rationality may be the abandonment by some individuals of in-
dividual epistemic rationality. However, it is possible that those individuals are overall ra-
tional. Something like this has been suggested by Kuhn (see, for example, his "Objectivity,
Value Judgment, and Theory Choice," in The Essential Tension [Chicago: University of
Chicago Press, 1977], pp. 320-39).
23. For a brief account, see chap. 10 of Nature of Mathematical Knowledge, and, for
more detail about the Leibnizians, Ivor Grattan-Guinness, The Development of the Foun-
dations of Analysis from Euler to Riemann (Cambridge, Mass.: MIT Press, 1970).
24. An especially clear example is furnished by the discussion among the Leibnizians
of the "result" that 1-1 + 1 - 1 + . . . = 1/2. See Leibniz's Mathematische Schriften,
ed. Gerhardt, 5 vols. (Halle, 1849-63), vol. 5, pp. 382ff., and vol. 4, p. 388. Euler was ex-
tremely dubious about the conclusions favored by Leibniz and Varignon. Nevertheless, his
own writings are full of inspired attempts to assign sums to divergent series that such later
writers as Abel would find appalling.
25. Important figures in the sequence are Benjamin Robins, Colin Maclaurin, and Simon
1'Huilier. The case of Maclaurin offers a clear contrast with the Continental tradition. When
Treatise on Fluxions is compared with any volume of Euler's works in analysis, one sees
two talented (though not equally talented mathematicians) proceeding by working on very
different problems. Maclaurin turns again and again to the question of finding an explana-
tion of the basic rules of the Newtonian calculus. Euler builds up a wealth of results about
integrals, series, maximization problems, and so forth, and is almost perfunctory about the
basic algorithms for differentiating and integrating.
26. In The Analyst (reprinted in The Works of George Berkeley, vol. 4, A. Luce and
T. Jessop, eds. [London: Nelson, 1950]). Berkeley's challenge provoked a number of
responses, some fairly inept (the essays of James Jurin, for example), others that helped
elucidate some important Newtonian ideas (the work of Maclaurian and, even more, the
papers of Benjamin Robins).
27. Plainly, this is simply part of a long and complicated story. The purpose of telling
it here is to show that a simplistic historiography is not forced on us by thinking about the
rationality of mathematical change. (I am grateful to Lorraine Daston for some penetrat-
ing remarks that raised for me the issue of whether my ascriptions of rationality to past
mathematicians commit me to Whig history. See her review of Nature of Mathematical
Knowledge, in Isis 75 [1984]: 717-21).
28. See Roger Cooke, The Mathematics of Sonya Kovaleskaya (New York: Springer,
1984), for some beautiful examples of the influence of physical problems on late-nineteenth-
century analysis.
29. See Penelope Maddy's "Perception and Mathematical Intuition," Philosophical
Review 89, (1980): 163-96. Related views have been elaborated by Michael Resnik in his
"Mathematics As a Science of Patterns: Ontology," Nous 15 (1981): 529-50.
324 Philip Kitcher
30. The geographical analogy stems from Frege; see Grundlagen, p. 108.
31. This is a point that has been emphasized by Karl Popper (see, for example, The Logic
of Scientific Discovery [London: Hutchinson, 1959], pp. 27-145); in the Popperian tradi-
tion, it leads to the notorious problems of constructing measures of verisimilitude (for re-
views, see I. Niiniluoto's "Scientific Progress," Syntheses [1980]: 427-62; and W. Newton-
Smith's The Rationality of Science [London: Routledge and Kegan Paul, 1981], chap. 2
and 8). I believe that the problem can be overcome if we break the spell of the idea that
the search for the significant is always the search for the general, but this is a long story
for another occasion.
32. Thus, for example, the account offered by Maddy in her "Perception and Mathematical
Intuition" seems at best to reveal how we are able to refer and to know about concrete sets.
It is not at all clear how this knowledge is supposed to provide us with a basis for reference
to and knowledge of abstract objects, where we are no longer in causal interaction with the
supposed objects. So even if we grant that our causal relation to an object provides us with
a basis for knowledge about the set whose sole member is that object, it is hard to see how
we obtain a similar basis when the sets under discussion do not have concrete objects as
members.
33. Nature of Mathematical Knowledge, chap. 6. This chapter has often been
misunderstood, and I have been taken to substitute one kind of abstract object (ideal agents)
for another (sets). But, as I took some pains to emphasize, there are no more any ideal agents
than there are such things as ideal gases. In both ideal gas theory and in mathematics, we
tell stories—stories designed to highlight salient features of a messy reality. I hope that my
present stress on storytelling will forestall any further misconceptions on this point.
34. See Nature of Mathematical Knowledge, pp. 128-29. It is crucial to appreciate that
some forms of human constructive activity consist in achieving representations of objects—as
when, paradigmatically, we cluster objects in thought. My claim is that the use of various
kinds of mathematical notation—designed to describe the properties of various con-
structions—makes possible new constructive activity. Thus we have constructive operations
that are iterated, sometimes to quite dizzying complexity, through the use of notation. Math-
ematics does not describe the notation but does provide (idealized) descriptions of the con-
structive acts that we can carry out with the help of the notation.
35. Mathematicians sometimes toy with the idea that mathematics is art—or is like art.
One consequence of my naturalistic epistemology for mathematics is that it enables us to
see what this idea might amount to and how it might apply to various parts of mathematics.
36. Since quaternions are now coming back into fashion, it may appear that the example
does not support my claims. However, what is now being done with quaternions is quite
distinct from what Hamilton did. For Hamilton, quaternions were to be treated in just the
ways that real and complex numbers had previously been treated. So, to cite only one exam-
ple, Hamilton set himself the task of defining the logarithm of a quaternion. So far as I
know, that perspective is a long way from the context of present discussion.
37. This is a common response from mathematicians who have read Nature of
Mathematical Knowledge. As I shall argue below in the text, the complaint seems to me
a very important one, and its justice can only be resolved by combining sophisticated under-
standing of contemporary mathematics with sophisticated understanding of the philosophi-
cal and historical issues. Here, I think, collaboration is clearly required
38. See H. M. Edwards, "The Genesis of Ideal Theory," Archive for the History of
the Exact Sciences 23 (1980): 321-78.
39. For Dedekind's argument, see his essay Sur la theorie des nombres entiers algebriques,
in GesammelteMathematische Werke, vol. 3, ed. E. Noether and O. Ore (Braunschweigh:
Vieweg, 1932). The crucial passage occurs on pp. 268-69.
40. In his presentation in Minneapolis, Garrett Birkhoff stressed the intertwining of threads
in "the tapestry of mathematics." It seems to me that there are numerous occasions in the
history of mathematics in which one area of mathematical practice is modified in response
to the state of others, and that concentration on the development of a mathematical field
can blind one to the ways in which fields emerge, modify one another, and are fused. For
MATHEMATICAL NATURALISM 325
some stimulating attempts to reveal these processes in concrete cases, see Emily Grosholz's
papers "Descartes' Unification of Algebra and Geometry," in Descartes, Mathematics and
Physics, ed. S. Gaukroger (Hassocks: Harvester Press, 1980); and "The Unification of Logic
and Topology," British Journal for the Philosophy of Science 36 (1985): 147-57.
41. See R. Thorn, Structural Stability and Morphogenesis, trans. D. Fowler (New York:
Benjamin, 1975); J. F. Traub and H. Wozniakowksi, Information and Computation, "Ad-
vances in Computers 23 (1984): 23-92; and M. Feigenbaum, "Universal Behavior in Nonlinear
Systems," Los Alamos Science (Summer 1980), pp. 3-27.
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IV. THE SOCIAL CONTEXT OF MODERN MATHEMATICS
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Judith V Grabiner
what may not be technical problems, and they underestimate the costs
of technical innovations while overestimating the benefits.
Upon listening to a debate like this, a historian might well ask whether
the claims of the partisans of the science—and of their critics—are solely
reactions to the current and probable future state of their science, or
whether they are drawing instead on the rhetoric and ideas of other debates
about somewhat different topics. If the latter, how did those earlier
debates—on which more of the returns are in—actually come out?
In this paper, I hope to provide some historical perspective on the im-
plications of Artificial Intelligence by describing a pattern followed by
similar debates about the implications of scientific breakthroughs earlier
in history. I will present several examples to show the frequency with which
this pattern has been followed. But the purpose of giving these examples
is not only to demonstrate that this pattern has occurred in the past. I
want also to suggest that causes similar to those that operated in the past
are working now; that some of the content of past debates has helped shape
the current one; and, finally, that recognizing the pattern will help clarify
what is really at issue now. In short, I hope to illuminate the present debate
over AI by showing that it shares many characteristics with past debates
between partisans of new scientific approaches and critics of those
approaches.
The pattern followed is roughly this: (1) The new methods solve some
outstanding set of problems despaired over in the past, and these successes
are impressive to outsiders and inspiring to those who have achieved them.
(I call this first stage technical success.) (2) The practitioners extrapolate
their successes to other fields (they have found the language of the book
of nature, or the secret of life); at the very least, they argue that those
who use their methods in other areas will succeed by these new means where
earlier investigators have failed. (I call this second stage extrapolation.)
(3) In reaction, antiscientific currents that always surround the sciences—
currents hostile even to successes of the delimited kind first mentioned—
are intensified by the extreme claims made in the second stage. (4) Finally,
more serious critics enter the fray—critics often from within the scientific
community itself, who perceive unfounded enthusiastic claims, claims
whose consequences—with no apparent scientific warrant—threaten
cherished values.
There are three other striking phenomena. First, the promises of the
first stage—successes in the original area and its nearby neighbors—are
PARTISANS AND CRITICS OF A NEW SCIENCE: AI 331
in fact fulfilled, but the claims of the second stage largely are not; and
there are costs even of the first. Second, the mature science growing out
of the initial success consolidates the earlier gains and continues. Third,
the enthusiasm and inspiration of the second stage seem valuable in con-
solidating the gains of the first stage of initial success, even though their
wide-ranging predictions are not fulfilled.
The debates also make reference to historical analogies taken from
similar debates in the past, and in so doing, they often blur the distinc-
tion between the stage of initial success and the second, more enthusiastic
stage. Critics opposing the second-stage extrapolation are often accused
of opposing the first, and are called Luddites or persecutors of Galileo.
Keeping the distinctions between the four stages clear, even though there
may be some blurring near the boundaries, would improve the current
debates over the support that society should give to fifth-generation AI
research, the human tasks that should be entrusted to computers, and the
information that artificial intelligence might give us about the nature of
man. It is important that these debates not be carried out with arguments
like "They laughed at Fulton" on the one hand and "AI is the new al-
chemy" on the other. Of course, history can help, but one must first un-
derstand that history.
To support these general points, I shall outline a set of examples from
the history of science so that we can then describe recent debates over AI
with this historical perspective. I will in particular discuss the methodo-
logical revolution in seventeenth-century science; the eighteenth-century
"spirit of systems" and the visions of society stemming from it; the In-
dustrial Revolution; and Darwinian evolution and the extrapolations—
social Darwinism, racism, atheistic materialism—from it. Finally, I will
take up the key ideas at the early stages of AI in the work of Alan Tur-
ing; the claims of AI pioneers like Herbert Simon, Alan Newell, John
McCarthy, and Marvin Minsky, and some of their successors; some reac-
tions to the overly enthusiastic claims; and, finally, the judicious, histor-
ically informed criticism of AI by computer scientist Joseph Weizenbaum.
The first example is to be found in the period of the scientific
revolution—in particular, the methodological revolution of seventeenth-
century science. Let us begin by measuring the first stage: its successes
in its own sphere. As recently as the sixteenth century, Michel de Mon-
taigne could speak of the futility of learning and could use the Coper-
nican system and the fact that it had ancient predecessors as an example
332 Judith V, Grabiner
of the way ideas about nature came and went as did other fashions (Defense
of Raymond Sebond, 86-87). According to Montaigne, the sciences had
established no real knowledge. But to Francis Bacon, this lack of past
success called not for despair but for a new approach: "Things which have
never yet been done can be done [only] by methods which have never yet
been tried" (Novum Organum, aphorism vi). And, indeed, the mechanical
philosophy did things that had never been done. Using the Cartesian
method of "analysis"—studying macroscopic phenomena by resolving
them into their component parts of bits of matter in motion—men like
Pascal, Torricelli, and Boyle were able to explain, by mechanical means,
a whole range of phenomena previously requiring notions like the abhor-
rence of a vacuum (Westfall 1971, 44-49). William Harvey, treating the
heart as if it were a pump and the blood as though it were an ordinary
fluid, demonstrated the circulation of the blood. Descartes saw Harvey's
work—though Harvey himself was far from being a mechanist—as a
triumph of the mechanical and analytic methods (Westfall 1971, 93;
Discourse on Method, 47-55). Cartesians saw the method as opening all
of nature to being reduced to Descartes's principles, with Descartes himself
believing that he had exhibited the mechanism of the solar system. Some
of his followers took the mechanical philosophy even further: LaMettrie
in 1747 spoke of man as a machine, and Hobbes (unlike Descartes) saw
even thought as a purely mechanical process (Leviathan', in Burtt 1939,
143-44).
But the outstanding predictive success of the mechanical philosophy
when applied to the vacuum or (with Borrelli) to the mechanical advan-
tage of muscles did not extend to psychology, or even to all of physics.
Descartes's prediction that a science of nature on his principles was almost
with his reach (Discourse on Method, 68) was an example of extrapola-
tion beyond the domain of success. So were the views that man was whol-
ly mechanical, and so were the views that the Cartesian principles suf-
ficed to explain the phenomena of the solar system. As is well known,
reactions even against the initial stage of predictive successes of science
occurred in the seventeenth century, from the fabled few who refused to
look through Galileo's telescope to the poets who grieved that "new
philosophy calls all in doubt" (Donne, "Anatomy of the World"; quoted
by Koyre 1957, 29). But there were other opponents of the new philosophy
who appreciated what the scientists had accomplished but who saw no
reason to believe that it extended as far as the scientists claimed. Blaise
PARTISANS AND CRITICS OF A NEW SCIENCE: AI 333
Pascal made clear that, though scientific reasoning had its place, there
was much in the situation of man in the world, and man's understanding
of himself, that was not amenable to reason. Pascal's esprit de finesse
recognized intuition and tacit ways of knowing as just as valid in their
sphere as mathematical reasoning is in its proper sphere (Pensees 1).
As for Newton and the Newtonians, they found the Cartesian insistence
that causes must be mechanical as the same sort of excess. Perhaps the
cause of gravity was not known; this did not mean that it was reducible
to mechanism, any more than the fact that a clock could be run by a spring
meant that a given clock was not actually moved by a weight. (For this
Turing test for pendulum clocks, see Roger Cotes's Preface to Newton's
Principia [pp. xxvii-xxviii].) Besides, as Newton had shown, Cartesian
mechanism had not been able to account for all the phenomena of the
solar system (notably Kepler's laws), whereas gravity seemed really to exist,
though its cause was unknown, and gravity could explain the motions of
the heavenly bodies, the fall of terrestrial objects, and the interaction be-
tween them in the movement of the tides (Newton's Principia, 547). Thus
the methodological imperialism of the Cartesians had not in fact produced
the promised understanding of the universe, much less of man. That critics
like Pascal and the Newtonians waged part of their battle on behalf of
their views of religion does not invalidate their conclusions about the limit-
ed scope of the successes of the mechanical philosophy.
My second example is found in the "spirit of systems" characteristic
of such thinkers of the Enlightenment as Voltaire, Condorcet, Comte, and
St.-Simon. Buoyed up by the successful application of reason in the realm
of nature, and by the analysis of macroscopic phenomena in terms of sim-
ple statements about the elements, they hoped to achieve in the same way
a science of society capable of remaking the world in a rational way. As
one opponent, A. Thibaudet, put it, they "ventured to create a religion
as one learns at the Ecole [polytechnique] to build a bridge or a road"
(Hayek 1955, 113). They had, however, left out of their calculations the
passions unleashed by, for instance, the French Revolution, and the
tenacious holding power of old institutions and ideas and the powerful
people who wanted those preserved. In particular, Voltaire's faith in the
power of reasonable arguments to persuade anybody who heard them,
based on the success of Newtonian science (Voltaire 1764, article "Sect"),
was not supported by events. The institutions of the past, as Burke pointed
out, and the forces preserving them, as Marx pointed out, had too much
334 Judith V, Grabiner
have brought it to the "state that it is in" include "the initial state of
the mind, say at birth," education, and other experiences. Education and
experience are not provided in the most efficient possible manner; this
could be done more rationally if we started with the initial state:
"Presumably the child-brain [has] so little mechanism that something like
it can be easily programmed" (2119).
Turing's views on computability and thought as both being formally
describable, rule-based processes appeared to some philosophers, notably
J. R. Lucas, to be similar to the old mechanistic ideas, though in a new
dress. Even if these new mechanists do not assert that minds come out
of matter, said Lucas, they do claim that minds "operate according to
'mechanical principles'"—that is, think that the whole is just the sum of
the operations of the separate parts and that the operation of each part
is determined by previous states or by random choices between states
(Lucas 1961, 126). I have discussed this controversy elsewhere (Grabiner
1984, 1986). For our present purposes, the key point is that Lucas's op-
position to Turing's ideas came from Lucas's desire to refute both "tradi-
tional materialism and its modern mechanist version" (Lucas 1968, 156),
a "mechanism" that is not implied by, but extrapolated from, Turing's
research in computability theory.
Beginning in 1956 with the Dartmouth conference that launched AI
as an organized discipline in the United States many more arguments
characteristic of the "extrapolation" stage began to appear. The original
proposal to the Rockefeller Foundation for the conference said: "The study
is to proceed on the basis of the conjecture that every aspect of learning
or any other feature of intelligence can in principle be so precisely des-
cribed that a machine can be made to simulate it" (McCorduck 1979, 93).
The key ideas used to "precisely describe" intelligence are that the human
mind is a system, either like a machine or like Laplace's world-system,
hierarchically organized and programmed by its environment; thought
takes place by the rule-generated manipulation of formal symbols. The
mind's organization, like the Tables of Discovery of Francis Bacon (Nov-
um Organum, aphorism cii), provides for empty slots into which pieces
of information from the world (disentangled from their context) can be
placed. For instance, John McCarthy: "One of the basic entities in our
theory is the situation. Intuitively, a situation is the complete state of af-
fairs at some instant in time. The laws of motion of a system determine
all future situations from a given situation" (McCorduck 1979, 213).
PARTISANS AND CRITICS OF A NEW SCIENCE: AI 339
lets us fully predict the future (1976, 221). But, as Aldous Huxley warned,
"The popular Weltanschauung of our times contains a large element of
what may be called 'nothing-but' thinking. Human beings . . . are nothing
but bodies, animals, even machines.... Mental happenings are nothing
but epiphenomena" (quoted by Weizenbaum 1976, 129). Max Horkheimer
sums this all up to Weizenbaum's satisfaction: too many people believe
that there is just one authority, namely science, and, furthermore, a science
conceived merely as "the classification of facts and the calculation of prob-
abilities" (Horkheimer 1947; quoted by Weizenbaum 1976, 252). Weizen-
baum summarizes his own position: "The computer is a powerful new
metaphor for helping us to understand many aspects of the world, but... it
enslaves the mind that has no other metaphors and few other resources
to call on." The computer scientist "must teach the limitations of his tools
as well as their power" (1976, 277).
Before ending our discussion of Weizenbaum's ideas, let me remind
the reader that there is nothing inevitable about the timing in the pattern
I have described. In particular, the fourth-stage critics may be rare, and
they do not necessarily win. As H. S. Jennings was unsuccessful in his
lone opposition to immigration restriction in 1924, so Weizenbaum now
is almost alone in the computer science community.
What, then, are we to conclude from the pattern common to all these
examples from the seventeenth-century mechanists to AI? It is possible,
as Max Weber has said, that scientific research must be a vocation and
that the researcher must be passionately committed to the belief that the
outcome of his next piece of research is the most important thing in the
world. But as the rest of us try to understand the implications of artificial
intelligence for our view of ourselves and for the future of our complex
world, we should remember that, though the sciences have over time greatly
enlarged the domain in which they can successfully predict and explain
phenomena, the enthusiastic predictions about extending methods suc-
cessful in one area to solve society's problems and to understand the world
have generally not come to pass.
References
Albus, James. 1981. Brains, Behavior, and Robotics. Peterborough, N.H.: Byte Books.
Bacon, Francis. 1620. Novum Organum. Reprinted in (Burtt 1939).
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Library.
Cauchy, A.-L. 1821. Cours d'analyse de I'ecole roy ale poly technique. Part 1. Analyse al-
gebrique. In Oeuvres completes. Paris: Gauthier-Villars, ser. 2, vol. 3, 1897.
344 Judith V. Grabiner
Orwell, George, 1937. The Road to Wigan Pier. Reprinted New York: Berkley, 1961.
. 1943. Looking Back on the Spanish War. Reprinted in A Collection of Essays. Garden
City, N.Y.: Doubleday, 1957, pp. 193-214.
Papert, Seymour. 1980. Mindstorms: Children, Computers, and Powerful Ideas. New York:
Basic Books.
Pascal, Blaise. [ca. 1660] 1958. Pensees. Reprinted New York: F. Button. (Numbering follows
the L. Brunschvicg edition).
Pursell, Carroll W., Jr., ed. 1969. Readings in Technology and American Life. New York:
Oxford University Press.
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The Sciences of the Artificial, 2d ed. Cambridge, Mass.: MIT Press.
Stanton, William. 1960. The Leopard's Spots. Chicago: University of Chicago Press.
Taylor, F. W. 1911. Scientific Management. New York: Harper. Excerpts from pp. 5-7,
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Turing, Alan. 1956. Can a Machine Think? Reprinted from Mind (1950) in The World of
Mathematics, vol. 4, ed. J. R. Newman. New York: Simon and Schuster, pp. 2099-2123.
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Viking Press, 1949, pp. 195-99.
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Language Communication between Man and Machine. Communications of the ACM
9: 36-45.
. 1976. Computer Power and Human Reason: From Judgment to Calculation. San
Francisco: Freeman.
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New York: Wiley.
William Aspray
1. Introduction
In 1896 the College of New Jersey changed its name to Princeton
University, reflecting its ambitions for graduate education and research.
At the time, Princeton, like other American universities, was primarily
a teaching institution that made few significant contributions to math-
ematics. Just four decades later, by the mid-1930s, Princeton had become
a world center for mathematical research and advanced education.1 This
paper reviews some social and institutional factors significant in this rapid
rise to excellence.2
The decade of the 1930s was a critical period for American research
mathematics generally, and for Princeton in particular. The charter of
the Institute for Advanced Study in 1930 and the completion of a univer-
sity mathematics building (Fine Hall) in 1931 frame the opening of the
period in Princeton; the completion of separate quarters (Fuld Hall) for
the institute mathematicians in 1939 and the entrance of the United States
into World War II effectively close it. During this decade, Princeton had
the unique atmosphere of an exclusive and highly productive mathematical
club. This social environment changed after the war with the increase in
university personnel and the move of the institute to separate quarters,
and the uniqueness was challenged by the improvement of mathematical
research and advanced education at other American institutions.
I appreciate the useful comments and corrections of Richard Askey, Saunders MacLane,
and V. Frederick Rickey to the conference presentation and of Garrett Birkhoff and Albert
W. Tucker to later drafts of the manuscript. The writing of this paper was stimulated by
my participation in the Princeton Mathematical Community in the 1930s Oral History Proj-
ect, which resulted from the efforts of Charles Gillespie, Frederik Nebeker, and especially
Albert Tucker and from the financial support of the Alfred P. Sloan Foundation.
346
THE EMERGENCE OF PRINCETON, 1896-1939 347
2. A Fine Start
Efforts to establish a research program in mathematics at Princeton
University began in the first decade of the twentieth century at the hands
of Henry Burchard Fine. Fine had completed an undergraduate major in
classics at Princeton in 1880 and remained until 1884, first as a fellow
in experimental physics and then as a tutor in mathematics. The latter
position brought him into contact with George Bruce Halsted, a math-
ematics instructor fresh from his dissertation work under J. J. Sylvester
at Johns Hopkins University.3 Fine wrote of their relationship in a
testimonial:
I am glad of this opportunity of acknowledging my obligations to Dr.
Halsted. Though all my early prejudices and previous training had been
in favor of classical study, through his influence I was turned from the
Classics to Mathematics, and under his instruction or direction almost
all of my mathematical training had been acquired. (Eisenhart 1950,
31-32)
On Halsted's advice, Fine traveled to Leipzig in the spring of 1884 to
study with Felix Klein. Halsted's ability to inspire proved greater than his
ability to teach Fine mathematics, for Klein found Fine to know no Ger-
man and little mathematics. Nevertheless, Fine was encouraged to attend
lectures. He progressed quickly and was awarded the Ph.D. after only
a year for his solution to a problem in algebraic geometry. In the summer
of 1885, and again in 1891, Fine visited Berlin to study with Leopold
Kronecker. Fine's first book and several of his papers are testimony to
the profound influence of Kronecker (Fine 1891, 1892, 1914).
Fine returned to Princeton in the fall of 1885 as an assistant professor
of mathematics with an admiration for the German system, which pro-
vided opportunities for young mathematicians to work closely with estab-
lished researchers. He progressed steadily through the ranks. In 1889 he
was promoted to professor and in 1898 to Dod Professor; by 1900, he
was the senior member of the department. During Woodrow Wilson's
tenure as university president, from 1903 to 1911, Fine's career and his
influence on Princeton mathematics advanced most rapidly.4 Fine was ap-
pointed chairman of mathematics (1904-28), dean of the faculty (1903-12),
and dean of the science departments (1909-28). When Wilson resigned to
run for governor of New Jersey in 1911, Fine served as acting president
of the university until John Grier Hibben was appointed president.
Fine published several research papers in geometry and numerical
348 William Aspray
valed that of the established world centers: Gottingen, Berlin, Paris, Cam-
bridge, Harvard, and Chicago.
3. The 1920s
Although individual members of the mathematics faculty carried on
intensive research activities, Princeton remained principally a teaching in-
stitution in the 1920s. As at most American universities during those years,
the Princeton faculty was saddled with heavy undergraduate teaching loads
and had little money to improve facilities or research opportunities. The
European mathematicians who came to Princeton recognized this clear-
ly. As Einar Hille remembers his first year there, in 1922-23: "Princeton
was somewhat of a disappointment. There were in power old under-
graduate teachers Gillespie, Mclnnes, Thompson. I think that during my
first term there I had two divisions of trigonometry with endless
homework" (Hille 1962). Solomon Lefschetz confirmed this situation:
When I came in 1924 there were only seven men there engaged in
mathematical research.9 These were Fine, Eisenhart, Veblen, Wedder-
burn, Alexander, Einar Hille and myself. In the beginning we had no
quarters. Everyone worked at home. Two rooms in Palmer [Laboratory
of Physics] had been assigned to us. One was used as a library, and
the other for everything else! Only three members of the department
had offices. Fine and Eisenhart [as administrators] had offices in Nassau
Hall, and Veblen had an office in Palmer. (Bienen 1970, 18-19)
The situation began to change around 1924 when an effort was made
to raise funds to support mathematical research. With the turnover in the
preceptorial rank and the disinterest of Wedderburn and others in institu-
tional matters, the responsibility for building the Princeton research pro-
gram devolved to Fine, Eisenhart, and Veblen. The first step was taken
by Veblen during his term in 1923-24 as president of the American Math-
ematical Society.10 In an effort to improve American mathematics national-
ly, he arranged for mathematicians to be included in the National Research
Council fellowship program already established for physicists and chemists.
He also established an endowment fund for the AMS and raised funds
to subsidize its publications.
Within Princeton, the move to improve the research environment was
spearheaded by Fine with the assistance of Eisenhart and Veblen. As dean
of the sciences, Fine assumed responsibility for helping Princeton Presi-
dent Hibben to raise and allocate funds for research in the sciences. In
350 William Aspray
a fund-raising document of 1926, Fine outlined the "means to the full
realization for the purposes of the Mathematics Department":
(1) Endowment for Research Professorships.
(2) Improvement and increase of personnel with schedules compati-
ble with better teaching and more research.
(3) A departmental research fund to meet changing conditions.
(4) A visiting professorship which might well bear the name of
Boutroux [in memorium].
(5) A group of offices and other rooms for mathematical work,
both undergraduate and advanced.
(6) Continued financial support for the Annals of Mathematics.
(7) A number of graduate scholarships. (Fine 1926)
It is instructive to compare this list of objectives outlined by Fine to
a plan for an Institute for Mathematical Research proposed by Veblen
in the period of 1924-26 to both the National Research Council and the
General Education Board of the Rockefeller Foundation. Veblen's plan
not only amplifies on the reasoning behind Fine's list, but also illustrates
the greater vision of Veblen—realized in the 1930s with the founding of
the Institute for Advanced Study.
Veblen's argument began with the premise that "the surest way of pro-
moting such research [in pure science] is to provide the opportunities for
competent men to devote themselves to it" (letter to H. J. Thorkelson,
21 October 1925, Veblen Papers). According to the American system,
Veblen noted, this is a "by-product of teaching. The consequence has been
that although our country has produced a great many men of high abili-
ties, very few of them have an output which corresponds to their native
gifts." Playing to the desire of funding organizations to build strong Amer-
ican research institutions, Veblen added that "men of considerably less
ability have been able to do greater things in the European environ-
ment . . . [because] their time and energy have been free for the prosecu-
tion of their research." Elaborating this argument elsewhere, Veblen noted
that his American colleagues taught nine to fifteen hours a week as com-
pared with three hours by a mathematician in the College de France;11
and that the American mathematician's primary task was the teaching of
elementary subjects to freshmen and sophomores. These subjects were
taught in the lycees and Gymnasia of Europe, and university research
mathematicians there could concentrate on the teaching of more advanced
subjects (letter to Vernon Kellogg, 10 June 1924, Veblen Papers).
THE EMERGENCE OF PRINCETON, 1896-1939 357
had raised the $2 million through alumni gifts, and one-fifth of the to-
tal amount ($600,000) was made available to the mathematics depart-
ment. It was used to buy library materials, support the journal Annals
of Mathematics,12 reduce teaching loads, and pay salaries of visiting
mathematicians.13
Most of the other objectives on Fine's list were also met. Soon after
Fine became involved with the Fund Campaign Committee in 1926, he
approached Thomas Jones, an old friend and former Princeton classmate
who had made a fortune through a Chicago law practice and his presidency
of the Mineral Point Zinc Company. Jones endowed the Fine Professor-
ship, the most distinguished chair in American mathematics at the time.
Together with his niece Gwenthalyn, Jones also provided $500,000 to the
research fund and endowed three chairs, including the Jones Chair in
Mathematical Physics, which was first held by Hermann Weyl in 1928-29.
Princeton was able to provide good financial support for doctoral and
postdoctoral mathematicians in the late 1920s and the 1930s. It attracted
more National Research Council fellows than any other U.S. university.14
British and French students were supported by the Commonwealth and
Procter Fellowship programs and American graduate students by univer-
sity funds.
Of the items on Fine's list, an endowment for research professorships,
a departmental research fund, a visiting professorship, support for An-
nals of Mathematics, and graduate scholarships were all met. Only two
items caused difficulty: increase in personnel and housing. Both needs were
met in the early 1930s.
4. Fine Hall
In the late 1920s, the University of Chicago began construction of
Eckart Hall for its mathematics department.15 Veblen, a Chicago Ph.D.
with continuing ties to his alma mater, kept closely informed about the
new mathematics building.16 He recognized its potential value in nurtur-
ing a mathematics community and "so he worked on Dean Fine to have
this as a goal in connection with the Scientific Research Fund" (private
communication from A. W. Tucker, 1985). Although Fine understood
the need for adequate space (item 6 on his list), he resisted Veblen's ex-
hortations because he knew that money was not available for similar
buildings for the other sciences. Psychology, in particular, Fine regarded
as having a greater space need than mathematics.
354 William Aspray
7. Conclusion
Institutional factors clearly helped to shape the development of the
mathematics programs in Princeton in the 1930s. Careful planning by Fine,
Eisenhart, and Veblen over the preceding quarter-century placed Princeton
in a position to establish a world-class center of mathematics once funds
started to become available in the late 1920s.
Funding for these purposes was fairly easily acquired. It may have been
fortunate that the Fuld and Bamberger families were willing to endow the
Institute in 1930, but it was no accident that Veblen was ready with a strong
plan to build it in Princeton and to devote it to mathematical research.
It is clear that the trend in the 1920s of the great foundations to support
American scholarship benefited Princeton mathematics. However, it must
be remembered that the bulk of the money for the university's program
in mathematics came through alumni gifts. The ease at raising these match-
ing funds is perhaps indicative of the general wealth of the United States
in the late 1920s.
The existing strength of the department and a workable plan for the
future were undoubtedly strong factors in attracting financial support.
Several principles were consistently applied by Fine, Veblen, and others
over the first forty years of the century to build excellence in the depart-
ment and later in the institute. Foremost was the emphasis on research,
as demonstrated in appointments and promotions, training offered to
graduate students, teaching loads, and many other ways. This ran counter
to the well-established tradition of American colleges as undergraduate
teaching institutions. Second was the attempt to build a community of
mathematical researchers so that the "old campaigner" and the "young
recruit" could exchange ideas. Third was the concentration on a few areas
(topology, differential geometry, mathematical physics, and logic), instead
of attempting to provide uniform coverage across all of mathematics.
Fourth was the adoption of an international perspective. More than any
other American university in the period 1905-40, Princeton sought out
students, visitors, and faculty from around the world. When the Nazi peril
disrupted European mathematics, Veblen and Weyl led the way in plac-
ing emigre mathematicians in American institutions, including some in
Princeton (Reingold 1981). Fifth was the decision to build up a research
community through the cultivation of young mathematical talent. Al-
though the institute took great advantage of the Nazi situation in attrac-
ting Einstein, von Neumann, and Weyl to its faculty, most of the Princeton
staff was hired at the junior level and promoted from within.
362 William Aspray
Finally, great attention was given to environmental factors that would
affect the research community. Prominent among these was Fine Hall.
It is hard to overemphasize the importance Princeton mathematicians of
the 1930s attached to their physical quarters. Soon universities through-
out the world came to recognize the value of a place where their mathe-
maticians could gather to discuss mathematics, with excellent support
facilities. Another factor was the editing of professional journals (Annals
of Mathematics and Studies, Journal of Symbolic Logic, and Annals of
Mathematical Statistics) at Princeton. They provided the faculty and their
students with an outlet for research and gave the faculty some control
over the direction of American research. These journals also provided ex-
tensive contacts with the wider mathematical community and a vehicle
for scouting new talent for appointments. Financial support for graduate
students and visitors and for reduced teaching loads of staff also promoted
the growth of a large community focused on mathematical research.
The success in Princeton is even more remarkable when it is considered
that it occurred at the same time as the Great Depression and the growth
of Nazism. General economic circumstances severely depressed academic
salaries, limited funds for graduate and postdoctoral support, and
restricted job placement for Princeton Ph.D.s and junior faculty. The
political disruption of European academics resulted in an influx of Euro-
pean mathematicians into the United States, further straining the appoint-
ment and promotion of American-bred and -trained mathematicians.
Notes
1. Others were even more lavish in their praise of Princeton: R. C. Archibald wrote of
Princeton as "the greatest center of mathematical activity in this country" (Archibald 1938,
169); the Danish mathematician Harald Bohr referred to Princeton as "the mathematical
center of the world" when addressing an international scientific audience in 1936 (Chaplin
1958).
2. This paper about Princeton tells part of a larger story of the emergence of mathematics
research in U.S. institutions in the period 1875-1940. The full story involves the rise of
mathematics at Brown, Chicago, Clark, Johns Hopkins, Harvard, and Yale universities in
the first half of this period and at Berkeley, MIT, Michigan, Stanford, and Wisconsin near
the end. Harvard and Chicago, in particular, have many parallels with Princeton. Some
information on this topic can be found in (Archibald 1938; Birkhoff 1977; Bocher et al.
1911; Lewis 1976; Reid 1976). Dr. Uta Merzbach of the Smithsonian Institution is prepar-
ing a history of American mathematics and mathematical institutions.
In this paper, I have focused on social and institutional issues. I am planning additional
papers on the contributions in the 1930s of Princeton mathematicians to topology and logic.
3. Halsted was widely influential in the early development of American mathematics,
e.g. inspiring Leonard Dickson and R. L. Moore as well as Fine (Birkhoff 1976; Lewis 1976).
4. Fine and Wilson were lifelong friends, having first become acquainted while working
on the editorial board of the student newspaper, the Princetonian.
THE EMERGENCE OF PRINCETON, 1896-1939 363
5. Harvard adopted the preceptorial system in 1910, using it to finance graduate students
and train them to teach, rather than hiring additional junior faculty members.
6. Other distinguished European research mathematicians that came to the United States
in its formative period include J. J. Sylvester at Johns Hopkins, R. Perrault at Johns Hopkins
and Clark, and Oskar Bolza and Heinrich Maschke at Clark and Chicago. The number of
foreign mathematicians willing to accept appointments in the United States was small. Many
of the senior American mathematicians in this period were not distinguished researchers.
Thus, Fine's appointment strategy appears sound.
7. Bliss became a leading figure in the Chicago department and in American mathematics
generally. After two years at the University of Illinois and a year as mathematics depart-
ment chairman at the University of Kansas, Young devoted many years to building up the
mathematics program at Dartmouth.
8. Note the interesting, but perhaps coincidental, pattern of junior appointments at the
university: 1905-10, American, trained elsewhere; 1910-25, mostly European; 1925-40, mostly
American, several having been trained at Princeton. Princeton was not the first experience
with U.S. institutions for some of these young European mathematicians. For example, Wed-
derburn first spent a year at Chicago and Hille a year at Harvard.
9. By the standards of the major European centers of the time, or of major American
universities when Lefschetz made this comment in 1970, seven was not a large number of
research mathematicians for an institution. But few other American universities, if any, had
that large a number in 1924.
10. Veblen's interest in fund-raising dates from after World War I, perhaps stemming
from his wartime administrative experience at Aberdeen Proving Grounds in Maryland.
11. At most American universities in the 1920s, mathematicians taught twelve or more
hours per week. According to Garrett Birkhoff, it was considered a great coup at Harvard
in 1928 when the weekly load for a mathematician was reduced to 4'/2 hours of lectures,
1 Vi hours of theoretical tutoring, and 3 hours of graduate student supervision (private cor-
respondence, 4 October 1985).
12. Princeton had assumed editorial responsibility for Annals of Mathematics in 1911.
Previously it was edited at Harvard, and before that at the University of Virginia.
13. Long-term visitors to the department in the late 1920s and 1930s included: Paul Alex-
androff and Heinz Hopf (1927-28); G. H. Hardy (1928-29); Thornton Fry, John von
Neumann, and Eugene Wigner (1929-30, the last two returning in subsequent years); J. H.
Roberts and J. H. van Vleck (1937-38); and C. Chevalley (1939-40).
14. Veblen was one of three members of the NRC Fellowship selection committee for
mathematics and was thus positioned to assist Princeton mathematics. This arrangement
is characteristic of the organization of American mathematics in the 1920s and 1930s, where
the power was concentrated in a small number of individuals, including G. D. Birkhoff of
Harvard, G. A. Bliss of Chicago, R. G. D. Richardson of Brown, Veblen, and perhaps a
few others.
15. Increased wealth in the United States in the latter 1920s redounded on the univer-
sities. Chicago's ability to construct Eckart Hall, Harvard's reduction of the teaching load,
and Princeton's ease at matching the Rockefeller grant through alumni contributions all
indicate this improved economic condition.
16. Veblen was at the University of Chicago from 1900 to 1905, receiving his Ph.D. in
1903 under the direction of E. H. Moore. Veblen's drive to build up American mathematics
may have been stimulated by his experience at the University of Chicago, where Moore had
built a strong program that produced many prominent research mathematicians, including
L. E. Dickson, G. A. Bliss, G. D. Birkhoff, and Veblen, as well as supplying several major
midwestern universities with their mathematics chairmen.
17. Veblen drew many ideas for Fine Hall from a visit to Oxford in 1928-29. As Savilian
Professor, G. H. Hardy was expected to lecture occasionally on geometry. To avoid this
responsibility, he exchanged positions that year with Veblen, whose principal mathematical
interest was geometry.
18. Physics and mathematics had shared quarters earlier in Palmer Laboratory. Pro-
364 William Aspray
fessor Condon of physics was a supporter and advisor on the planning of Fine Hall; there
are many references to Condon's role in the Oswald Veblen Papers.
19. Weyl reported to his former colleagues in Go'ttingen that German was spoken as much
as English in the institute, then located in Fine Hall (Reid 1976, 157). Garrett Birkhoff reports
that in the latter 1930s the official language of the institute was jokingly said to be "broken
English" (private communication, 24 October 1985).
20. Harvard did have a common room, Room O, in Widener Library, but it did not
have the facilities or receive the use that Fine Hall did.
21. Although Tucker may attach more significance to the environment of Fine Hall than
others might care to, more than twenty mathematicians commented on the amenities of Fine
Hall in the Princeton Mathematics Community of the 1930s Oral History Project.
22. The Conference Board of the Mathematical Sciences used Fine Hall as a model in
a book on mathematical facilities (Frame 1963).
23. The most accessible account is in (Flexner 1940, chap. 27 and 28).
24. Flexner and Veblen held similar views about education and research. Both were
enamored with the European university systems (see Flexner 1930), and both felt a need
"to emphasize scholarship and the capacity for severe intellectual efforts" (Flexner 1927,
10). Both saw a need for an environment where the research faculty would be free from
"routine duties ... —from administrative burdens, from secondary instruction, from distract-
ing tasks undertaken to piece out a livelihood" (Flexner 1930, 10-11). Both saw the research
institute as a "specialized and advanced university laboratory" (Flexner 1930, 35). However,
they disagreed on the addition of schools of study to the institute, support of European
mathematicians, a new institute building off the university campus, administrative respon-
sibilities for institute faculty, and other matters (see Reingold and Reingold 1982, chap. 13).
25. There is little firm evidence of who else was considered or offered an original ap-
pointment at the institute. An offer to G. D. Birkhoff was declined. Harvard countered
the offer by making Birkhoff a Cabot Fellow. The address pages of Veblen's 1932 diary
list five groupings of names that may have been candidates for institute positions or people
whose advice Veblen sought about candidates during a 1932 tour of Europe and North
America. The names (separated by semicolons the way Veblen grouped them) are as follows:
Dirac, Artin, Lefschetz, Morse; Alexandroff, Wiener, Kolmogorov, von Neumann; Albert,
R. Brauer, GOdel, Douglas; Bernays, Peterson, Kloosterman, Heyting, Chapin; Deuring,
McShane, Whitney, Mahler.
26. Von Neumann was teaching half-time at Berlin and half-time at Princeton. After
wavering for a considerable time, the university offered him a full-time position. He chose
the appointment at the institute instead.
27. The 1933-34 directory of members of the mathematics department lists professors
L. P. Eisenhart, W. Gillespie, S. Lefschetz, J. H. M. Wedderburn, and E. P. Wigner; associate
professors H. P. Robertson and T. Y. Thomas (on leave); assistant professors H. F.
Bohnenblust, A. Church, and M. S. Knebelman; and instructors E. G. McShane, J. Singer,
A. W. Tucker, and S. S. Wilks. There were others listed as part-time instructors, research
assistants, and advanced fellows.
28. Additional information about mathematics journals edited at Princeton in the 1930s
and their contributions to the research community can be found in an oral history I con-
ducted with Albert Tucker on 13 April 1984 (Princeton Mathematical Community in the
1930s, Oral History PMC 34).
29. In 1932-33, Eisenhart headed the Princeton administration as dean of the faculty—
between the death of President Hibben and the appointment of President Dodds. Dodds
appointed Eisenhart to replace the physicist Augustus Trowbridge as dean of the Graduate
School and made Robert Root the new dean of the faculty.
THE EMERGENCE OF PRINCETON 1896-1939 365
Bibliography
The following bibliography includes only published and widely distributed printed
materials. Biographies of the many distinguished Princeton mathematicians and physicists
discussed herein are excluded, except those that have been quoted. Bulletin of the American
Mathematical Society, Dictionary of Scientific Biography, Proceedings of the National
Academy of Sciences, and Princeton Alumni Weekly have carried biographies of many of
these mathematicians.
Two fertile archival sources for this research are the Oswald Veblen Papers and the
Princeton Mathematical Community in the 1930s Oral History Project. The Veblen Papers
are held by the Manuscripts Division of the Library of Congress, Washington, D.C. The
Oral History Project records are held by the Seeley G. Mudd Manuscript Library of Prince-
ton University. Copies are on deposit at the American Philosophical Society in Philadelphia
and the Charles Babbage Institute, University of Minnesota, Minneapolis.
Archibald, R. C. 1938. A Semicentennial History of the American Mathematical Society:
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Society.
Bienen, Leigh Buchanan. 1970. Notes Found in a Klein Bottle. Princeton Alumni Weekly
(April 21), pp. 17-20.
Birkhoff, Garrett. 1977. Some Leaders in American Mathematics: 1891-1941. In The Bicenten-
nial Tribute to American Mathematics 1776-1976. ed. J. Dalton Tarwater. Providence,
R. I.: American Mathematical Society.
Bocher, M., Curtiss, D. R., Smith, P. F., and Van Vleck, E. B. 1911. Graduate Work in
Mathematics. Bulletin of the American Mathematical Society 18: 122-37.
Chaplin, Virginia. 1958. A History of Mathematics at Princeton. Princeton Alumni Weekly
(May 9).
Eisenhart, Katherine. 1950. Brief Biographies of Princeton Mathematicians and Physicists.
Unpublished, Princeton.
Eisenhart, L. P. 1931. The Progress of Science: Henry Burchard Fine and the Fine Memorial
Hall. Scientific Monthly 33: 565-68.
Fine, H. B. 1891. The Number System of Algebra Treated Theoretically and Historically.
Boston: Lench, Showall, and Sanborn.
. 1892. Kronecker and His Arithmetical Theory of the Algebraic Equation. Bulletin
of the New York Mathematical Society 1: 173-84.
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Bulletin of the American Mathematical Society 20: 339-58.
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Princeton Fund Committee, Princeton University Press.
. 1927. Calculus. New York: Macmillan.
Fine, H. B., and Thompson, Henry Dallas. 1909. Coordinate Geometry. New York:
Macmillan.
Flexner, Abraham. 1927. Do Americans Really Value Education? The Inglis Lecture. Cam-
bridge, Mass.: Harvard University Press.
. 1930. Universities: American, English, German. New York: Oxford University Press.
. 1940.1 Remember: The Autobiography of Abraham Flexner. New York: Simon and
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Frame, J. Sutherland. 1963. Buildings and Facilities for the Mathematical Sciences.
Washington, D.C.: Conference Board of the Mathematical Sciences.
Grabiner, Judith V. 1977. Mathematics in America: The First Hundred Years. In The Bicenten-
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Hille, Einar. 1962. In Retrospect. Unpublished notes dated 16 May, Yale University Math-
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Lewis, Albert. 1976. George Bruce Halsted and the Development of American Mathematics.
In Men and Institutions in American Mathematics, ed. J. Dalton Tarwater, John T. White,
and John D. Miller. Lubbock, Tex.: Texas Tech Press, Graduate Studies 13, pp. 123-30.
Montgomery, Deane. 1963. Oswald Veblen. Bulletin of the American Mathematical Society
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pp. 111-13.
Reid, Constance. 1976. Courant in GOttigen and New York. New York: Springer-Verlag.
Reingold, Nathan. 1981. Refugee Mathematicians in the United States of America, 1933-1941:
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History, 1900-1939. Chicago: University of Chicago Press.
Richardson, R. G. D. 1936. The Ph.D. Degree and Mathematical Research. American
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Contributors
377
572 Index
Askey, Richard: on hypergeometric series, Bessel functions, 206
208; mathematician as historian, Beth, E. W., 82, 83
37-39, 47-51 passim, 208 Betti numbers, 245
Aspray, William, 46, 47, 48, 50 Bianchi, Luigi, 159
Assessment: epistemic, 311; pragmatic, Bibliographies, value of, 26
312 Bieberbach conjecture, 39, 209
Associationism, 231 Binomial coefficients, 205, 206, 207
Astronomy, 221 Biographies of mathematicians, 24, 28
Automorphic function, 161, 162, 163 Birkhoff, G. D., 348, 363nn.l4, 16,
Axiom: of Choice, 117, 118; of Com- 364n.25
pleteness, 105, 106, 108; of Infinity, Birkhoff, Garrett: on Erlanger Prog-
111, 187; of Reducibility, 113, 115, ramm, 36, 48; at Princeton, 360,
118, 119, 128; of Symbolic Stability, 363n.ll, 364.19; Workshop on the
69; used for systematization, 318 Evolution of Modern Mathematics, v,
"Axiomatic Thinking" (Hilbert), 113 51, 187, 188
Bishop, Errett: and Kronecker, 35, 140,
Babylonians, 265 141; nonstandard analysis, 177, 178,
Bacon, Francis, 332, 334, 338, 341 187-92
Baer, Reinhold, 360 Blackley, G. R., 192
Ball, Sir Robert, 169 Blaschke, W., 168
Bardeen, John, 360 Bliss, Gilbert A., 348, 363nn.7, 14, 16
Bargmann, Valentine, 360 Blumenthal, Otto, 241, 253
Barnes, J. L., 360 B6cher, M., 163, 164
Bateman, G., 170 Bochner, Salomon, 360
Baxter, R. J., 214 Bohnenblust, H. F., 360, 364n.27
Bayes, T., 228 Bohr, Harald, 362n.l
Begriffsschrift (Frege), 42, 101, 102, Bohr, Niels, 286, 288
129n.4, 246 Bolletino di storia della scienze matema-
Bell, E. T., 21, 140, 268 tiche, 26
Belousov-Zhabotinskii chemical reaction, Bolyai, John, 148
282 Bolyai-Lobachevskyan geometry, 241
Beltrami, Eugenio, 147 Bolza, Oskar, 363n.6
Benacerraf, Paul, 13, 14, 15, 248 Bolzano, Bernhard, 141
Bennett, M. K., 36, 48 Bolzano-Weierstrass Theorem, 106
Berkeley, George, Bishop, 185 Boole, George: algebra for logic, 34;
Berlin Akademie der Wissenschaften der Frege's logic, 101, 102, 103,
DDR, 25, 182 129nn.4,8; invariant theory, 146;
Berlin University, 347, 349, 357, 364n.26 logical calculus, 96, 97, 98; Russell's
Bernays, Paul: Habilitationsschrift, 116; logic, 110, 112
logic, 116, 119, 126, 127, 128; at Boolos, G., 16
Princeton, 360, 364n.25 Borel, fimile, 185, 225
Bernoulli, Daniel, 226, 227, 233, 308 Borrelli, G. A., 332
Bernoulli, Jakob, 225, 226, 230, 233, 308 Bossut, Charles, 223
Bernoulli, Nicholas, 226, 228, 233, 308 Bourbaki, N., 201, 240, 278
Bernoulli's theorem, 229 Boutroux, Pierre, 348, 350
Bernstein, Felix, 122 Boyer, Carl B., 21, 27
Bertrand, Joseph, 228, 264 Boyle, Robert, 332
Bertrand Russell Archives, 26 Brauer, Richard, 364n.25
Bessel, Friedrich Wilhelm, 147 Brill, Alexander von, 147
Bessel equation, 213 Brioschi, F., 159
INDEX 373
Dirichlet, Peter Gustav Lejeune: influence Engel, Friedrich, 149, 155-57, 169
of later mathematicians, 41, 241-51; Enlightenment, 40, 221, 333, 341
number theory, 317, 318 Epistemic ends, 304-17
Dirichlet Principle, 167, 241 Epistemology: apriorist, 294, 296; Frege,
Discontinuous group, 145, 158, 159, 162 4; vs. methodology, 274, 275;
Discovery, mathematical, 18 modifications, 298; Quine, 10; of
Discrete mathematics, 284, 287 science, 301, 303; twentieth-century,
Disquisitiones Arithmeticae (Gauss), 240, 16, 17
242, 243 Epsilon-axiom, 119
Dixon, A. C., 204 Epsilon-delta methods, 186
Dixon's sum, 204 Equitable Assurances, 234, 235
Dodds, Harold, 364n.29 Equitable Society for the Assurance of
Dodson, James, 233 Lives, 233
Donaldson, Simon, 287 Erlanger Programm, 25, 36, 145-76
Douglas, Jesse, 364n.25 Euclid: Elements, 238, 261, 262, 265,
Dowker, Hugh, 359 273; geometry, 241, 244, 263; primes,
Dreyfus, Hubert, 339 267; proofs, 268; quantity, 242; rigor,
du Bois-Reymond, Paul, 152 269
Duhem, Pierre, 263-73 passim Euclidean group, 151, 169
Dyck, Walther von, 159, 160, 169 Euclidean plane, 290
Dynamical systems, 287 Euclidean spaces, 288
Eudoxos, 238, 244, 289
East, Edward M., 336 Eugenics, 336, 342
Economics, 40 Euler, Leonhard: calculus, 182, 202;
Eddington, Arthur, 348 "comfortable" series, 207; complex
Edwards, Charles, 178 numbers, 270; Heer, Vorsselman de,
Edwards, Harold, 36-48 passim, 319 214; hypergeometric functions, 206,
Einleitung in die Hohere Geometric 210; intuition, 298; Kline, 212;
(Klein), 153 polyhedra, 264, 268; proof, 295;
Einstein, Albert: general relativity, 171, reconstructed, 178, 180
290; Munich, 160; at Princeton, 357, Euler-Cauchy conjecture, 18
360, 361; special relativity, 170 Eves, Howard, 21
Einstein metrics, 280 Excluded middle, law of, 7, 117
Eisenhart, Luther P., 348-62 passim, Existence: and consistency, 65, 125; of
364nn.27, 29 model, 105, 107; nonconstructive
Elementarmathematik vom HOheren proof, 249-52; "there exists," 117
Standpunkt aus (Klein), 151, 153, 167 Existence proof, 249-50
Elementary Calculus: An Approach Using Expectation, 225, 226, 227
Infinitesimals (Keisler), 189 Experience, 222, 228, 236, 315
Elements (Euclid), 261, 262, 265, 273 Expert systems, 285, 329
ELIZA, 340 Explanation, 17, 18
Elliptic plane, 152
Empiricism: Carnap, 82; Frege, 5, 8; Factual content, 10
Lakatos, 269; mathematical Falsification, 273
methodology, 43, 274; J. S. Mill, 298; Famous Problems in Elementary
Quine, 10, 12; simplistic, 294 Geometry (Klein), 165
Encyclopedic (d'Alembert), 222 Fano, G., 162, 166, 170
Encyklopadie der mathematischen Fee, E., 24
Wissenschaften, 21, 164, 166 Fehr, Henri, 165
End, epistemic, 316, 317, 320 Feigenbaum, Edward, 319, 320, 341
376 Index
Feigenbaum cascades, 281 Fricke, Robert, 162, 164
Feminist historians, 24 Friedman, Michael, 32, 33, 48-50, 287
Fermat, Pierre de, 267, 295 Frobenius, F. George, 140
Fermi, Enrico, 282 Fry, Thornton, 363n.l3
Fiedler, Otto Wilhelm, 162 Fuchs, Lazarus, 161
Field, H., 14, 15, 16 Function: complex-analytic, 241; elliptic,
Fifth Scandinavian Congress of Mathe- 159, 160, 162
maticians, 123 "Function and Concept" (Frege), 101
Fine, Henry Burchard: and F. Klein, Functional analysis, 287
162-68 passim; at Princeton, 46, Fundamental Laws of Arithmetic (Frege),
347-62, 362nn.3, 4, 363n.6 101, 102, 109, 110
Fine Hall, 47, 346, 353-56, 358-62 Funktionentheorie (Courant and Klein), 160
passim, 363n.l7, 364nn.l8, 19, 20, 21, Fuschsian function, 161
22
Finite fields, 240 Galen, 263
First-order languages, 11 Galilei, Galileo, 290
First-order logic, 95-135 Galois, Evariste, 240, 244, 303
Fisher, R. A., 214, 215 Galois's theory, 244
Fiske, Thomas, 348 Gambling, 225, 233
Flexner, Abraham, 356, 357, 364n.24 Gasper, George, 208, 210, 214
Flexner, William, 359 Gauss, K. F.: algebra, 266; complex-
Formalism: Carnap, 83, 87; first-order, analytic functions, 241; cyclotomic
34; Hilbert, 16, 65, 89, 172, 177, 239; equation, 302; Disquisitiones
mathematical methodology, 269, Arithmeticae, 243; Gauss's sum, 204,
274-75; vs. neo-Fregeanism, 12; 213; Heer, Vorsselman de, 214;
Robinson, 186; truth vs. meaning, hypergeometric function, 206, 210;
178, 253 Kronecker, 251; Lagrange, 240;
"Formalism 64" (Robinson), 186 primes, 267; proof, 295; quantity,
Formation rules, 88 242; space 290
Foster, A. L., 360 Gautschi, Walter, 209, 210
Foundations of Arithmetic (Frege), 101 General relativity, 170, 171, 290
Fourier, Joseph, 264, 265, 267, 320 Gentzen, Gerhard, 112
Fourier series, 279 Geometric function theory, 147, 158, 159,
Fourier transforms, 212 160
Fox, Ralph, 359 Geometrie der Dynamen (Study), 169
Fraenkel, Abraham, 124-26, 186, 195, Geometry: analytic, 302; continuity, 244;
257n.33 elliptic, 152; Euclidean, 11; Grass-
Franco-Prussian war, 149 mannian, 169; Hilbert, 106, 107, 108,
Free creation, 249 113, 114, 253; hyperbolic, 152, 158; of
Frege, Gottlob: apriorism, 44, 294; Can- lines, 146, 153, 168, 169; Newtonian,
tor, 194; consistency, 182, 255; 308, 315; parabolic, 152; projective,
Dedekind, 42; first- vs. second-order 146-48, 152, 170, 172; proofs, 300;
logic, 101-29, 129n.7, 254; founda- rigor, 270
tions of mathematics, 320n.l; Hilbert, Geometry of Screws (Ball), 169
239; logicism, 48, 49, 82-94, 245; on- Gibbs, Josiah Willard, 193, 266
tology, 247, 248; Poincar6, 63-69; Gillespie, William, 349, 364n.27
quantifier, 97 Girard, Albert, 270
French Revolution, 333, 342 Givens, Wallace, 360
Frend, P., 264 Global analysis, 281
INDEX 377
McCarthy, John, 46, 331, 338, 339 Montel, Paul, 183, 184
McCorduck, Pamela, 341 Montgomery, Deane, 360
McCoy, Neal, 360 Montucla, Jean-Etienne, 223, 224, 229, 231
McMaster University, 26 Moore, E. H., 172, 363n.l6
McShane, E. J., 360, 364nn.25, 27 Moore, Gregory, 25-51 passim
Meaning: formalism, 178, 186, 187, 253; Moore, R. L., 362n.3
Frege, 9; Hilbert, 254; intuition of, Morgan, William, 235
296; metamathematics, 116; nonstan- Morrey, Charles B., 360
dard analysis, 188, 189, 192; physics, Morse, Marston, 356, 357, 364n.25
255; Quine, 10, 37; stipulative fixing, Mostowski, A., 128
294; truth-in-virtue-of-meaning, 90, 91
Meaningfulness, 10, 37, 85 Nagel, Ernest, 264, 270
Mechanical philosophy, 332 Napier, John, 270
Mechanics, 243, 251 Narasimhan, R., 207
Mechanism, 338 National Research Council, 349-62
Merton, Robert, 22 passim, 363n.l4
Merzbach, Uta, 22, 362n.2 Natural science: belief, 295; mathematics
Metalanguage: Carnap, 83, 89, 92, 93; and, 280-82; mixed mathematics, 224;
Frege, 34; Russell and Whitehead, philosophy of, 17, 18; von Neumann,
111; Wittgenstein, 85 279
Metamathematics: Hilbert, 65, 69, 116; Naturalism, 32, 44, 70, 293-325
Lakatos, 179; LOwenheim, 120; New- Naturalistic fallacy, 274
tonian, 300 Nature of Mathematical Knowledge, The
Metaphysical Foundations of Natural (Kitcher), 264, 270, 316
Sciences (Kant), 290 "Naturerkennen und Logik" (Hilbert),
Metaphysics, 16, 17, 188, 240 254
Metaphysics (Aristotle), 222 Neo-Fregeanism, 16
Metatheory, 65, 69 Netto, Eugen E., 164
Methoden zur Theorie der Ternaerien Neue Geometric des Raumes (Pllicker),
Formen (Study), 169 146
Methodology of mathematics, 271, 274, Neugebauer, O., 202
275 Neumann, Carl, 147, 156, 163
Metric, Laguerre-Cayley-Klein projective, Nevanlinna, Rolf, 183, 184
166 Newell, Alan, 46, 331, 339
Meyer, W. F., 164 Newton, Sir Isaac: analysis, 267, 319;
Michigan, University of, 21, 362n.2 calculus, 289, 299; clarity, 308;
Milin conjecture, 209 hypergeometric series, 207; infinite
Mill, John Stuart, 4, 12, 228, 298, 321n.9 series, 302; infinitesimals, 181, 184;
Minkowski, Hermann, 164, 170, 241, intuition, 298; laws of dynamics, 229;
242, 244 Leibniz, 307; limit, 183; mechanical
Minnesota, University of, 26 cause, 333; proof, 295; transitions
Minsky, Marvin, 331, 339 from, 300; world-machine, 289
Mittag-Leffler, G., 142, 161 Neyman, J., 29
Mixed mathematics, 39, 40, 221-36 Nieuwentijt, Bernard, 185
Mobius, Augustus Ferdinand, 145,149, 172 Noether, Emmy, 244, 245, 317
Modal logic, 13, 15 Noether, Max, 153, 154, 155, 164
Model theory, 36, 187, 254 Nominalism, 11-16 passim
Moderne Algebra (van der Waerder), 317 Non-Euclidean geometry, 147, 148, 150,
Montague, R., 126 263, 312
Montaigne, Michel de, 331, 332 Nonstandard analysis, 177-200, 269
382 Index
Normal families, 183, 184 Peirce, Charles Sanders: abduction, 252;
Number, 243 division of logic, 110; Euclidean
Number systems, 194 geometry, 262; Grand Logic, 100;
Number theory: and computer science, identity, 104, 111, 115; length of ex-
285, 287; epistemic appraisal of, 315; pressions, 127; LOwenheim, 34, 120,
foundation, 62; nineteenth-century 121, 128; quantifier, 97-102, 103, 107,
transformation, 240-53 passim; and set 109; Russell, 112
theory, 319 Peirce, J. M., 163
Numerical methods, 287 Perception, 222, 312
Perceptual knowledge, 314
Omega-axiom, 118 Perrault, R., 363n.6
Omega-order logic, 96, 102, 119, 128 Peter-Weyl theory, 172
Omega-rule, 120, 126 Pfaff, Johann Friedrich, 203, 204
On the Geometric Interpretation of Pfaff-Saalschiitz sum, 203
Binary Forms (Dedekind), 159 Philosophy of language, 17
Ontology: Frege, 12, 247; Hilbert, 255; Philosophy of Natural Science (Hempel),
Parsons, 16; Platonism, 13, 311-13 261
Orwell, George, 335 Philosophy of science, 317
Osgood, W. F., 163 Physics, 221, 301, 310, 316, 319
Ostrowski, O., 168 Picard, Charles Emile, 156, 183
Over en Classe Geometriske Transforma- Plato, 238, 239, 255, 288, 290
tion (Lie), 149 Platonism: difficulties, 314; explanation,
Oxford University, 26, 354, 363n.l7 221; Frege, 11; intuition, 294, 295;
mathematicians, 202; methodology,
P-adic numbers, 194, 195 274-75; ontological, 12, 13, 311;
P-rules, 88 philosophical option, 43; Platonic
Fade, H., 162 realism, 186; progress, 312; truth, 45,
Paradox: largest cardinal, 110; largest or- 187
dinal, 110; Richard, 71, 72, 77; Playfair, John, 264
Russell, 6, 34, 109, 110, 115; set- Plucker, Julius, 145-69 passim
theoretic, 6, 7, 25, 103; Skolem, 123, Poincard, Henri: Cantor, 194; definition
124, 126 of mathematics, 286; Erlanger Pro-
Parallel postulate, 11 gramm, 145, 169, 170; existence, 107;
Parallel processing, 285 Klein, 160, 161; Kronecker, 140;
Paris, University of, 349, 354, 356 logicism, 31, 48, 49, 61-81; philosophy
Paris Academic des Sciences, 231 of mathematics, 6; polyhedra, 268;
Parsons, Charles D., 16, 68, 321n.7 range of knowledge, 357
Pascal, Blaise, 332, 333, 342 Poinsot, Louis, 228, 231
Pasta, John, 282 Poisson, Simeon-Denis, 227, 228, 230,
Peacock, George, 97 231
Peano, Guiseppe: Frege, 129n.8; length P61ya, George, 18, 268
of formulas, 100; linear language, Popper, Karl, 17, 18, 268
101; membership relation, 123; natural Positivism, 11, 13
numbers, 95; Peirce-Schroder tradi- "Possibilities in the Calculus of Rela-
tion, 34; Poincard, 61, 65; Russell and tions, On" (LOwenheim), 121
Whitehead, 112; second-order logic, Post, Emil Leon, 255
127; universal quantifier, 109 Post completeness, 116
Peano Postulates: and first-order logic, Post-Weierstrassian analysis, 310
95, 108, 109, 119, 120; set theory, 296 Posterior Analytics (Aristotle), 265
Peirce, B. O., 163 Prandtl, Ludwig, 165
INDEX 383
manuscripts, 25; vicious circle prin- Skolem, Thoralf: first-order logic, 48,
ciple, 73, 74; Wittgenstein, 86; zigzag 95-129 passim; LOwenheim, 34, 35;
theory, 71 nonstandard analysis, 186; paradox,
123, 124, 126
SaalschOtz, L., 203, 204 Smith, Adam, 334, 341
Saccheri, Gerolamo, 266 Smith, David Eugene, 21, 165
St. Petersburg problem, 40, 225-27, 236 Smith, H. J. S., 269
Saint-Simon, Claude-Henri de Rouvroy, Snapper, Ernst, 360
Comte de, 333 Snyder, V., 164
Sarton, George, 21, 23 Sociology of knowledge, 48
Schonfinkel, M., 122 Soliton theories, 282
Schouten, J. A., 171, 172 Sommer, J., 106
Schroder, Ernst: Axiom of Symbolic Sommerfeld, Arnold, 165
Stability, 69; Bernays, 116; Boole, 97; Sonine, Nikolai J., 212
Frege, 102; infinitary logic, 124; in- Space, 14, 171, 243, 253, 290
finitely long expressions, 127; Space analysis, 266
Lowenheim, 34, 121; Peano, 109; Spherical harmonics, 206
Peirce, 99; Peirce-Schroder tradition, Staatsbibliothek Preussischen
107, 120, 128; quantifiers, 122; Kulturbesitz, 25
Russell, 112; second-order logic, 103 Stanford University, 362n.2
Schroder-Bernstein theorem, 73 Steenrod, Norman, 360
Schur, Fr., 157, 158, 163 Stein, Howard, 41, 42, 48-51
Schwarz, H. A., 142, 159, 162, 164 Steiner, Jacob, 18, 145, 149
Scientific management, 335 Steiner groups, 147
Scientific Revolution, 22, 288, 331 Stetigkeit und irrationale Zahlen
Scopes trial, 336 (Dirichlet), 244
Screw motions, 148, 169 Stolz, Otto, 147, 148
Scripta Mathematica, 21 Stone, Marshall, 360
Searle, John, 339, 340 Streenrod, Norman, 359
Seidel, Philipp L., 36 Strodt, Walter, 360
Semantics: algebraic techniques, 103; con- Structuralism, 14, 15
flate with syntax, 112; definability, 71; Study, Eduard, 145-68 passim
first-order languages, 11; GOdel's In- Sturm, Charles, 147
completeness Theorem, 96; Hilbert, 9; Sullivan, Kathleen, 190, 191
logicism, 10; metamathematics, 120; Superstring theory, 280
quantifiers, 99; second-order logic, Survey of Symbolic Logic, A (C. I.
126; set theory, 129; truth, 127 Lewis), 112
Set theory: arithmetic, 84; Cantor, 11, Siissmilch, Johann, 235
35, 127, 139; emergence, 317; first- Syllogism, 96, 97, 270
order logic, 95-135 passim; founda- Sylvester, J. J., 146, 170, 347, 363n.6
tion, 12, 254, 303; number theory, Symmetry (Weyl), 172
318-19; ontology, 16; rigor, 270; set- Syntax: Carnap, 33, 83-93 passim; vs.
theoretical identifications, 13, 15; metamathematics, 116; vs. semantics,
transfinite, 185; Wittgenstein, 86; 96, 99, 112, 126, 129
Zermelo, 295; Zermelo-Fraenkel, 126 SzegO, G., 208, 211
Siegel, Carl Ludwig, 140
Signs, 82 Tarski, Alfred: model theory, 187;
Simon, Herbert, 46, 331, 339 semantics, 11, 89, 93, 127; truth, 14;
Simpson, Thomas, 232, 264 Upward, LOwenheim-Skolem
Singer, J., 364n.27 Theorem, 95, 128
INDEX 385