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Joseph Kouneiher
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Foundations of
Mathematics and
Physics One Century
After Hilbert
New Perspectives
Foundations of Mathematics and Physics One
Century After Hilbert
Joseph Kouneiher
Editor

Foundations of Mathematics
and Physics One Century
After Hilbert
New Perspectives

123
Editor
Joseph Kouneiher
Nice and Sophia Antipolis University
Nice
France

and

Côte d’Azur University and Lab.

ARTEMIS UMR 7250
(OCA, UCA, CNRS)
Nice
France

ISBN 978-3-319-64812-5 ISBN 978-3-319-64813-2 (eBook)

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In memory of Elie, the other me,
To the twins Ellie and Jezabel
Preface

From 1891 to 1933, David Hilbert gave a series of lectures on the foundations of
mathematics and physics. Those unpublished lectures became available in a
six-volume edition released by Springer Verlag. Hilbert’s lectures and his personal
interactions exercised a profound influence on the development of twentieth-
century mathematics and physics.
In his address to the second International Congress of Mathematicians on
Wednesday, the August 8, 1900, in Paris at the turn of the century, Hilbert began
with the following words [1]:
Wer von uns wüurde nicht gern den Schleier luften, unter dem die Zukunft verborgen liegt,
um einen Blick zu werfen auf die bevorstehenden Fortschritte unsrer Wissenschaft und in
die Geheimnisse ihrer Entwickelung wäahrend der küunftigen Jahrhunderte! Welche
besonderen Ziele werden es sein, denen die fuhrenden mathematischen Geister der kom-
menden Geschlechter nachstreben? welche neuen Methoden und neuen Thatsachen werden
die neuen Jahrhunderte entdecken - auf dem weiten und reichen Felde mathematischen
Denkens?

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a
glance at the next advances of our science and at the secrets of its development during
future centuries? What particular goals will there be toward which the leading mathematical
spirits of coming generations will strive? What new methods and new facts in the wide and
rich field of mathematical thought will the new centuries disclose?

Hilbert then went on to deliver a list of 23 problems for the twentieth century.
The sixth problem is of particular concern for us. Indeed, it is in problem number 6
that Hilbert outlined his program for axiomatizing physics with the intent of putting
it on the same level as axiomatized geometry:
The investigations on the foundations of geometry suggest the problem: To treat in the
same manner, by means of axioms, those physical sciences in which mathematics plays an
important part . . .

If geometry is to serve as a model for the treatment of physical axioms, we shall try first, by
a small number of axioms, to include as large a class as possible of physical phenomena,
and then, by adjoining new axioms, to arrive gradually at the more special theories. At the

vii
viii Preface

same time, Lie’s principle of subdivision can perhaps be derived from the profound theory
of infinite transformation groups. The mathematician will also have to take account not only
of those theories that approach reality, but also, as in geometry, of all logically possible
theories. He must be always alert so as to obtain a complete survey of all conclusions
derivable from the system of axioms assumed.

Hilbert’s work on the foundations of mathematics has its roots in his work on
geometry of the 1890s, culminating in his influential textbook Foundations of
Geometry (1899). Hilbert believed that the properly rigorous way to develop any
scientific subject required an axiomatic approach. Through provision of an axio-
matic treatment, the theory would be developed independent of any need for
intuition, and it would facilitate an analysis of the logical relationships between the
basic concepts and the axioms. Moreover, Hilbert’s view of physics from a
mathematician’s perspective becomes quite explicit in remarks he made regarding
the relationship between physics and geometry. Hilbert regarded geometry as a
genuine branch of mathematics, so it had become mathematized, arithmetized, and
eventually axiomatized, and was no longer subject to experimental examination.
For Hilbert, this development was the proper advancement of science, and not
simply an account of the factual historical development. An advancement should be
furthered wherever possible.
Thus, as early as 1894, in a lecture on geometry that he gave while still in
Königsberg, Hilbert said:
Geometry is a science that essentially has developed to such a state that all its facts may be
derived by logical deduction from previous ones [2, 3].

Later in this lecture, in the course of discussing the axiomatic foundations of

geometry, he presented the axiom of parallels and discussed the alternatives of
Euclidean, hyperbolic, and parabolic geometries. In this context, he remarked:
Now, all other sciences are also to be treated following the model of geometry, first of all
mechanics, but then optics and electricity theory as well [2, 3].

Many of the world’s great scientific truths are based totally upon mathematical
formulation. The extraordinarily results have left the originators obliged to admit to
some mysterious and intimate connection between the physical world and its
abstract mathematical counterpart. To quote Einstein himself:
Here arises a puzzle that has disturbed scientists of all periods. How is it possible that
mathematics, a product of human thought that is independent of experience, fits so
excellently the objects of physical reality? [4].

These quotations demonstrate that, while the fields of mathematics and physics
were considered separate, there was still a strong conjunction between them. The
great upheavals in Physics in the first quarter of the twentieth century only deep-
ened the relation between physics and mathematics. In his stunning 1931 paper (in
which he predicted the existence of three new particles), Dirac was both eloquent
and exuberant at the very outset:
Preface ix

The steady progress of physics requires for its theoretical formulation a mathematics that
gets continually more advanced … What, however, was not expected by the scientific
workers of the last century was the particular form that the line of advancement of the
mathematics would take, namely, it was expected that the mathematics would get more and
more complicated, but would rest on a permanent basis of axioms and definitions, while
actually, the modern physical developments have required a mathematics that continually
shifts its foundations and gets more abstract … It seems likely that this process of
increasing abstraction will continue in the future . . . [5].

Around the same time, Einstein expressed similar sentiments:

Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of
mathematical simplicity. It is my conviction that pure mathematical construction enables us
to discover the concepts and the laws connecting them which give us the key to the
understanding of the phenomena of Nature. Experience can, of course, guide us in our
choice of serviceable mathematical concepts; it cannot possibly be the source from which
they are derived; experience, of course, remains the sole criterion of the serviceability of a
mathematical construction for physics, but the truly creative principle resides in mathe-
matics. In a certain sense, therefore, I hold it to be true that pure thought is competent to
comprehend the real, as the ancients dreamed [6].

Concerning the atomic physics and the fact that Quantum Mechanics, using
radically new concepts, such as the linear superposition of states and the uncertainty
principle, required an entirely new mathematical framework, Dirac wrote:
Quantum mechanics requires the introduction into physical theory of a vast new domain of
pure mathematics -the whole domain connected with non-commutative multiplication.
This, coming on top of the introduction of the new geometries by the theory of relativity,
indicates a trend which we may expect to continue. We may expect that in the future,
further big domains of pure mathematics will have to be brought in to deal with the
advances in fundamental physics [7].

Mathematics and Physics: A Common Matter?

Since Hilbert, conferences, physics, and mathematics have experienced great

upheavals, with new ideas invading the two areas of study. Several ideas from
physics have allowed for a better understanding of certain mathematical problems
and their resolution. Indeed, over the past 50 years, a new type of interaction has
taken place, as has happened frequently in the past, in which physicists, while
exploring their new and still speculative theories, have stumbled across a whole
range of mathematical discoveries.
The physicists’ approach was derived by physical intuition and heuristic argu-
ments, which are beyond the reach, as yet, of mathematical rigour, but which have
withstood the tests of time and alternative methods. The impact of these discoveries
on mathematics has been profound and widespread. Areas of mathematics such as
topology and algebraic geometry, which lie at the heart of pure mathematics and
appear very distant from the physics frontier, have been dramatically affected.
x Preface

This development has led to many hybrid subjects, such as topological quantum
field theory, quantum cohomology, and quantum groups, which are now central
topics of research in both mathematics and physics. Remarkably, modern physical
constructions such as quantum field theory and string theory, which are very far
removed from everyday experience, have proven to be a similarly fertile setting for
mathematical problems. Indeed, in many ways, quantum theory has turned out to be
an even more effective framework for mathematics than classical physics. Particles
and strings, fields and symmetries, they all have a natural role to play in mathe-
matics. Understanding this is the great problem of our time.

Truth, Depth, and Beauty

Many mathematicians and physicists share the view that the beauty of mathematics
is a guide toward a theory whose coherence and simplicity aids our comprehension
of nature. Beauty is what guides the mathematician, while the physicist searches for
truth, consistent with experiment. The mystery of the effectiveness of mathematics
in fundamental physics is much deeper than just the miracle of its astonishing utility
[9]. We aim to describe the microscopic laws in terms of simple mathematics, but,
as we probe deeper, to microscopic scales, we require deeper mathematical struc-
tures. But beyond that, these mathematical structures are not just deep—they are
also interesting, beautiful, and powerful. As Dirac put it:
It seems to be one of the fundamental features of nature that fundamental physical laws are
described in terms of great beauty and power and, as time goes on, it becomes increasingly
evident that the rules that the mathematician finds interesting are the same as those that
Nature has chosen [10].

On the relation between mathematics and Nature, Hermann Weyl wrote:

There is inherent in nature a hidden harmony that reflects itself in our minds under the
image of simple mathematical laws. That then is the reason why events in nature are
predictable by a combination of observation and mathematical analysis. Again and again in
the history of physics, this conviction, or should I say this dream, of harmony in nature has
found fulfillments beyond our expectations [11, 12].

To appreciate mathematical beauty may require, as in music, extensive education

and training, and it is always a subjective judgment. Nonetheless, there tends to be a
large degree of consensus among mathematicians and physicists that the beautiful
parts are those that explain the forces of nature as arising from principles of
symmetry.
These are beautiful to physicists since, from a simple principle of symmetry, we
deduce, in an almost unique fashion, via gauge theories, the nature of the funda-
mental forces and the existence of the carriers of these forces. The ugly parts are
those that describe the strange spectrum of matter, which does not appear to follow
from any symmetry principle. To agree with experiment, one requires far too many
parameters to be put in by hand. Einstein’s dream was that the ugly should be made
Preface xi

beautiful, and that geometry should totally unify spacetime and matter. This is the
task for us all, a task that may yet take the whole twenty-first century and beyond.
Supported by prominent scientists in mathematics and physics, this book cele-
brates the centenary of Hilbert’s work on the foundations of physics and mathe-
matics, and explores the rich new perspectives resulting from the deep interplay
between mathematics and physics during the twentieth century. The result is a
broad journey through the most recent developments in both mathematics and
physics.
In mathematics, the journey takes us through differential and algebraic geometry,
to topology, noncommutative geometry, and twistor theory.
In physics, the journey takes us through gauge and quantum field theories to
string theory and quantum gravity.

Edinburgh, UK Michael Atiyah

2018 Joseph Kouneiher

References

1. D. Hilbert, Mathematische Probleme. Vortrag, gehalten auf dem internationalen

Mathematiker-Kongress zu Paris 1900. Königliche Gesellschaft der Wissenschaften zu
Göttingen. Mathematisch-physikalische (Klasse, Nachrichten, 1900), pp. 253–297
2. Niedersächesische staats und Universitätsbibliothek Göttingen Handschriftenabteilung, Cod.
Ms. Hilbert 541, p.7; Tilman Sauer, The Relativity of Discovery: Hilbert’s First Note on the
Foundations of Physics. arXiv:physics/9811050
3. D. Hilbert, Uber die Grundlagen der Geometrie (Gottinger Nachrichten, 1902), pp. 233–241
4. A. Einstein, Sidelight on Relativity, Forgotten Books ed., (1921), www.forgottenbooks.com
5. P.A.M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133,
p. 60
6. A. Einstein, On the Method of Theoretical Physics, Philosophy of Science, vol. 1, no. 2, (The
University of Chicago Press on behalf of the Philosophy of Science Association, 1934),
pp. 163–169. http://www.jstor.org/stable/184387
7. P.A.M. Dirac, The Relation between Mathematics and Physics, Proceedings of the Royal
Society (Edinburgh) vol. 59, 1938–39, Part II pp. 122–129, 1938–39.
8. M. Atiyah, in Hermann Weyl Biographical Memoirs, vol. 82, Nat. Acad. Sci. 2002.
9. D. Gross, Mathematics and the Sciences, Proc. Nati. Acad. Sci. vol 85, (USA, 1988),
pp. 8371–8375
10. P.A.M. Dirac (1939) Proc. R. Soc. Edinburgh Sect. A 59, p. 122.
11. H. Weyl, Philosophy of Mathematics and Natural Science (Princeton Press, 1949)
12. J. Kouneiher, Symmetry and Cohomological foundations of Physics (Towards a new
Philosophy of Nature, Hermann Sciences editions, 2010)
Acknowledgements

First of all, I would like to express my deep gratitude to all my friends contributors,
who kindly accepted to be part and encouraged the project of this special book,
which will remain a reference in the domain of mathematics and mathematical
physics: Michael Atiyah, Jeremy Attard, Jeremy Butterfield, Ali Chamsddine, Alain
Connes, Leo Corry, Jordan François, Misha Gromov, Sebastian de Haro, Serge
Lazzarini, Colin MacLarty, Matilde Marcolli, Thierry Masson, Roger Penrose, Lee
Smolin, John Stachel, and Edward Witten.
I wish to thank all my friends from the group geometry and physics for the many
years of collaborations and discussions on mathematical physics issues and which
found echo through the lines of the contribution “Where We stand today”: Frédéric
Hélein, Daniel Bennequin, Volodya Robtsov, Paul Baird, Franz Pedit, and Cécile
Barbachoux.
I would like to express my very great appreciation for the fruitful encounters and
discussions with my friends: Abhay Ashtekar, Robert Dijkgraaf, Carlo Rovelli,
Alain Herreman, Jean Michel Alimi, Newton da Costa, Michel Paty, Jean Jacques
Szczeciniarz, Dominique Lambert, Jean Luc Gautero, and Raffaele Pizano.
I want to extend my thanks to Salvatore Capozziello for his encouragement to
undertake this project. I want to warmly thank also Nary-Catherine Man and Michel
Boer from Artemis for their support.
I would like to thank Springer editions for their supports. I am particularly
grateful to Kirsten Theunissen for her assistance and almost daily encouragement.
Special thanks to Aldo Rampioni, for his help and collaboration from the beginning
of the project of the book.
I wish to acknowledge the help and support provided by Cécile Barbachoux
during the preparation of the manuscript. Special thanks should be given to my
students for their active interactions during the elaboration of the book. I’m greatly
indebted to my family for their indulgence.

xiii
xiv Acknowledgements

Finally, I wish to express my sincere gratitude to Michael Atiyah, Alain Connes,

Ali Chamsddine, Misha Gromov, Roger Penrose, and Edward Witten, for the long
e-mails exchanges, encouragements, and support and especially the daily exchange
with Michael about mathematics and physics and our heroes Hermann Weyl, James
Clerk Maxwell, David Hilbert, Albert Einstein, and others.
Contents

Where We Stand Today . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Joseph Kouneiher
Mie’s Electromagnetic Theory of Matter and the Background to
Hilbert’s Unified Foundations of Physics . . . . . . . . . . . . . . . . . . . . . . . . 75
Leo Corry
Hilbert and Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Joseph Kouneiher and John Stachel
Grothendieck’s Unifying Vision of Geometry . . . . . . . . . . . . . . . . . . . . . 107
Colin McLarty
Understanding the 6-Dimensional Sphere . . . . . . . . . . . . . . . . . . . . . . . . 129
Michael Atiyah
A Dozen Problems, Questions and Conjectures About Positive
Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Misha Gromov
Geometry and the Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Alain Connes
What Every Physicist Should Know About String Theory . . . . . . . . . . . 197
Edward Witten
Quanta of Space-Time and Axiomatization of Physics . . . . . . . . . . . . . . 211
Ali H. Chamseddine
Twistor Theory as an Approach to Fundamental Physics . . . . . . . . . . . 253
Roger Penrose
What Are We Missing in Our Search for Quantum Gravity? . . . . . . . . 287
Lee Smolin

xv
xvi Contents

A Schema for Duality, Illustrated by Bosonization . . . . . . . . . . . . . . . . . 305

Sebastian De Haro and Jeremy Butterfield
The Dressing Field Method of Gauge Symmetry Reduction, a Review
with Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
J. Attard, J. François, S. Lazzarini and T. Masson
Syntactic Phylogenetic Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Kevin Shu, Sharjeel Aziz, Vy-Luan Huynh, David Warrick
and Matilde Marcolli
Summary

The project of this book is the result of the desire of prominent scientists in
mathematics and physics to, first, celebrate the centenary of Hilbert’s work on the
foundations of physics and mathematics, and second, to explore the rich new
perspectives resulting from the deep interplay between mathematics and physics
during the twentieth century. The papers published in this volume provide insight
into their works, and analyze the impact of the breakthrough and the perspectives
of their own contributions.
In his contribution, Leo Corry describes the motivations of Hilbert’s Unified
Foundations of Physics. He presents the two main pillars on which Hilbert’s built
his own theory as presented in Göttingen in November 1915. To do so, he discusses
the contents of Mie’s electromagnetic theory of matter and explains the context in
which the theory needs to be understood as part of contemporary debates on
gravitation in which also Einstein took part. He reviewed also the way in which
Max Born mediated between Mie and Hilbert by presenting the formers work in a
way that would be amenable to Hilbert’s current scientific interests. Finally, he
gives a brief account of Hilbert’s talk of November 1915 and explains its contents
against the background of the ideas of Gustav Mie’s electromagnetic.
The common paper of Kouneiher and Stachel highlights the twenty-odd-year
relationship between Einstein and Hilbert and traces the relationship between the
two men during this period in the form of encounters, each of which characterizes a
particular aspect of their relationship and their impact on the final form of Einstein
equation and the way to derive it.
Colin Maclarty’s contribution is a fine analysis of Grothendieck’s vast unifying
vision which provided new working and conceptual foundations for geometry, and
even led him to logical foundations. Maclarty clarifies Grothendieck attitude of
mind by favoring the words and commutative diagrams over pictures and refusing
to think the geometry pictorially. This contrasts with the mainstream attitude where
the majority prefer the pictures as illustrations of the geometry. We can see how, by
using such approach, Grothendieck construct his mathematical and geometrical
universe of topos and schemes.

xvii
xviii Summary

In his contribution A Dozen Problems, Questions and Conjectures about

Positive Scalar Curvature (SC > 0), Gromov invites us to explore and discover new
paths and challenges us to open windows that give access to new facts in geometry
of the domain (SC > 0). One of the beauties of the author’s approach to geometry is
his gritty hands-on method, dealing with basic concepts that one could explain to a
nonexpert, rather than heading on a long trek of increasing abstraction. It is at the
same time simple, but extraordinarily difficult to put into practice, and of course
requires great insight to know where to look for answers. The author deals primarily
with the difficult domain of positive scalar curvature. The paper reminds us that in
spite of the remarkable advances in Riemannian geometry in recent years, there is
still a wealth of fundamental unresolved problems.
In his contribution, Michael Atiyah presents a proof of a long-standing con-
jecture concerning the six-dimensional sphere and the possibility to have a complex
structure. To do so, he uses two aspects: a hypothetical complex structure without
any symmetry assumptions and he considers the case of conformal sphere S6 and
not the round sphere S6. More precisely, he uses the fact that S6 is a homogeneous
space of the conformal group Spiny(7; 1), which preserves future and past. The
proof is a master class move in mathematics. The author suggests also to shed new
light on many problems of physics: In the future I expect these ideas will provide a
different perspective, with substantial benefits in all areas.
The subject of Edward Witten’s contribution1 is one of the big breakthrough
ideas in mathematical physics in the twentieth century, the strings theories, with a
new paradigm based on the conception of elementary particle as one-dimensional
string.2 The aim of this contribution is to describe the minimum that any physicist
should know about string theory, focusing on a few basic questions. How does
string theory generalize standard quantum field theory? Why does string theory
force us to unify general relativity with the other forces of nature, while standard
quantum field theory makes it so difficult to incorporate general relativity? Why are
there no ultraviolet divergences in string theory? And what happens to Einstein’s
conception of spacetime? For instance, as we know, in general, a string theory
comes with no particular spacetime interpretation. The spacetime M emerges
through the link between its metric tensor GIJ(X) and a particular 2D conformal
field theory. That is the only way that spacetime entered the story.
Witten tries to answer all these questions with clarity and simplicity whenever it
is possible.
In his contribution, Roger Penrose Twistor Theory as an Approach to
Fundamental Physics describes the original motivations underlying the introduction
of twistor theory which has been pioneered by him and others since the 1960s. The
primary objective of twistor theory originally was—and still is—to find a deeper
route to the working nature; so the theory should provide a mathematical

1
An earlier and restraint version of this paper was published in physics today.
2
Later on, people understood that some objects as Branes play also an essential and fundamental
role in the theory conception.
Summary xix

framework with sufficient power and scope, to help us toward resolving some of the
most obstinate problems of current physical theory. One of the original motivations
was to unify general relativity and quantum mechanics in a nonlocal theory based
on complex numbers. The application of twistor theory to differential equations and
integrability has been an unexpected spin-off from the twistor program.
In capturing both relativity and quantum mechanics, twistor theory demands
some modifications of both. For example, it allows for the introduction of nonlinear
elements into quantum mechanics, which are in agreement with some current
interpretations of the measurement process: The collapse of the wave function
contradicts the principle of unitary time evolution, and it has been proposed that this
failure of unitarity is due to some overtaking nonlinear gravitational effects. The
main two ingredients of twistor theory are non-locality in spacetime and analyticity
(holomorphy) in an auxiliary complex space, the twistor space.
Alain Connes’s contribution has the ambition of answering the questions posed
by the divers temptations to create a theory founded on the principle of quantum
mechanics and general relativity and which goes beyond their limit to integrate the
gauge theories and matter. The point of view adopted in this essay is to try to
understand from a mathematical perspective, how the perplexing combination
of the Einstein–Hilbert action coupled with matter, with all the subtleties such as the
Brout–Englert–Higgs sector, the V-A and the see-saw mechanisms, etc., can
emerge from a simple geometric model. The new tool is the spectral paradigm, and
the new outcome is that geometry does emerge on the stage where quantum
mechanics happens, i.e., Hilbert space and linear operators. In his contribution,
Alain Connes introduce the noncommutative geometry and the spectral paradigm
developed by the author since 1980s. It is based on the Hilbert space formalism of
quantum mechanics and on mathematical ideas coming from K-theory and index
theory. This new paradigm of geometry provided a new perspective on the geo-
metric interpretation of the detailed structure of the standard model and of the
Brout–Englert–Higgs mechanism.
With Ali Chamseddine, they understood that they could obtain the full package
of the Einstein–Hilbert action of gravity coupled with matter by a fundamental
spectral principle. In the language of NCG, this principle asserts that the action only
depends upon the “line element”, i.e., the inverse of the operator D. The presence
of the other fields forces, due to renormalization, the addition of higher derivative
terms of the metric to the Lagrangian and this in turns introduces at the quantum
level an inherent instability that would make the universe blow up. The approach
used in this contribution is based on the idea of “particle picture” for geometry,
allowing to stay very close to the inner workings of the standard model coupled to
gravity.
Ali Chamsddine’s contribution forms a logical continuation to Alain Connes’s
one in this volume. Notice that all the material covered in Chamsddine review is a
result of a long-time collaboration with Alain Connes. The author shows that
starting with the axioms of noncommutative geometry supplemented by a minimal
number of physical assumptions would result, unambiguously, in a unified theory
of all fundamental interactions and matter content of spacetime. And so they will be
xx Summary

able to establish a link between the quantization of volume of space at Planck

energy and the constituents of matter and their symmetries. In addition, he uncovers
the origin of the Higgs fields and symmetry breaking, and indicates possible
solutions to long-standing problems such as resolving the singularities in GR, dark
matter, and dark energy.
The author of What are we missing in our search for quantum gravity?, Lee
Smolin, starts his contribution by considering the various approaches to quantum
gravity and asks the question why, in spite of many efforts, we have not yet found
the true theory of quantum gravity. He makes a succinct analysis of the causes
of the failures of different approaches and suggests to consider them as different and
complementary models of a single theory to be found by a common effort and an
explicit choice of a scientific approach based on a more general physical principle.
The idea is that, in the absence of a real experience at the Planck scale to guide us
and inspired by various developed models, we can at least get back and make some
reflections on what we may be missing in our search for quantum gravity.
In their contribution, Attard, François, Lazzarini, and Masson propose a review
of gauge theory, one of the most profound breakthrough ideas of twentieth century
in mathematical physics. More precisely, they describe another way to perform
gauge symmetry reduction which they call the dressing field method. It is for-
malized in the framework of the differential geometry; it has a corresponding BRST
differential algebraic formulation. The method boils down to the identification of a
suitable field in the geometrical setting of a gauge theory that allows to construct
partially of fully gauge invariant variables out of the standard gauge fields. This
formalizes and unifies several works and approaches which found origins in Dirac’s
pioneering works.
Butterfield and De Haro’s contributions propose a schema to understand duality
between models in physics. Notice that the idea of the duality is one of the chal-
lenging ideas of the twentieth-century physics and mathematics. This paper is
written for physicists as well as for philosophers of sciences. The approach uses a
formalization of the notions of theories, models and what the mean of a duality in
this framework. Discussions are given to illuminate some crucial points of this
formal approach. The main proposed example concerns the duality known as
“bosonization”, which establishes an equivalence between two physical models:
one based on bosons and the other on fermions. The authors insist, on one hand, on
the fact that this boson duality, by contrast with other dualities in physics, is exact
and on the other hand its role in both cases of isomorphic and non-isomorphic
models.
In their contribution “Syntactic Phylogenetic Trees”, M. Marcolli, K. Shu,
S. Aziz, V. Y. Huynh, and D. Warrick try to apply some methods that came from
mathematics and computational methods developed in the context of mathematical
biology in the linguistic domain. They start by identifying several serious problems
that arise in the use of syntactic data from the SSWL database for the purpose of
computing phylogenetic trees of language families in the context of the field of
historical linguistics.
Summary xxi

They show that the most naive approach fails to produce reliable linguistic
phylogenetic trees and identifies some of the sources of the observed problems.
They describe also how the use of phylogenetic algebraic geometry can help in
estimating to what extent the probability distribution at the leaves of the phylo-
genetic tree obtained from the SSWL data can be considered reliable, by testing it
on phylogenetic trees established by other forms of linguistic analysis. They remark
that after restricting to smaller language subfamilies and considering only those
SSWL parameters that are fully mapped for the whole subfamily, the SSWL data
match extremely well-reliable phylogenetic trees, according to the evaluation of
phylogenetic invariants. This is a promising sign for the use of SSWL data for
linguistic phylogenetics, which was their first motivation.
Where We Stand Today

Joseph Kouneiher

1 Hilbert and the Foundations of Mathematics and Physics

In his work of 1918, Hermann Weyl extended the general theory of relativity, which
Albert Einstein had set forth in the years 1915–1916, to unify the two field phenomena
known at this time, namely those described by electromagnetic and gravitational
fields. But more was at stake. At the beginning of the paper in which Weyl worked
out the mathematical foundations of the theory, he observed that:
According to this theory, everything real, that is in the world, is a manifestation of the
world metric; the physical concepts are no different from the geometrical ones. The only
difference that exists between geometry and physics is, that geometry establishes, in general,
what is contained in the nature of the metrical concepts, whereas it is the task of physics
to determine the law and explore its consequences, according to which the real world is
characterized among all the geometrically possible four-dimensional metric spaces. [124]

This work sounds like an echo of a work undertaken by Hilbert many years earlier.1
Indeed, when Hilbert started studying the analysis of integral equations, he sought
to achieve Poincaré’s program unifying different aspects of mathematical analysis
and physics. For him, the aim was to expose and simplify the known results, just like
“formal” physicians such as Dirac, who sought to give physics a coherent mathemat-
ical basis. It was on this occasion that he developed the theory of quadratic forms to
an infinite number of variables, work that would later lead to the birth of “Hilbert
spaces” (and to spectral theory), and that consists in interpreting equations as terms
of linear transformation of this space. He permitted, via his theory of spaces, a new

1 See Corry’s contribution in this volume.

J. Kouneiher (B)
Côte d’Azur University and Lab. ARTEMIS UMR 7250 (OCA, UCA, CNRS),
Nice, France
e-mail: Joseph.KOUNEIHER@unice.fr
J. Kouneiher
Nice and Sophia Antipolis University, Nice, France

© Springer International Publishing AG, part of Springer Nature 2018 1

J. Kouneiher (ed.), Foundations of Mathematics and Physics One Century After Hilbert,
https://doi.org/10.1007/978-3-319-64813-2_1
2 J. Kouneiher

“geometrization” of physics, thanks to the invention of a new geometry that J. Von

Neumann and F. Riesz axiomatized, and that became a powerful tool in mathematical
physics.
In fact, Hilbert’s approach concerning unification in physics can be seen as a problem
of finding a consistent and satisfactory mathematical unification of the gravitational
and electromagnetic fields, be it through modified field equations, a modification of
the space-time geometry, or by increasing the number of space-time dimensions.
Hilbert’s view of physics from a mathematician’s perspective becomes quite explicit
in remarks he made regarding the relationship between physics and geometry. Hilbert
regarded geometry as a genuine branch of mathematics, so it had become mathema-
tized, arithmetized and eventually axiomatized [59], and was no longer subject to
experimental examination. However, originally, geometry was a natural science.
Hilbert’s work on the foundations of mathematics and physics has its roots in his work
on geometry from the 1890s, culminating in his influential textbook Foundations
of Geometry (1899). Hilbert believed that the properly rigorous way to develop
any scientific subject required an axiomatic approach. In providing an axiomatic
treatment, the theory would be developed independent of any need for intuition, and
it would facilitate an analysis of the logical relationships between the basic concepts
and the axioms.
Thus, as early as 1894, in a lecture on geometry that he gave while still in Königsberg,
Hilbert said:
Geometry is a science that essentially has developed to such a state that all its facts may be
derived by logical deduction from previous ones [58, 108].

Later in this lecture, in the course of discussing the axiomatic foundations of geom-
etry, he presented the axiom of parallels and discussed the alternatives of Euclidean,
hyperbolic and parabolic geometries. In this context, he remarked:
Now, all other sciences are also to be treated following the model of geometry, first of all
mechanics, but then optics and electricity theory as well [58, 108].

According to Hilbert, physics was but a four-dimensional pseudo-geometry,

whose metric was connected, via his theory, to electromagnetic quantities, i.e., to
matter. And with this knowledge, an old geometric problem could now be solved:
whether and in what sense Euclidean geometry - about which we only know from
mathematics that it is a logically consistent structure - is also valid in reality. After
discussing Gauss’s inability to verify empirically a non-Euclidean physics through
angle measurement in a large triangle, Hilbert talked about how the physics of Ein-
stein’s general theory of relativity had a totally different relationship to geometry.
The new physics started neither with Euclidean nor with any other fixed geometry
in order to deduce the actual laws of physics. Instead, general relativity yielded, in
one blow, the laws of geometry and physics through one and the same Hamiltonian
principle, i.e., through the fundamental equations of his theory. Hilbert’s conclusion
was:
Where We Stand Today 3

Euclidean geometry is an action-at-a-distance law alien to modem physics: while the theory
of relativity rejects Euclidean geometry as a general presupposition for physics, it teaches
furthermore that geometry and physics are of a similar kind and rest, as one science (Wis-
senschaft), on a common foundation.

Weyl commented Hilbert’s work on general relativity and unified field theories,
noting that: “Hopes in the Hilbert circle ran high at that time: the dream of a universal
law accounting both for the structure of both the cosmos as a whole and of all the
atomic nuclei seemed near fulfillment.”
So, the idea of unification for Weyl and his contemporaries was understood not merely
as a synthesis of the electromagnetic and gravitational fields, but also as a unification
of geometry and physics and as the quest for a universal world law accounting for
the structure of both cosmos and matter.
Later in the mid-1940s Weyl felt that it was insufficient to unite electromagnetism and
gravitation, and that quantum and nuclear phenomena had to be taken into account as
well. By focusing on the unification of the physical fields known at that time, Weyl
continued the re-definition of the aims of the unification project that he had begun in
the late 1920s with the advent of the new quantum mechanics. As the above passage
from the Hilbert obituary shows, in the mid-1940s Weyl would not even discuss the
union between geometry and physics that seemed so attractive in the 1910s.

Recall that the mathematical scene in Gottingen2 in the first decade of the twentieth
century was dominated by Felix Klein (1849–1925), David Hilbert (1862–1943), and
Hermann Minkowski (1864–1909). Klein described the spirit that dominated at that
time :

Speaking, as I do, under the influence of our Göttingen traditions, and dominated somewhat,
by the great name of Gauss, I may be pardoned if I characterize the tendency outlined in these
remarks as a return to the general Gaussian program. A distinction between the present and
the earlier period evidently lies in this: that what was formerly begun by a single mastermind,
we now must seek to accomplish through united efforts and cooperation. [65]

As we know, many contemporary mathematicians envisaged a unified science

at the time. Felix Klein’s History of the Development of Mathematics in the 19th
century3 , Kaluza, Einstein, Weyl and others are examples of this [109].

Note that Hilbert’s perspective on the mathematical sciences as an integrated

whole can be seen as an attempts to come to grips with the philosophical implica-
tions of an ever-increasing specialization in the natural sciences. So, by invoking
the axiomatic method (im Sinne der axiomatischen Methode’), Hilbert was calling
attention to a specifically epistemological method of investigation of mathematical

2 Inthe early twentieth century, Göttingen was the location of an exceptionally vibrant community
within which a belief in the mathematical comprehensibility of nature was widespread and facilitated
very free exchanges between mathematicians.
3 This can also be seen as a most interesting attempt to understand the inner organic unity of the

corpus of mathematical knowledge [66].

4 J. Kouneiher

theories (including those of physics) that he pioneered, and which he saw as being
closely tied to the nature of thought itself [10]. Therefore, this term implicates more
than a merely typical mathematical concern with the rigorous, explicit statement
of a theory; it also connotes a specifically logical and epistemological method of
investigation for deepening the foundations of the theory.
In Hilbert’s mind, this is tributary to how cognition arises from the distinct sources
of intuition, concepts and ideas. Therefore, the axiomatic method is conceived as
a logical analysis that begins with certain facts’ presented for our finite intuition
or experience. Indeed, both pure mathematics and natural science alike begin with
facts’, i.e., singular judgments about something already given to us in representation
(in der Vorstellung): “certain extra-logical discrete objects that are intuitively present
as an immediate experience prior to all thinking”.
In his 1930 paper entitled Knowledge of Nature and Logic’, Hilbert commented
on how modern science had led to the judgment that Kant had far overestimated
the role and extent of a priori elements in cognition, and carried on to endorse the
conception of such elements as nothing more and nothing less than a basic point
of view (Grundeinstellung) or expression of certain unavoidable preconditions of
thinking and experience’. He concluded that what remains of Kant’s synthetic a
priori is just this a priori intuitive point of view’ that is presupposed in all theoretical
concept construction in mathematics and physics. But Hilbert stressed that this was
in full agreement with the basic tendency of Kantian epistemology:
Thus, the most general and fundamental idea of Kantian epistemology retains its signifi-
cance: namely, the philosophical problem of determining that a priori intuitive point of view
(jene anschauliche Einstellung a priori) and thereby of investigating the conditions of the
possibility of all conceptual knowledge and of all experience.

So, through this citation, we discover Hilbert’s conviction of the existence of a third
source of cognition (Erkenntnisquelle) outside of deduction and experience, what he
called the “a priori intuitive viewpoint”. Hilbert describes this intuitive viewpoint
(anschauliche Einstellung) as an a priori insight . . . that the applicability of the
mathematical way of reflection over the objects of perception is an essential condition
for the possibility of an exact knowledge of nature’, an epistemological position,
Hilbert goes on to state, that seems to me to be certain’ [10, 37].
As an observation on Hilbert’s program on the axiomatization of physics, Einstein4
wrote:
Our experience hitherto justifies us in believing that nature is the realization of the simplest
conceivable mathematical ideas. I am convinced that we can discover by means of pure
mathematical constructions the concepts and the laws connecting them with each other,
which furnish the key to the understanding of natural phenomena. Experience may suggest
the appropriate mathematical concepts, but they most certainly cannot be deduced from it.
Experience remains, of course, the sole criterion of the physical utility of a mathematical
construction. But the creative principle resides in mathematics. In a certain sense, therefore,
I hold it true that pure thought can grasp reality, as the ancients dreamed [50].

4 For
the relation between Hilbert and Einstein, see Stachel and Kouneiher’s contribution in this
volume.
Where We Stand Today 5

For Einstein, the problem of finding a mathematical representation that would provide
a unification of the gravitational and electromagnetic fields was more than just a tech-
nical problem. This aspect of his work is expressed most convincingly in Einstein’s
own account of his lifelong research concerns, as given in his 1949 Autobiographical
Notes [49]. Einstein, in his later work, followed a path that is not at all dissimilar to
Hilbert’s. Hilbert himself perceived Einstein as sharing his concern. Both Einstein
and Hilbert belong to a tradition that attempts to integrate our human knowledge and
to perceive an inner unity in science.

2 The Rise of Mathematical-Physics

A real interaction between mathematics and physics began to open up in the nine-
teenth century5 . For example, in volume 2 of Nature, from 1870, we read of the
following challenge from the pure mathematician Sylvester [86, 119]:
What is wanting (like a fourth sphere resting on three others in contact) to build up the ideal
pyramid is a discourse on the relation of the two branches (mathematics and physics) to,
and their action and reaction upon, one another - a magnificent theme with which it is to be
hoped that some future president of Section A will crown the edifice, and make the tetrology
…complete.

James Clerk Maxwell, as president of the British Association, took up the challenge
in a very interesting address in [83]. He modestly recommended his somewhat-
neglected dynamical theory of the electromagnetic field to the mathematical com-
munity. According to [47], not many mathematicians paid attention, constituting one
of the greatest Missed Opportunities of all time. Hertz commented on Maxwell’s
approach:
Maxwell’s theory consists of Maxwell’s equations. One cannot escape the feeling that these
equations have an existence and intelligence of their own, that they are wiser than we are,
wiser even than their discoverers, that we get more out of them than was originally put into
them.
In his address to the very first International Congress of Mathematicians in Zürich
in 1897, Henri Poincaré chose as his topic Sur les rapports de l’analyse pure et de la
physique mathématique, (On the relation of pure analysis to mathematical physics).
He was particularly impressed by Maxwell’s achievement:
How was this triumph attained?

Maxwell succeeded because he had become imbued with the idea of mathematical symmetry.
Would he have triumphed so well had others before him not explored this symmetry for its
own sake? […] Analysis was perhaps not among Maxwell’s skills, but to him, it would have
only been cumbersome and useless baggage. On the contrary, he was gifted with a profound
sense of mathematical analogy. This is why he produced good mathematical physics. [96].

5 For the history of the mathematization and geometrization of physics and the role of Euclid, Aris-

totle, Archimedes and the Greek philosophers, followed by Gallileo, Descartes, Newton, Leibniz,
and, even later, Grassmann, Hamilton and Elie Cartan, see [73].
6 J. Kouneiher

It is this realm of fundamental physics that is intimately intertwined with mathemati-

cal research at the frontiers of mathematical study. The relation between mathematics
and physics is one with a long tradition going back thousands of years. This has been
true from the beginning of modern physics, when Galileo first enunciated the propo-
sition that the natural language of physics was mathematics. Newton, one of the
greatest mathematicians of his day, invented the calculus of infinitesimals to calcu-
late planetary orbits, as well as to solve pure mathematical problems. His universal
law of gravitation explained everything from the fall of an apple to the orbits of the
planets.
In the following centuries, there was little distinction between theoretical physics
and mathematics, with many of the greatest contributors - Laplace, Legendre, Hamil-
ton, Gauss, Fourier - being regarded as physicists and mathematicians at the same
time.
The sophistication of Maxwell’s equations in the nineteenth century in including the
behaviour of electromagnetism induced an analogue process in the twentieth century
through Einstein’s theory of special relativity and then of general relativity. Einstein’s
choice to privilege symmetry over the laws of mechanics naturally implies the refor-
mulation of gravitation and electromagnetism as field theories in four-dimensional
space-time. This fusion of geometry and classical physics provided a strong stimulus
to mathematicians in the field of differential geometry.
However, the twentieth century has witnessed two revolutions in physics and the
completion of a theory of ordinary matter and its interactions. Once again, we have
called on mathematics to supply the tools and framework for this task. When Ein-
stein created general relativity, the dynamical theory of space and time, in 1915, the
necessary tools of differential geometry were available. They had been created by
Gauss and Riemann in the previous century. Riemannian geometry thus became a
central topic of geometry.
By the 1920s, it had been realized that atomic physics in the form of quantum me-
chanics, and the use of radically new concepts, such as the linear superposition of
states and the uncertainty principle, required an entirely new mathematical frame-
work. Physics appeared to be diverging from classical mathematics and the hope
of capturing the fundamental physical laws in terms of deep and elegant mathe-
matics faded away. The development of quantum mechanics built on understanding
of Hilbert spaces influenced the development of functional analysis. Early particle
physics drew heavily on the theory of continuous groups, which itself was partly
motivated by the desire to understand the spatial symmetry of crystalline structures.
Nonetheless, during the middle part of that century, mathematics and fundamental
physics developed in very different directions, with little significant interaction be-
tween them. This was due, in part, to an atmosphere of increased abstraction in the
mathematics community, as well as an insistence on rigid formal rigor, as exemplified
by the famous Bourbaki school.
However, much of the reason for this separation was due to developments in physics.
First, the early development of quantum mechanics and the early applications of
quantum mechanics to elucidating the structure of matter required little mathematical
sophistication. During the first decades after World War II, the vistas of particle
Where We Stand Today 7

physics rapidly expanded. These times were dominated by experimental surprises,

and theoretical model building required little more than traditional mathematical
tools.

3 Gauges Theories, Dualities and Fiber bundles

The advent of the Yang-Mills equations in 1955 showed that particle physics could
be treated with the same kind of geometry as Maxwell’s theory [132], but with quan-
tum mechanics playing a dominant role. Later, in the 1970s, it became clear that
these non-Abelian gauge theories were indeed at the heart of the standard model of
particle physics, which describes the known particles and their interactions within
the context of quantum field theory. These non-Abelian gauge theories of strong,
weak, and electromagnetic interactions are now universally accepted as yielding
a complete description of all the interactions of matter at energies and distances
that are experimentally accessible at present. This development was surely one of
the most remarkable accomplishments of twentieth century science. Attention has
more recently turned to even more ambitious attempts to construct unified theories
of all the interactions of matter, together with gravity. In the development of these
gauge theories, it has happened that many significant physical problems have led
to equally significant concepts in modem mathematics.6 Many of these concepts,
in fact, were invented independently by physicists and mathematicians. It is a re-
markable achievement that all the building blocks of this theory can be formulated
in terms of geometrical concepts such as vector bundles,7 connections, curvatures,
covariant derivatives and spinors. This combination of geometrical field theory with
quantum mechanics worked well for the structure of matter, but seemed to face a brick
wall when confronted with general relativity and gravitation (for the next sections
see [20, 93]).

3.1 Connections in a Fiber Bundle (Elie Cartan)

A notion that includes both Klein’s homogeneous spaces and Riemann’s local ge-
ometry is Cartan’s generalized spaces (espaces generalisés). In modern terms, this

6 Formore details, see Masson et al’s contribution in this volume.

7 Paul Dirac in 1931 discussed the possible existence of elementary magnetic charges-magnetic
monopoles [41]. He showed that in quantum mechanics, such magnetic monopoles made sense if,
and only if, the product of their charge, g, with the electric charge of the electron, e, was an integer
multiple of Planck’s constant , precisely: ge = n. This was very exciting, since it meant that as
long as there existed one magnetic monopole in the universe, all charges had to be quantized in units
of /g. In mathematical terms, Dirac had discovered an integer that characterized the topological
classification of vector bundles, mathematical constructs that were being invented at about the same
time by mathematicians. These concepts have come to play a role of increasing importance in
modern gauge theories.
8 J. Kouneiher

is called “a connection in a fiber bundle.” It is a straightforward generalization of

the Levi-Civita parallelism, which is a connection in the tangent bundle of a Rie-
mannian manifold. In general, we have a fiber bundle π : E −→ M, whose fibers
π −1 (x), x ∈ M are homogeneous spaces acted upon by a Lie group G. A connection
is an infinitesimal transport of the fibers compatible with the group action by G.
In the case of a complex vector bundle, the fibers are complex vector spaces
Cn of dimension n and G = G L(n; C) [18]. The importance of complex numbers
in geometry has a profound implication. It is well organized and complete. One
manifestation is the simple behaviour of the group, G L(n; C): its maximal compact
subgroup U (n) has no torsion and has, as a Weyl group the group of all permutations
on n letters.
We shall call a frame an ordered set of linearly independent vectors e1 , . . . , en ∈
π −1 (x), x ∈ M. In a neighborhood U , where a frame field e1 (x), . . . , en (x), x ∈ U ,
is defined, a connection is given by the infinitesimal displacement

Deα = ωαβ eβ , 1 ≤ α, β ≤ q, (1)

where ωαβ are linear differential forms in U . We call ωαβ the connection forms and
the matrix
ω = (ωαβ ) (2)

the connection matrix. Under a change of the frame field


eα = aαβ eβ , A = (aαβ ), (3)

the connection matrix is changed as follows:

ω A = d A + Aω . (4)

We introduce the curvature matrix

 = dω − ω ∧ ω , (5)

which is a matrix of exterior two-forms. Through exterior differentiation of (4),

we get
 = AA−1 , (6)

It follows that the exterior polynomial

 
i
det I+  = 1 + c1 () + · · · + cn (), (7)
2π

in which cn () is a 2α-form, is independent of the choice of the frame field, and
is hence globally defined in M. Moreover, each cα is closed, i.e.,
Where We Stand Today 9

dcα = 0. (8)

The form cα () has been called the αth Chern form of the connection, and its
cohomology class [cα ()], in the sense of de Rham cohomology, is an element of
the cohomology group H 2α (M; Z) and is called the αth Chern class of the bundle
E. These characteristic classes are the simplest and most fundamental global in-
variants of a complex vector bundle. They have the advantage of possessing a local
representation, by curvature.
As in the Gauss-Bonnet formula, such a representation is of great importance,
because the forms cα () themselves have a geometrical significance. Moreover, let
π  : P −→ M be the bundle of frames of the complex vector bundle. Then, the
pull-back π ∗ cα becomes a derived form, i.e.,

π ∗ cα = dT cα , (9)

where T cα , a form of degree 2α − 1 in P, is uniquely determined by certain

properties. This operation is called transgression, and T cα have been called the
Chern-Simons forms [17]. These forms have played a role in three-dimensional
topology and in the works of E. Witten on quantum field theory [128].
This theory can be developed for any fiber bundle (see [19]). The above provides
the geometrical basis of gauge field theory in physics. Here M is a four-dimensional
Lorentzian manifold, so that the Hodge ∗-operator is defined, and we define the
codifferential
δ = ∗d∗, (10)

There is a discrepancy of terminology and notation, as given by the following

table:

Mathematics Physics
Connection ω Gauge potential A
Curvature  Strength F

Maxwell’s theory is based on a U (1)-bundle over M, and his field equations can
be written as
d A = F, δ F = J, (11)

where J is the current vector. Actually, Maxwell wrote the first equation as

d F = 0, (12)

which is a consequence. For most applications, (12) is sufficient. But a critical

study of an experiment proposed by Bohm and Aharanov and performed by Chambers
shows that (11) are the correct equations [131]. A generalization of (11) to an SU (2)
bundle over M gives the Yang-Mills equations
10 J. Kouneiher

D A = F, δ F = J. (13)

It is indeed remarkable that developments in geometry have been consistently

parallel to those in physics.

The following quote from Raoul Bott captures the spirit of the time8 :
Although we still often do not understand each other, the push and pull relationship of our
two points of view has never been stronger and has invigorated both of us. Certainly in math-
ematics, the physically inspired aspects of the Yang-Mills theory has had a profound effect
on our understanding of the structure of 4-manifolds, and I also think we mathematicians
are only now learning to appreciate the rich mathematical structure of the Dirac sea - and
indeed of the whole Fermion-inspired world of the physicists, as well as their mystical belief
in supersymmetry. And on the other hand, the most modern achievements of mathematics
- from cobordism to index theory and K theory - have by now made their way into some
aspects of present day physics - I think to stay. [13].
We will illustrate this exchange between physics and mathematics described by Bott
starting with the case of the three-dimensional topology [8].

3.2 The Dawn of Mathematical Physics

The prime example of a topological problem is that of knots in three-dimensional

space,9 In 1984, the world of knot theory underwent a remarkable new development
when Vaughan Jones discovered a polynomial knot invariant (now named after him)
that was different from the Alexander polynomial [64]. Crucially, it was chiral, that is,
it could distinguish knots from their mirror images, which the Alexander polynomial
could not. It soon emerged that this new invariant was part of a grand family of
invariants based on Lie algebras and their representations. Shortly afterwards, Witten
showed how to interpret the Jones polynomial in terms of a quantum field theory in
three dimensions [126].
In fact, this relation between knot invariants and particles goes to the very beginning
of relativistic quantum field theory as developed by Feynman and others in the 1940s.
The basic idea is that, if we think of a classical particle moving in space-time, it will
move in the direction of increasing time. However, within quantum theory, the rules
are more flexible. Now, a particle is allowed to travel back in time. Such a particle
going backwards in time can be interpreted as an anti-particle moving forwards in
time. Once it is allowed to turn around, the trajectory of a particle can form, as it

8 Supersymmetry, the link between bosons and fermions, is a closely related concept from physics
that has also influenced differential geometry. As first noted by Edward Witten, supersymmetry
applied within quantum mechanics is an elegant way to derive the basic principles of Morse theory
[125]. Another application is in the development of hyper-Kähler geometry - the curved manifes-
tation of Hamilton’s quaternions. Although the definition has been in the differential-geometric
literature since the 1950s, it was 30 years later, as a result of the infiltration of ideas from the
supersymmetric sigma model, that a mechanism for constructing good examples was found.
9 Henceforth in this section, we follow [8].
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Here is an ancient receipt for making a Christmas game pie, found in
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or forced meats) and eyren (eggs) made ynto balles. Caste p’to
(thereto) poudre of pepyr, salte, spyce, eysell,[14] and funges
(mushrooms) pykled; and panne (then) take pᵉ boonys and let hem
seethe ynne a pot to make a gode brothe p’ for (for it) and do yᵗ ynto
pᵉ foyle of paste and close hit uppe faste, and bake yᵗ wel, and so s’ue
(serve) yᵗ forthe: wt pᵉ hede of oone of pᵉ byrdes, stucke at pᵉ oone
ende of pᵉ foyle, and a grete tayle at pᵉ op’ and dyvers of hys longe
fedyrs sette ynne connynglye alle aboute hym.’
Marrow bones seem to have been in favour at an early date. 2,000
marrow bones were among the requisites for the Goldsmiths’
Company’s feast, on St. Dunstan’s day, 1449.
In the reign of Edward VI., 1548, a time of plague and scarcity, the
king thought it prudent to fix the price of cattle, &c., sold in the
several seasons of the year:—
£ s. d.
The best fat ox, from Midsummer to
Michaelmas, at 2 5 0
 One of inferior sort 1 8 0
The best fat ox, from Hallowmas to
Christmas 2 6 8
One of inferior sort 1 19 8
The best fat ox, from Christmas to
Shrovetide 2 8 4
One of inferior sort 2 6 8
The best fat wether, from shearing time
to Michaelmas 0 4 0
One of ditto, shorn 0 3 0
The best fat ewe 0 2 6
One ditto, shorn 0 2 0
The best fat wether, from Michaelmas
to Shrovetide 0 4 4
One ditto, shorn 0 3 0
Essex barrelled butter, per pound 0 0 0¾
All sorts of other barrelled butter, per
pound 0 0 0½
Essex cheese, per pound 0 0 0¾
All other sorts 0 0 0½
We are not quite such prodigious devourers of eggs as our French
neighbours, having a greater amount of meat or solid animal food to
fall back upon, and fewer fast days. Another reason is, that we
cannot, like the French, get them so fresh and cheap; but as an
alimentary substance, eggs are always in demand at a ratio
proportionate to the prices at which they can be obtained. In Paris
the consumption of eggs is at least 175 per annum to every head of
the population; in the departments it is more than double that
amount; eggs entering into almost every article of food, and
butchers’ meat being scarce and dear. If we only use, in London, half
the number of eggs the Parisians do, there must be a sale of about
173 millions a year; and the consumption throughout the kingdom
would be fully 2,000 millions. Although smaller in size, and not equal
to a new-laid egg, the French eggs arrive in pretty good condition,
and, if sold off quickly, are well adapted for ordinary culinary
purposes. Few are wasted, for even when not very fresh, they are
sold for frying fish, and to the lower class of confectioners for pastry.
Fried eggs, boiled eggs chopped up with salad, egg sauce for fish,
&c., eggs for puddings, for omelets, and pancakes, all contribute to
the sale. Omelets, sweet or flavoured with herbs, are much less
patronized in this country than they are in France.
The sixty wholesale egg merchants and salesmen in the metropolis,
whose itinerant carts are kept constantly occupied in distributing their
brittle ware, might probably enlighten us as to the extent and
increasing character of the trade, and the remunerative nature of the
profits. Railways and steamers bring up large crates, and carefully
packed boxes of eggs, for the ravenous maws of young and old, who
fatten on this dainty and easily digested food. The various city
markets dispose of two millions of fowls, one million of game birds,
half a million of ducks, and about one hundred and fifty thousand
turkeys, every year. But even if we doubled this supply, what would it
be among the three million souls of the great metropolis requiring
daily food.
Ireland and the continent contribute largely to our supply of poultry
and eggs. Immense pens of poultry, purchased in the Irish market,
are shipped by the steamers to Glasgow and Liverpool. Commerce
owes much to the influence of steam, but agriculture is no less
indebted to the same power. Taking everything into account, and
examining all the advantages derived by cheap and rapid transit, the
manufacturer of food is quite as much indebted to the steam-ship
and the locomotive as the manufacturer of clothing.
There is no difficulty whatever in testing eggs; they are mostly
examined by a candle. Another way to tell good eggs is to put them
in a pail of water, and if they are good they will lie on their sides,
always; if bad, they will stand on their small ends, the large ends
always uppermost, unless they have been shaken considerably,
when they will stand either end up. Therefore, a bad egg can be told
by the way it rests in water—always end up, never on its side. Any
egg that lies flat is good to eat, and can be depended upon.
An ordinary mode is to take them into a room moderately dark and
hold them between the eye and a candle or lamp. If the egg be good
—that is, if the albumen is still unaffected—the light will shine
through with a reddish glow; while, if affected, it will be opaque, or
dark.
In Fulton and Washington market, New York, a man may be seen
testing eggs at almost any time of the year. He has a tallow candle
placed under a counter or desk, and taking up the eggs, three in
each hand, passes them rapidly before the candle, and deposits
them in another box. His practised eye quickly perceives the least
want of clearness in the eggs, and suspicious ones are re-examined
and thrown away, or passed into a ‘doubtful’ box. The process is so
rapid that eggs are inspected perfectly at the rate of 100 to 200 per
minute, or as fast as they can be shifted from one box to another, six
at a time.
The preservation of eggs for use on ship board has always occupied
a large share of attention. They have been usually smeared with oil
or grease, and packed in bran or sawdust. A plan recommended by
M. Appert for preserving eggs is to put them in a jar with bran, to
prevent their breaking; cork and hermetically seal the jar; and put it
into a vessel of water, heated to 200 degrees Fahrenheit, or 12
degrees below boiling. The vessel with water being taken from the
fire, the water must cool till the finger may be borne in it; then
remove the jar. The eggs may then be taken out, and will keep for six
months.
Salted ducks’ eggs are an article in great demand in some parts of
the East, for transport by the trading junks. The Malays salt them as
they do their meat; but the Chinese mix a red unctuous earth with
the brine, which no doubt stops the pores of the shell, and preserves
them better. They are put into this mixture at night, and taken out
during the day to be dried in the sun, which is, in fact, a half roasting
process in a tropical climate.
Pickled eggs, while they constitute a somewhat novel feature in the
catalogue of condiments, are at the same time particularly relishing.
When eggs are plentiful, farmers’ wives, in some localities, take four
to six dozen of such as are newly laid, and boil them hard; then,
divesting them of the shells, they place them in large-mouthed
earthen jars, and pour upon them scalded vinegar, well seasoned
with whole pepper, allspice, ginger, and a few cloves of garlic. When
this pickle is cold, the jars are closed, and the eggs are fit for use in
a month afterwards. Eggs thus treated are held in high esteem by all
the farm-house epicures.
Fowls’ eggs, variously coloured, and having flowers and other
matters upon them, formed by the colouring matter being picked off
so as to expose the white shell of the egg, are a part of all the Malay
entertainments in Borneo. The eggs eaten by the Dyaks are
frequently nearly hatched when taken from the nest, as they enjoy
them just as well as when fresh.
An article called ‘condensed egg’ is now sold in the shops. It consists
of the whole substance of the fresh uncooked egg, very delicately
and finely granulated by patent processes, after the watery particles,
which the egg naturally contains, have been completely exhausted
and withdrawn, without further alteration of its constituents. It
contains all the nutritious properties of the egg in its natural state,
and must be valuable to shipmasters, emigrants, and others. One
ounce of it is said to be equal to three eggs.
The ancient Romans, though not great beef-eaters, were particular
as to poultry. Dr. Daubeny, in his Lectures on Roman Husbandry,
says—‘The ancient Romans had large preserves, not only of poultry
and pigeons, but even of thrushes and quails enclosed in pens which
were called ‘ornithones,’ from which they could draw their supply for
the table at pleasure. We are told, indeed, of two sorts of ornithones,
the one merely aviaries stocked with birds for the amusement of the
proprietor; the other kind, constructed with a view to profit, which
were often of vast extent, to supply the demands of the Roman
market for such articles of luxury. In the Sabine country particularly,
we read of extensive pens, filled with birds for the latter purpose. For
thrushes alone there were large rooms provided, each capable of
holding several thousand birds. As they were put in to be fattened,
the place had only just light enough to enable the birds to see their
food, but there was a good supply of fresh water accessible. And I
may remark that, whilst nothing is said by the Roman writers about
the fattening of oxen and sheep, particular directions are given for
fattening poultry and other birds—a strong additional argument of the
little importance they attached to the larger animals as articles of
food.’
The following may be enumerated as the sportsman’s game in
Jamaica:—
1. The pintado, or wild guinea fowl (Numida Meleagris), a bird now
domesticated in our poultry-yards. In its wild state the flesh is
considered by many persons to equal that of the pheasant.
2. The quail (Perdix coturnix).
3. Wild pigeons, namely, ring-tail, bald-pate, pea-dove, white-breast,
white-wing, mountain-witch, ground dove, and red-legged partridge.
4. Snipe (Scolopax gallinago).
5. Wild duck, or mallard (Anas boschas).
6. Gray, or Gadwall duck (Anas strepera).
7. The common teal (Anas crecca), the flesh of which was so much
prized by the Roman epicures, and is still in request for the table.
8. Widgeon (Anas Penelope).
9. Gray and ring plover (Charadrius minor, and hiaticula).
If we are out shooting in Canada we may easily add to our mess the
ruffled grouse (Tetrao umbellus), although these, like many other
birds, are partridges with the settlers—this variety being termed the
birch partridge. Another species, the spruce partridge of the colonists
(T. Canadensis), is less palatable, for, unfortunately, it has a habit of
feeding upon laurel leaves. But here is something to make amends
—a fine Esquimaux curlew, as large as an English partridge, and a
mud-sucker, id est snipe.
Let me note a Canadian receipt for cooking a partridge, which may
be useful to sportsmen and travellers:—
‘Expedition is the maxim of all sylvan cookery, and as plucking the
feathers of a partridge would be too great a tax on the time and
patience of the voyageur, the method most in vogue is to run your
hunting knife round his throat and ancles and down his breast, when,
taking a leg in each hand, and pressing your thumb into his back,
you pop him out of his skin, as you would a pea from its pod. Then
make a spread-eagle of him on a forked twig, the other extremity of
which is thrust into the ground, and after wrapping a rasher of bacon
around his neck and under his wings, as ladies wear a scarf, you
incline him to the fire, turning the spit in the ground, and you will
have a result such as Soyer might be proud of. When your other
avocations will not afford time even for the skinning process, an
alternative mode is to make a paste of ashes and water, and roll up
your bird therein, with the feathers, and all the appurtenances
thereof, and thrust the performance in the fire. In due time, on
breaking the cemented shell (which is like a sugared almond), the
feathers, skin, &c., adhere to it, and then you have the pure kernel of
poultry within.’
The red-legged partridge is common in the Greek islands, on the
continent of Asia, and in the southern countries of Europe. In some
of the Cyclades, where the inhabitants are too poor to expend
money on powder, they chase the birds on foot, till they are so
wearied, as to be easily taken with the hand.
Of all the European birds, the quail (Coturnix vulgaris) is the most
remarkable, on account of the vast numbers which congregate on
the shores of the Mediterranean in the spring, coming from Asia
Minor and Northern Africa, to avoid the excessive heat. For a few
weeks in the month of April, when they first begin to arrive in Sicily,
everybody is a sportsman. Arriving always in the night, although not
a quail could be seen the evening before, the report of guns the next
morning, in all directions, attests their number and the havoc that
has begun upon them. Such prodigious numbers have appeared on
the western coasts of the kingdom of Naples, that a hundred
thousand have been taken in a day, within the space of four or five
miles.
The flesh of the turtle dove is considered much superior to that of the
wild pigeon.
The passenger pigeon (Columba migratoria) of America, is a very
large and well flavoured variety, being 16 inches long, and 24 inches
in the spread of its wings; its hue chiefly slate-colour. They migrate at
certain seasons in millions, and feed on acorns and fresh mast. They
travel in the morning and evening, and repose about mid-day in the
forests. Their passage, whether in spring or autumn, lasts from 15 to
20 days, after which they are met with in the centre of the United
States. The Indians often watch the roosting places of these birds,
and knocking them on the head in the night, bring them away by
thousands. They preserve the oil or fat, which they use instead of
butter. There was formerly scarcely any little Indian village in the
interior, where a hundred gallons of this oil might not at any time be
purchased.
These pigeons spread over the whole of North America, abounding
round Hudson’s Bay, where they remain till December. They arrive in
the fur countries in the latter end of May, and depart in October. They
are met with as far south as the Gulf of Mexico, but do not extend
their range westward of the Rocky Mountains. Stray passenger
pigeons have been taken both in Norway and in Russia; and this bird
has found a place in the British fauna, from a solitary bird having
been shot in Westhall, Fifeshire, on the 31st December, 1825. Like
other pigeons, this genus makes a slender platform nest of sticks
and straws, but, unlike other pigeons, prolific as it is, it lays but one
egg. The female builds the nest, the male bird fetches the materials.
The time of incubation is 16 days, and the male relieves the female
in sitting during that period. The immense number of these birds
baffles all computation. Those eminent ornithologists, Wilson and
Audubon, describe flocks seen by them to contain respectively from
thousands of millions to upwards of a billion in each, the daily food
required to sustain which would be at least 60,000 bushels; and the
New York Evening Post informs us that, on one day, seven tons of
these pigeons were brought into the New York market by the Erie
railroad.
In their breeding places, herds of hogs are fed on the young pigeons
or ‘squabs,’ which are also melted down by the settlers, as a
substitute for butter or lard. The felling a single tree often produces
200 squabs, nearly as large as the old ones, and almost one mass of
fat. When the flocks of full grown pigeons enter a district, clap-nets
and guns are in great requisition. Pennant, in his Arctic Zoology,
says, Sir William Johnstone told him, that at one shot, he brought
down with a blunderbuss above a hundred and twenty pigeons.
Wagon loads of them are poured into the towns, and sold as cheap
as a half-penny up to two-pence the dozen. The flesh tastes like the
common wild blue pigeon, but is, if anything, better flavoured. Why (it
has been asked) could not this large pigeon, whose migratory habits
are principally caused by search for food, be introduced into this
country as a tame variety, or by crossing with our native breeds
enlarge the size; or, in the same way as fresh mutton was sent from
Australia, be sent in casks potted in their own fat, to supply us with
cheap pigeon pies? And the same with a cross with the large Texan
rabbit, or the wild American turkey, the latter being far superior in
size and appearance to its degenerate descendant, the tame turkey,
sometimes as much as four feet in length, and five feet from wing to
wing? The canvas-back ducks of America are there boasted of
exceedingly as a delicacy, yet, although a great variety of useless
water-fowl has been introduced merely as an ornament to the ponds
and streams of our gentry, no attempt has been made to bring this
kind to our farm-yards and tables; and even if it was found
impossible to tame the pure breed, a cross with our own might be
effected. In the capercailzie, or cock of the wood, a bird of the
grouse species, but nearly as large as a turkey, once indigenous to
Scotland, but now only found in the north of Europe, and in the
bustard, the largest European land-bird, the cock weighing from 25
to 27 lbs., we have examples of two fowls well worth the trial of
domesticating by the amateur or intelligent agriculturist, a trial which,
if successful, would probably repay quite as well as competition
about the colour of a feather, or the shortness of a tail, and in time
would be the means of affording a constant, certain, and moderately-
priced supply, which is never the case while animals remain in a wild
or half-wild state.
Although the forests of New Zealand are not thickly inhabited by the
feathered tribes, there are many birds to be met with. Among others
are the following, which are excellent eating.—The wild pigeon,
which is very large and common; the parrot or ka-ka; and the tui or
mocking bird, which is about the size of the English black-bird, and
of the same colour, but with two bunches of white feathers under the
neck—his notes are few, but very melodious, resembling the tinkling
of small bells, which harmonize together as they are delivered. The
bronze-winged pigeon of Australia is most delicate eating. It abounds
in summer, when the acacia seeds are ripe.
 GRALLATORES.
From the order of grallatores, waders or stilt-birds, we find many
which yield choice dainties, whether it be the ostrich or emu for their
eggs, the bustard and bittern, the flamingo for its tongue, the plover,
dotterel, curlew, snipe, woodcock, rail, &c., for the table.
An ostrich egg is considered as equal in its contents to 24 of the
domestic hen. When taken fresh from the nest, they are very
palatable, and are wholesome, though somewhat heavy food. The
best mode of cooking them is that practised by the Hottentots, who
place one end of the egg in the hot ashes, and making a small orifice
at the other, keep stirring the contents with a stick till they are
sufficiently roasted; and thus, with a seasoning of salt and pepper,
you have a very nice omelet. The nest sometimes contains as many
as 24 eggs, and the difficulty the sportsman has is how to carry away
his spoil. The usual plan is to denude himself of his upper or lower
garments, and, tying up the orifices of leg-holes or arm-holes, to
make an impromptu sack, in which he can bear away his prize. If he
leaves them, he will be sure to find on his return that the ostrich has
broken the eggs, because they have been disturbed.
The eggs of the emu of South America are large, and, although the
food which they afford is coarse, it is not unpalatable.
The emu, or New Holland cassowary, is becoming rarer as
settlements advance. The same remark applies also to the kangaroo
and other animals against whom a war of extermination seems to
have been declared.
The emu is easily domesticated when taken young, and becomes
very familiar with, and attached to, the dogs, which generally leads to
the death of a tame one. A full-grown one, when erect, stands seven
feet high. The natives creep on them and spear them. The eggs are
of a tea green colour, with a watered appearance on the surface.
There is a singularity in the growth of the feathers—two of them
spring from one quill. The bird is principally valued for its oil. The skin
of a full-grown bird produces six or seven quarts of oil, clear, and of a
beautiful bright yellow colour. The method of extracting or ‘trying’ out
the oil is to pluck the feathers, cut the skin into pieces, and boil it; but
the aborigines prefer the flesh with the skin upon it, regarding it as
the Esquimaux do the flesh of whales and seals, as a highly luscious
treat. The flesh is eaten by Europeans, and preferred by some to the
kangaroo; the rump part is considered as delicate as fowl; the legs
coarser, like beef, but still tender.
Bustards are plentiful in many parts of the Cape Colony, and the
smaller sorts, called koerhans, are approachable in a bush country;
but the larger kinds, called paws, are a great prize, as they are found
on plains, and are generally shot with ball. In Australia, the bustard is
called, colonially, the wild turkey. It is a fine large bird, frequently
weighing 12 to 15 lbs., and extending full six feet, from tip to tip of
the wings. There it is declared excellent for eating, but its flesh is
much too gamey for ordinary palates.
Don Pernetty, in his Historical Journal of the Voyage to the Falkland
Islands, under the command of M. de Bouganville, says, they found
the bustard ‘exquisite either boiled, roasted, or fricasseed. It
appeared from the account we kept that we ate 1,500. It is, indeed,
hardly to be conceived that the ship’s company of our two frigates,
consisting of a hundred and fifty men, all in perfect health and with
good stomachs, should have found a quantity of these birds
sufficient for their subsistence during a stay of more than two
months, within a tract of country not exceeding three leagues.’
But they also tried other descriptions of feathered game. The wild
ducks were found, in general, to have the taste of mussels. Of a kind
of grey goose, weighing about 19 or 20 lbs., it is reported: ‘Its flesh
was oily, had a disagreeable smell and a fenny taste; but it was
eaten by the ships’ companies when no bustards were given them.’
The clucking hen of Jamaica (Ardea scolopacea), on the authority of
Browne and Robinson, is looked upon as the best wild fowl in the
country, although the latter writer tells us it feeds upon snakes,
toads, and lizards, as well as wood snails and gully crabs. The
flavour is, however, represented to be remarkably fine—a compound
of ham, partridge, and pigeon. The flesh is of a peculiarly close and
compact texture, and very tender.
The mangrove hen (Rallus Virginianus), indigenous to the watery
marshes of Jamaica, greatly resembles the dappled grey variety of
the common fowl. At the pullet age, the young birds are run down,
when feeding on the mud, with great facility. At this time, I have
found them to be delicious eating. Persons, on whose taste reliance
may be placed, say that, though a plover be undoubtedly a fine bird
for the table, and the sanderling a great delicacy, the young
mangrove hen exceeds both, as it combines all their peculiarity of
flavour with the fleshiness of the quail. This is no small
commendation. But much depends upon your cuisinier; if he is a
good artiste—a man of undoubted talents, it matters little what the
materials be.
The Rallus crex is another esteemed dainty of no ordinary kind, and
a most delicious bird.
In the reign of Henry the Eighth, the bittern was held in great esteem
at the tables of the great. Its flesh has much the flavour of hare, and
is far from being unpleasant.
Snipe of all kinds, from the ‘teeterer,’ that hovers about the edge of
the surf, to the jack snipe (Scolopax gallinula), half-brother to the
woodcock, are in high esteem for the table. The ‘green’ sportsman
finds these birds the most perplexing of all feathered game when on
the wing. Their catter-cornered, worm-fence line of flight renders
them very difficult to hit, until long practice has rendered the
marksman’s eye familiar with their erratic movements. Some
sportsmen take them at an angle; others after they have made their
tack; and others, again, seem to blaze away at them without any
particular aim, and yet always bring down their bird. The yellow-
legged snipe is in America considered the best species for the table.
They should be larded and roasted in bunches of three, and served
in gravy made from their own unctuous drippings. There are few
side-dishes more popular with epicures than snipe on toast. Some
cooks stuff them with a composition of bread crumbs and egg, highly
seasoned; but, in my opinion, they are far better without this kind of
‘trimmings.’
While the trail of the woodcock is a choice morsel with the English
epicure, the inhabitants of the North of Europe, to whose forests the
woodcocks retire in the summer, never eat the birds, esteeming their
flesh unwholesome, from the circumstance of their having no crops.
But they are particularly fond of the eggs, which the boors offer for
sale in large quantities in the principal markets, and this contributes,
possibly, to make the birds so scarce.
The semi-palmated snipe, better known by its common sobriquet of
‘pill-will-willet,’ the loud shrill note which it emits, is at certain periods
of the year esteemed an excellent bird in America. It ought to be
served up in the mode that snipes usually are, and for these
delicious viands it is esteemed a tolerable substitute when in good
order.
Dampier, nearly two centuries ago, speaking of the flamingo, says:
‘Their flesh is lean and black, but not ill-tasted. They have large
tongues, and near the root of them a piece of fat, which is accounted
a great dainty.’
The flamingo was much esteemed by the Romans at their
sumptuous entertainments. Their flesh is thought tolerably good
food, and the tongue was looked upon by the ancients as among the
most delicate of all eatables. Pliny, Martial, and many other writers,
speak of it in the highest terms of commendation. Many who have
tried it, consider the flesh extremely rich, much like that of the wild
duck, but with a strong fishy taste. The tongue is certainly delicate,
but scarcely worthy the high encomiums bestowed on it by the
ancients.
During the surveying expedition of Captain Owen, on the east coast
of Africa, the sailors used to shoot hundreds of these beautiful birds
for the purpose of making a dish of the tongues alone. The
remainder of the bird—in imitation of the Roman epicures—being
thrown away.
 NATATORES.
The Natatores, or swimming birds, supply us with very choice food.
Even many of the coarse sea fowl are not rejected by voyagers.
The Chinese shoot sea-gulls in large numbers, which add to their
stock of food. A man is constantly engaged in the bay of San
Francisco, California, shooting sea-gulls, which he sells to the
Chinese at the rate of 25 cents each. The San Francisco Evening
News says,—‘This bird is a slow and steady mover, of large size,
and flies at a convenient distance over the head of the sportsman.
The man in the skiff was armed with a double-barrelled shot gun,
both barrels of which he would load, and taking a dead gull would
throw it high in the air and allow it to fall at some distance from the
boat. This would naturally attract a flock of gulls, and as they made
their slow circuit around the spot, the gunner raised his piece and
generally succeeded in bringing down a bird for both barrels. He
would then re-load as fast as possible, and if a gull was in range,
another shot was fired and another trophy won.’
The flesh of sea-fowl is generally too rancid to find much favour with
fastidious palates. Sailors indeed eat the livers and hearts of the
penguins, which are exceedingly palatable, but the black flesh of the
body is rank and oily, and has rather a perfumed taste. Some
voyagers, however, tell us, that eaten in ragouts, they are good as
that made from a hare.
The young puffins, having gorged themselves with sprats and
crustacea, when pickled with spices, are by some considered
dainties, and they are, occasionally, potted in the North. But when it
has attained its ugly full developed bill, like a short, thick plough
coulter, this bird does not look very prepossessing. Besides making
use of them for food, some of the islanders use them for fire-wood.
They split them open, dry them, and then burn them feathers and all.
There is a species of puffin, the Puffinus urinatrix or P. brevicaudis,
popularly termed the mutton-bird by Tasmanian colonists, which is
met with on some of the New Zealand islands. It forms the principal
food for the native inhabitants of Foveaux Straits, and by them is
called the titi. It is a sea bird of black colour,[15] in its usual condition
smaller than the common duck. Like all sea birds it has thin, slender
legs, with webbed feet: the wings are long, with many joints, I forget
how many: the bill is a little hooked at the point. They are generally
in large flocks, covering the ocean as far as the eye can reach;
sometimes flying all in the same direction, at other times crossing
through each other like swarming bees. They breed on the small
uninhabited islands scattered round the coasts of Stewart’s Island.
These islands have a loose, dry, peatish soil, on a stony bottom.
Their being exposed to the stormy winds, loaded with the salt spray
of the sea, prevents the growth of a forest, except patches of stunted
bushes intermixed with a sort of soft, light green fern. The loose soil
is perforated with numberless birdholes, like a piece of worm-eaten
wood, running from two to four feet underground in a horizontal
direction, at the farthest end of which is the nest. Each female lays
only one egg, which is nearly as big as a goose egg, on which they
sit—it is believed male and female alternately—many weeks. The
young bird is full grown in the month of April, which corresponds to
October in Europe. At that time, almost all the inhabitants of Foveaux
Straits, old and young—the infirm only excepted—repair to the Titi
Islands, and take the young birds out of their nests, which amount to
many thousands, and a great many still escape. They put a stick in
the hole to feel where the bird is, which generally betrays itself by
biting the stick. If the hole is so long that the bird cannot be reached
by the hand, a hole is dug over it, the bird taken out and killed by
breaking its head, and the broken hole covered with rubbish and
earth, so that it may be used again the next year. Afterwards the
birds are plucked, and, to clean the skin from the hairy down, it is
moistened and held over the fire, when it is easily wiped quite clean.
Then the neck, wings, and legs are cut off, the breast is opened, the
entrails are taken out, and the body is laid flat, either to be salted or
to be boiled in its own fat, and preserved in air-tight kelp bags.
Though it cannot be said that the young birds suffer, they being killed
so quickly, yet it might seem cruel to rob the parents of their young
ones on so large a scale, and one would fancy a great deal of
fluttering and screaming of the old ones, bewailing the bereavement
of their offspring. But that is not the case. None of the old birds make
their appearance in the day-time. They are all out at sea, and come
only to their nests in the evening when it gets dark, and are off again
at day-break. But yet it would seem the parents would be distressed
at finding their nests robbed. Not so. It would seem as if Providence
had ordered it so that man should go and take the young birds for his
food without hurting the feelings of the parents. When the young
birds are full grown, then they are neglected by their parents, in
order to starve them to get thin, else they would never be able to fly
for the heaviness of their fat. It seems that at the time when they are
taken by men, they are already forsaken by the old birds; and those
that are not taken are compelled by hunger, when they have been
starved thin and light, to leave their holes and go to sea. The old
birds are tough and lean, but the young ones, which are nearly twice
as big, contain, when the legs, wings, neck, and entrails are taken
off, three-fourths of pure white fat, and one-fourth of red meat and
tender bones. The flavour is rather fishy, but, if once used to it, not
bad at all, only rather too fat. They eat best when salted and smoked
a little, and then boiled a short time, and afterwards eaten cold. If
properly salted, they might make an article of trade, like herrings in
Europe. The fat when clean is quite white, and looks just like goose
fat, but the taste is rather oily; however, it may be used for a good
many other purposes than for food. It burns very well on small
shallow tin lamps, which get warmed by the light and melt the fat.
The feathers are very soft, and would make excellent beds if they
could be cured of the oily smell, which it is likely they can.
The following remarks on the articles of food found in the arctic
regions are by one of the officers of the Assistance:—
‘To the feathered tribe we are chiefly indebted, and foremost in the
list for flavour and delicacy of fibre stand the ptarmigan (Lagopus
mutus) and the willow grouse (Tetrao saliceti). The flesh is dark-
coloured, and has somewhat the flavour of the hare. These may be
used in pie, stewed, boiled, or roast, at pleasure, and are easily shot.
Next in gustatory joys, the small birds rank, a kind of snipe, and a
curlew sandpiper; both are, however, rarely met with, and do not
repay the trouble of procuring them.
‘The brent goose (Anser torquatus) is excellent eating, and its flesh
is free from fishy taste. Then follow the little auk or rotge (Alca alle),
the dovekey, or black guillemot (Uria grylle), the loon, or thick-billed
guillemot (Uria Brunnichii). The first two are better baked with a
crust, and the last makes, with spices and wine, a soup but little
inferior to that of English hare. All these are found together in flocks,
but the easiest method of obtaining them is either to shoot them at
the cliffs where they breed, or as they fly to and fro from their feeding
ground.
‘The ducks now come upon the table, and are placed in the following
order by most Polar epicures. The long-tailed duck (Fuliluga
glacialis), the king duck (Anas spectabilis), and the eider duck (A.
mollissima). They require to be skinned before roasting or boiling,
and are then eatable, but are always more or less fishy.
‘The divers are by some thought superior for the table to the ducks,
but the difference is very slight. The red-throated diver was most
frequently seen, but few were shot; and of the great northern divers
(Colymbus glacialis) none were brought to table, two only having
been seen. Some of the gulls were eaten, and pronounced equal to
the other sea birds; they were the kittiwake, the tern, and the herring
or silver gull.
‘The denizens of the sea have fallen little under our notice, and they
may be dismissed with the remark, that curried narwhal’s skin can be
tolerated, but not recommended. Some fresh-water fish were caught,
and proved to be very good; they are said to be a kind of trout.’
The eggs of sea-fowl, although much eaten on the coasts, are
seldom brought to market for consumption in our large English
towns, and yet they form a considerable article of traffic in several
parts of the world, and are procured in immense quantities about the
lands near the North and South Poles.
The precipitous cliffs of England are occasionally searched for the
eggs of the razor-billed auk, which are esteemed a delicacy, for
salads especially.
A correspondent at San Francisco informs me that an important
trade is carried on in that city in the eggs of sea birds. He states, that
the Farallones de los Frayles, a group of rocky islets, lying a little
more than twenty miles west of the entrance to the Bay of San
Francisco, are the resort of innumerable sea-fowl, known by the
fishermen as ‘murres.’ These islands are almost inaccessible, and,
with a single exception, are uninhabited. They, therefore, very
naturally afford a resort for great multitudes of birds. Some time
since a company was organized in San Francisco for the purpose of
bringing the eggs of the murres to market. An imperfect idea of the
numbers of these birds may be formed from the fact, that this
company sold in that city the last season (a period of less than two
months, July and parts of June and August) more than five hundred
thousand eggs! All these were gathered on a single one of these
islands; and, in the opinion of the eggers, not more than one egg in
six of those deposited on that island was gathered. My
correspondent informs me that he was told by those familiar with the
islands that all the eggs brought in were laid by birds of a single kind.
Yet they exhibit astonishing variations in size, in form, and in
colouring. There is no reason to suppose that he was misinformed in
regard to these eggs being deposited by a single species. The men
could have had no motive for deception, and similar facts are
observable on the Labrador coast and in the islands north of
Scotland. Besides, the writer ascertained from other sources, that all
the eggs brought to the market were obtained from a limited portion
of the island, known as the Great Farallone—called the Rookery,
where a single species swarm in myriads, and where no other kind
of bird is found. Naturalists, who have received specimens of these
birds, pronounce them to be the thick-billed or Brunnich’s guillemot,
or murre, of Labrador and Northern Europe. The eggs are three and
a half inches in length, and are esteemed a great delicacy.
There is a small island off the Cape of Good Hope, named Dassen
Island, about six miles from the mainland, which is one and a half
mile long by one broad, from which 24,000 eggs of penguins and
gulls are collected every fortnight, and sold at Cape Town for a half-
penny each.
The late Lieut. Ruxton, R.N., speaking of the Island of Ichaboe, on
the Western Coast of Africa, says, ‘Notwithstanding that the island
had been occupied for nearly two years, during which time
thousands upon thousands of penguins had been wantonly
destroyed, on the cessation of work these birds again flocked to their
old haunts, where they had again commenced laying their eggs. The
rocks round the island are literally covered with penguins,
cormorants, and albatrosses. The former, wedged together in a
dense phalanx, have no more dread of man than ducks in a poultry-
yard, although they have met with such persecution on the island;
and any number might be taken by the hand without any difficulty.
The sailors eat the livers and hearts, which are exceedingly
palatable, but the flesh of the body is rank and oily.’[16]
Captain Morrell, also writing of Ichaboe (Nautical Magazine, vol. 13,
p. 374), tells us, ‘Eggs may be obtained here in great quantities. In
the months of October and November this island is literally covered
with jackass-penguins and gannets, which convene here for the
purpose of laying and incubation. The nests of the gannets are
formed like those of the albatross, but are not so much elevated;
while the jackass-penguins lay their eggs in holes in the ground from
twelve to thirty inches in depth, which they guard with the strictest
vigilance. They frequently lay three or four eggs, but the gannet
seldom lays more than two.’
A correspondent, writing from Tristan d’Acunha, in September, gives
an account of his adventures in taking penguins’ eggs. ‘This is now
the time for penguins’ eggs. They get great numbers of them. There
are two rookeries, as they call them; one on the east, and one on the
west, of us. To the one on the west, they go over land, beyond
Elephant Bay. I went there last year, when I saw the great sea
elephant and the penguins for the first time. But this year I have
been disappointed, the weather has been so unsettled. But
yesterday was a fine day, and they were going in the boat to the
other, to which they can go only by water; so I went with them. It was