Natures of Mathematics
Natures of Mathematics
Natures of Mathematics
SECUYA
BEED-II
Symbolical mathematics.
In earlier times, mathematics was in fact, fully verbal. Now, after the dramatic advances
in symbolism that occurred in the Mercantile period (1500s), mathematics can be practiced in an
apparent symbolic shorthand, without really the need for very many words. This, however, is
only a shorthand. The symbols themselves require very careful and precise definition and
characterization for them to be used, computed with, and allow the results to be correct.
The modern language of working mathematics, as opposed to expository or pedagogical
mathematics, is symbolic, and is built squarely upon the propositional logic, the first order
predicate logic, and the language of sets and functions.2 The symbolical mode is one which
should be learned by the student and used by the practitioner of mathematics. It is the clearest,
most unambiguous, and so most precise and therefore demanding language. But one might say it
is a “write only” language: you don’t want to read it. So, once one has written out one’s ideas
carefully this way, then one typically switches to one of the other two styles: direct or expository,
these being the usual methods of communicating with others.
Evolution Through Dialectic
Mathematical definitions, mathematical notions of correctness, the search for First
Principles (Foundations) in Mathematics and the elaboration of areas within Mathematics have all
proceeded in a dialectic fashion, alternating between periods of philosophical/foundational
contentment coupled with active productive work on the one hand, and the discovery of
paradoxes coupled with periods of critical review, reform, and revision on the other. This
dialectical process through its history has progressively raised the level of rigor of the
mathematics of each era.
The level of precision in mathematics increased dramatically during the time of Cauchy, as those
demanding rigor dominated mathematics. There were simply too many monsters, too many
pitfalls and paradoxes from the monsters of functions in the function theory to the paradoxes and
strangeness in the Fourier analysis and infinite series, to the paradoxes of set theory and modern
logic. The way out was through subtle concepts, subtle distinctions, requiring careful
delineations, all of which required precision.
A Culture of Precision
Mathematical culture is that what you say should be correct. What you say should have a
definition. You should know the definition and limits of what you are saying, stating, or claiming.
The distinction is between mathematics being developed informally and mathematics being done
more formally, with necessary and sufficient conditions stated up front and restricting the
discussion to a particular class of objects.