Princeton, 1950

Karlis Podnieks

University of Latvia

ON THE NATURE OF MATHEMATICS

IX scientific conference
"Modern Logic: Problems of Theory, History, and Applications in Science"
Saint Petersburg, Russia,
June 22-24, 2006

Picture from: MacTutor History of Mathematics archive - Godel Portraits

 This work is licensed under a Creative Commons License and is copyrighted © 2006-2008 by  me, Karlis Podnieks.

The human mind has first to construct forms, independently, before we can find them in things.
Albert Einstein
Quotations by Albert Einstein, The MacTutor History of Mathematics archive

Is it worth knowing what mathematicians think about the nature of mathematics?

Strangely, most of their opinions are provably equivalent to the following sentence attributed to David Hilbert:

"Die Mathematik ist das, was kompetente Leute darunter verstehen."

"Mathematics is what competent people understand by it.", or, "Mathematics is what competent people think it is." (Sorry, my own English translations.)

Gordon Fisher. Marshack & When begins mathematics? Historia Matematica Mailing List Archive, August 1999: "... This reminds me of an old story about R L Moore, the University of Texas topologist. It is said that when a newspaper reporter asked him what topology is, he replied by saying that it's what topologists do. So maybe mathematics is what mathematicians do? Just joking ... :-)"

"Mathematics is what mathematicians do." By Reuben Hersh? Or, by Henry Poincare?

Thus, it seems, mathematicians would prefer determining themselves the "meaning" of their occupation. But why can't they define it in a more sensible way than just saying that "it is what we are doing"? Could this be because that - as noted by John von Neumann (see Quotations by John von Neumann at ):

"In mathematics you don't understand things. You just get used to them. "

Or, this attitude is caused by a kind of "misere de la philosophie" around mathematics? Indeed, as noted by Harvey Friedman, a really useful philosophy of mathematics should yield "observable consequences" for the mathematical practice:

"For me, the productive questions are oblique, and very exciting.
1. What observable consequences ensue from adopting any of the usual isms, or various modifications of the usual isms?
2. What kinds of exact findings would bear on the merits of various isms?
...
So if there are no observable consequences from taking one point of view or another, then what is the fuss about?" (full text: H. Friedman. Formalism/Platonism, FOM posting, October 2003)

Thus, is it worth knowing what mathematicians think themselves about the nature of mathematics?

Mainly, no. There are only a few exceptions. Probably the most outstanding one was Andrey Nikolaevich Kolmogorov - see, for example, the following passage:

"... the process of cognition of the concrete [des Konkreten - K.P.] proceeds as a struggle of two tendencies; on the one side, by extracting the form of the phenomena and the logical analysis of this form, on the other side, by discovering the aspects that are not covered by the established forms, by considering new, more flexible forms covering the phenomena in a more complete way. If the difficulties in the study of a set of phenomena belong to the second tendency, if each new step of investigation must take into account qualitatively new aspects of the phenomena, then the mathematical method steps back to the rear, in this case a dialectical analysis of the full concreteness of the phenomenon can only be obscured by a mathematical schematization. But if, otherwise, relatively simple and stable basic forms of the phenomena cover these phenomena with high precision and completeness, and if within these fixed forms hard and complicated problems appear requiring a specific mathematical investigation, in particular, creating of a specific symbolic notation or of a specific algorithm for their solution, then we arrive in the sphere of the supremacy of the mathematical method." (Sorry, my own English translation.)

The original text:
A. N. Kolmogorov. Mathematics, BSE, 1938/1954, in Russian, online copy available).

See also Opinion 69 (December 2005) by Doron Zeilberger.

Three approaches to mathematics

In this lecture, I will try approaching to mathematics from three different directions:

1. Mathematics as a strange social phenomenon. The ritual aspect of mathematics - to what extent is it essential? The most complicated mathematical proofs - are they really (un)reliable? Using computers in mathematical proofs. "Professional" computerized mathematics against "amateur" human mathematics? Continue...

2. Is mathematics an "ordinary" branch of science, or, its position among other branches of science is absolutely specific? Plato - Kant - Hilbert. Does infinity exist in the natural world? Formalism and platonism. The source of the "surprising efficiency of mathematics" - the ability of mathematicians to draw a maximum of conclusions from a given set of premises. Continue...

3. Mathematics and modeling. The distinguishing feature of mathematical models - it is worthwhile to investigate these models without any reference to the modeled objects. The task of mathematics is developing methods of creating and exploring of this kind of models. Left and right hemispheres of the human brain - and the two dimensions of mathematics. Continue...

Remark. By natural numbers mathematicians understand the numbers 0, 1, 2, 3, ...