Diploma

What is Mathematics:
Gödel's Theorem and Around

Hyper-textbook for students
by Karlis Podnieks, Professor
University of Latvia
Institute of Mathematics and Computer Science

An extended translation of the 2nd edition of my book " Around Goedel's theorem" published in 1992 in Russian (online copy).

Table of contents

Diploma

 This work is licensed under a Creative Commons License and is copyrighted © 1997-2010 by  me, Karlis Podnieks.

Horizons of Truth:
Gödel Centenary 2006

Wolfgang Haken and Kenneth Appel, see pictures:
European Mathematical Society,
Newsletter No. 46, December 2002, pp. 15-19.
http://unhmagazine.unh.edu/sp02/mathpioneers.html
http://www.ehu.es/~mtwmastm/PresentacionSCTM06.pdf

My Hall of Fame

Plato    I. Kant   G. Cantor    C. S. Peirce    H. Poincaré
   D. Hilbert   E. Zermelo   K. Gödel   P. Cohen

See portraits of these brilliant people in the MacTutor History of Mathematics archive
at the University of St Andrews. And in Wikipedia.

What is Mathematics? (My Main Theses)

I define mathematical theories as stable self-contained (autonomous?) systems of reasoning, and formal theories - as mathematical models of such systems. Working with stable self-contained models mathematicians have developed their ability to draw a maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient (my solution to the problem of "The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" - as put by Eugene Wigner).

For me, Goedel's results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are either non-universal (i.e. they cannot express the notion of natural numbers: 0, 1, 2, 3, 4, ...), or they are universal, yet then they run inevitably either into contradictions, or into unsolvable problems.

For humans, Platonist thinking is the best way of working with imagined structures. (Another version of this thesis was proposed in 1991 by Keith Devlin on p. 67 of his Logic and Information.) Thus, a correct philosophical position of a mathematician should be: a) Platonism - on working days - when I'm doing mathematics (otherwise, my "doing" will be inefficient), b) Formalism - on weekends - when I'm thinking "about" mathematics (otherwise, I will end up in mysticism). (The initial version of this aphorism was proposed in 1979 by Reuben Hersh (picture) on p. 32 of his Some proposals for reviving the philosophy of mathematics.)

Next step

The idea that stable self-contained system of basic principles is the distinctive feature of mathematical theories, can be regarded only as the first step in discovering the nature of mathematics. Without the next step, we would end up by representing mathematics as an unordered heap of mathematical theories!

In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).

Thus, we should think of mathematics as a "two-dimensional" activity. Sergei Yu. Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).

Do we need more than this, to understand the nature of mathematics?

What is Mathematics?

Four definitions of mathematics provably equivalent to the above one (see Section 1.2):

Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.
This elegant definition was put on the web by Dave Rusin. Who might be the author?

... the human mind has first to construct forms independently, before we can find them in things. ... knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact.
Albert Einstein, see The Schiller Institute (English translation by Bruce Director)

In mathematics you don't understand things. You just get used to them.
John von Neumann, see Quotations by John von Neumann).

Mathematicians are mad tailors: they are making "all the possible clothes" hoping to make also something suitable for dressing...
Stanislaw Lem, "Summa Technologiae" (sorry - my own English translation, the initial version of this aphorism may be due to David van Dantzig, see Quotations by David van Dantzig)

Why is Einstein much more popular than Gödel?
Kurt Godel in Blue Hill, by Peter Suber

Wir muessen wissen -- wir werden wissen!
David Hilbert's Radio Broadcast, Koenigsberg, 8 September 1930
(audio record published by James T.Smith, and translations in 7 languages published by Laurent Siebenmann).

"Hilbert and Goedel never discussed it, they never spoke to each other. ... They were both at a meeting in Koenigsberg in September 1930. On September 7th Goedel off-handedly announced his epic results during a round-table discussion. Only von Neumann immediately grasped their significance..." (G.J.Chaitin' s lecture, 1998, Buenos Aires)