Introduction to
Mathematical Logic

Hyper-textbook for students
by Vilnis Detlovs and Karlis Podnieks
University of Latvia

Vilnis Detlovs. Memorial Page

What is Mathematics:
Gödel's Theorem and Around

Hyper-textbook for students
by Karlis Podnieks
Russian version available

Quote of the Day

Niels Bohr, before 1963.

"It is wrong to think the task of physics is to find out how Nature is. Physics concerns what we can say about Nature."

Quoted after:
Melvin Goldstein. Physics Foibles. A Book for Physics, Math and Computer-Science Students. Trafford Publishing, 2003, 410 pp.

Nancy Cartwright, 1983, How the Laws of Physics Lie

“My basic view is that fundamental equations do not govern objects in reality; they only govern objects in models.”

See p. 129 of:
N. Cartwright. How the Laws of Physics Lie. Oxford University Press, 1983, 232 pp.

Previous quotes

What is Mathematics?

Four provably equivalent definitions of mathematics:

Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.
I do not know the author of this elegant definition put on the web by Dave Rusin.

... 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

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)

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 stable self-contained systems. 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) Advanced Formalism - on weekends - when I'm thinking "about" mathematics (otherwise, I will end up in mysticism). (The initial version of this aphorism is due to Reuben Hersh / picture).

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?

About Me

In 1974, during a Soviet army training course, I discovered a simple (almost trivial) extension of Goedel's theorem - my double incompleteness theorem. Since that time, I'm an amateur philosopher of mathematics. My education up to Ph.D. in 1979 was purely mathematical, but I was elected Professor of Information Technologies (second class computer science). However, reading all the funny things about mathematics written even by the most prominent philosophers and mathematicians, I'm feeling at least as their kind of person.

Karlis Podnieks, March 25, 2007