Subject: Re: What is Mathematics?
Date: Tue, 08 Jun 2000 13:30:41 -0400
From: Alex Bogomolny

Dear Ben:

> ... even Godel proved
> ... that you must take some things on intuition

This is not what Godel proved. That you have to take some things on intuition is an indisputable fact valid for all branches of knowledge.

Godel proved that we can't formalize all of mathematics. But regardless, the axioms are not God given. They come from somewhere, and if the source is not our intuition, I do not know what it is.

As an example, the fact that the straight line can be extended indefinitely is definitely based on our intuition and not on our observation.

Peano axioms that claim that any number has a successor are also based on intuition rather than observation.

> I hate to feel

Why? Look around, you probably have a lot of knowledge about many things. You were able to turn on a computer, to get on line, to find and read my page, to write a response. Have you any notion of how did it happen that you speak, write and understand a certain language - yours native? I am pretty sure that you do not. You just go around using it. And note that knowldge of language is pretty basic. Can you imagine life without this means of communication? And, yet, you definitely have no definition of that wonderful tool - language - you use on a daily basis. Or, rather, any definition you may give, won't be any better than the definitions we encounter in mathematics.

The difference is in that mathematics explains why something must be taken on intuition.

> ... my mind starts seeing things in terms of

What's wrong with that? The life and science is full of contradictions, though I do not know exactly what you mean by this. Disagreements, misunderstandings, variations of viewpoints, inconsistencies, etc. are abound.

> I am just so utterly confused as to why mathematics is so successful
> in everything it does yet so "definably impaired".

You statement will remain true if you replace "mathematics" with any other piece of knowledge. You are right in that mathematics has unusually broad applications in which it is remarkably successful.

Look at "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by E.P.Wigner (Comm Pure and Appl Math 13 pp 1-14 1960).

The breadth of applicability comes from a high degree of abstraction. But abstraction does not mean lack of intuition.

> Sometimes I feel like I'm fishing with an imaginary
> rod when I'm using mathematics.

Do not you have the same feeling about language, driving car, etc?

I personally got some ideas about effectiveness of mathematics from reading Holland's book on "Adaptation in Natural and Arificial systems." (I make a reference to it in my June's MAA Online column.) The operation of crossover between two chromosomes features a built-in "implicit parallelism" - an awful numbers of genes and their combinations get promoted into the future generation at once. I can't explain this in a short reply. He talks about schema: a combination of a group of genes that is indifferent to other genes.

Schema, in my view, is an exact analog of abstraction: ignoring other genes one simply ignores what is irrelevant for the study at hand or phenomenon under discussion, etc. Mathematics is this kind of schemata: it only picks a few memes at a time. Concrete knowledge comes from schemata with the addition of specific genes. But look into my column: Goldberg's book and Mitchell's are quite popular.

All the best,
Alexander Bogomolny