Mathematics and Philosophy

Date: 6/5/96 at 21:3:57
From: Anonymous
Subject: Maths-phylosophy

Hello,

I have heard that maths is the universal language. I was wondering if
mathematicians viewed maths as being 'transcendental' of the universe?
The laws of physics apply specifically to this universe - does the
same apply to mathematics? Is it really a universal language, or 
would an alien race have a completely different kind of logic?

Thanks for your time.

Date: 6/13/96 at 21:55:11
From: Doctor Tom
Subject: Re: Maths-phylosophy

It's difficult to know what an alien race might do or think.
In fact, our understanding of mathematics and the foundations
of mathematics has changed radically in just the last 60 or
70 years.

There is a field called "metamathematics", which is the study
of mathematics.  It's "outside" of mathematics the same way
"metaphysics" is outside of physics.

But we can reason in metamathematics about the sorts of things
that can and cannot be done with mathematics.

For example, let's look at a "simple" problem.  Given Peano's
postulates that describe the natural numbers 0, 1, 2, ..., is
it true that any theorem about them can be shown to be true or
false given enough time, and given a smart enough mathematician?

Up until 1930 or so, people thought that this was the case -
that there might even be a mechanical way to decide (given
enough time) whether any theorem about the natural numbers is
true or false.

But then Kurt Godel proved (using meta-mathematics) his so-called
"incompleteness theorem".  It states that there is a theorem
about the natural numbers that can neither be proved true nor
proved false.  That you can add it or its negation as an axiom,
and either will give a consistent theory of the natural numbers.

What's more, even if you add this axiom, there will be another
undecidable theorem, and no matter how many axioms you add, there
will always be another.

Some theories can be proven complete; others can't.  The study
of this is very deep, and very interesting.

Mathematics is clearly more powerful (in a sense) than physics.
For example, in physics in our universe, light always travels at
the same speed.  A mathematician, however, would be able to tell
you what the physics would be like in a universe where light
travels at a different speed, or where the electric field dropped
off as 1/r^3, or where there were only protons and no electrons
to balance the charges.

I believe that mathematics is quite universal, but I don't think
it's possible to "prove" that!

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/