Is it possible to calculate in physics with different sets of axioms?

[silenzer]

I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.

 

[Jorriss]

Can you clarify what you mean?

There is generally more than one set axioms that produces the same theory.

 

[silenzer]

Thanks for the reply. What I mean is that I'm debating with someone about whether mathematics is an invention or discovery, and he said that there is only one set of axioms in mathematics that produces results in physics. Is this true? So that, if we were to alter some of those mathematical axioms, the results would be incorrect.

 

[micromass]

Thanks for the reply. What I mean is that I'm debating with someone about whether mathematics is an invention or discovery, and he said that there is only one set of axioms in mathematics that produces results in physics. Is this true? So that, if we were to alter some of those mathematical axioms, the results would be incorrect.

Well, what he said is clearly false.

Do you know which axioms he's referring to?

 

[micromass]

And I'll say it again. Thinking that mathematics is based on axioms is an illusion. The axioms are merely a method of presenting the material and to put it on a rigorous basis.

In actual mathematical research and discovery, axioms are rarely used. What happens is, we look at some basic examples and derive some general theory for that. Then we notice that our theory is very similar to some other theories that are developed. Finally, we abstract those theories to some more general theory. In order to present that abstract theory, we invent some axioms for them.

Everything we do in mathematics is in some way or another tied to nature. For example, the natural numbers are based on counting as we know it. The Peano axioms for the natural numbers were not put forward as some abstract entity. It's not that Peano said: "let's assume these axioms" and then suddenly found out that "hey, these are the natural numbers!". He made the axioms because he wanted to get the natural numbers. If they didn't give the natural numbers, then the axioms were wrong.

So mathematics is, in that sense, an experimental science. We see something that is interesting. And then we abstract this to a mathematical theory. I highly agree with Arnold when he says that: " Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. " See here for the interesting points of view of Arnold: http://pauli.uni-muenster.de/~munsteg/arnold.html

Anyway, given a certain theory, there are many possible axiom systems that will give you the theory. In fact, a mathematical theory consists of some physical examples that we want to abstract. If we can find axiom systems that encompass this, then we're done.

So, focussing on axioms is wrong. Axioms are a language, and not part of mathematics itself.

 

[Bacle2]

I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.

This is mostly a philosophy question, so that I don't believe it has a clear yes/no answer. There are many good books on it; some good ones are the "What is Mathematics", and many of the books by Ian Stewart. Antonio Damasio, a Neurologist ( or some type of brain scientist) put out a good book called "Where Mathematics Comes From". There is another good one by a UCBerkeley linguist whose name I can't remember now, but I'll think about it and get back to you. The good thing about Ian Stewart's book is that , on top of his being a great expositor, he has done research in just-about every area of Math one can think of.

 

[Chronos]

No math = magic.