guenther vonKnakspot wrote: > Stephen Montgomery-Smith wrote:
>>Now, one of the respondents said that Norm had rejected the axiom of >>infinity, and so how is he supposed to do number theory. But nowhere in >>his article has he rejected the axiom of infinity. He has rejected the >>whole notion that axioms are the way to go. I would paraphrase what he >>said slightly differently - axioms (might) describe the natural numbers, >>but they don't define them. There is clearly a problem that there are >>numbers between 1 and googolplex that we can never write down or >>meaningfully describe. What makes us think that they are really there? >> The evidence is at best empirical. > > Think about your above paragraph for a minute. Do you really think that > the Number 2 is more real than some natural number which has never been > mentioned or even thought of? Do you really think that "2" is the > Number 2 ? "2" is the representation in one particular language of the > name in one particular language of the Number 2. The Number 2 is > neither its name in any language nor the representation of any such > name in any particular language. The Number 2 is something which does > not exist physically and can not be experienced by the senses. It is an > abstract concept as are all other numbers.
I think your opening "think about it" sums it up best. The answer is not obvious to me. I do personally hold some Platonic belief that each of the natural numbers, even all of those between 1 and googolplex, have an abstract existence. But I regard my viewpoint as more a statement of faith (perhaps something that is quote self evident unquote) than anything else. I think that questioning my viewpoint is a reasonable thing to do and not at all crankish.
>>The responses have led me to think that some people here believe in the >>modern axiomatic system like a religion. > > > This is also one of Wildberger's assertions, one that has caused much > anger. I don't think any serious mathematician "believes" the axioms or > in the axioms. Mathematicians agree to use a set of axioms as the > foundation of the formal system they are concerned with.
My statement is aimed more at people reponding to this thread than mathematicians in general. Some very serious and talented mathematicians have very little idea about the foundations of math. And I say all the power to them.