> 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.
I believe worrying about the existence of any given element of the set of all natural numbers is missing the point. The difference I perceive between infinite sets and finite sets is structural. Even if we to admit the concept of infinite sets into our weltanschauung, we should understand that they are structurally different from finite sets. With a well ordered finite set, we can count from either end. Reversing the order makes sense, etc. We can't do that with infinite sets.
> 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.
I believe there is some confusion regarding the meaning of "axiom". There are at least two distinct connotations that I am aware of. One is "a self-evident truth neither in need of, nor admitting proof". That is the original meaning of the term. Another meaning is "an arbitrarily chosen propositional form admitted into a system without derivation." When people speak of the axioms of logic they typically mean the former. When they speak of the axioms of a group, they mean the latter. As I've said elsewhere, I believe it is a misuse of the term 'axiom' to call the properties of a group 'the axioms of a group'. We can meaningfully call them the axioms of group theory when talking about the closed system of group theory. When we observe that the Lorentz transformations form a group, we would be in error to claim that they satisfy the axioms of a group. In that context, the properties we are erroneously calling axioms are derived.
For myself, I'm still not convinced Euclid's axioms are down for the count. Before anybody jumps on this in haste, bear in mind that my first exposure to Euclidian geometry was in the context of learning about non-Euclidian geometry. -- Nil conscire sibi