Patricia Shanahan wrote: > Stephen Montgomery-Smith wrote: > > 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. > > Isn't there also a simplicity argument here? The set of all natural > numbers, with no gaps and no end, is in many ways an easier thing to > reason about than the set of natural numbers that someone, somewhere, > has used or will use.
You may be right, but your argument is incompletely formed, and I don't trust it. It's not clear what you mean by "easier to reason". How do you measure "ease of reasoning"? And it's not clear why ease of reasoning (whatever that is) should be a priority.
My main question is this: If we accept your simplicity argument, how do we draw the line between orgies of metaphysical speculation (on the one hand) and actual science (on the other)? It seems that the metaphysicians would argue (according to your approach): Our aim is ease of reasoning, i.e. *simplicity*. We're building vast castles of useless logic and nested infinities which have no relevance whatsoever to the real world, because it's *simpler* to proceed that way. That seems wrong to me.