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.
I have always regarded this kind of occum's razor type argument as a rule of thumb rather than a hard and fast method of deduction. But I think you make a good point - if we restrict ourselves to only saying that computable numbers exist, we leave a lot of very awkward gaps in the natural numbers. (I mean, how do you even properly define 'computable' - Cantor type diagonal arguments are always going to say any definition is incomplete, like 'the smallest number that cannot be described in less than 13 words.')
His argument about his number having 'in effect' no prime factorization is also interesting. Presumably he has some number theory argument that proves it has no small prime factors. But then he is implicitly accepting all the results that number theory produces. (I know that many of you exclaimed - "how can a university teacher say this kind of stuff" - but fully understanding the import of his statement means that one has already accepted the formal proof of the fundamental theorem of arithmetic - he is not saying the proof is wrong, rather he is saying that maybe the proof doesn't mean anything, or at least doesn't apply universally.)
I also disagree with him that thinking of the real numbers as equivalence classes of Cauchy sequences of rational numbers is wrong (or a joke, as he puts it). While I concede the possibility that the underlying set theory in which this definition is made might be all screwed up, I feel that if or when we get set theory right, that the real numbers will still be equivalence classes of Cauchy sequences, only perhaps the word "equivlence class" will no longer represent a partition of sets, and similarly the phrase "Cauchy sequences" might be a more restrictive term. But conceptually, it really does capture what real numbers are all about.
I don't know if the real numbers really do exist, but I think they do at least represent something concrete rather more than just computable real numbers. The concepts of completeness and compactness - in their more abstract forms - really do become important if one is studying differential equations and dynamical systems, particularly the kind of qualitative long term behavior described by things like strange attractors. I like the example of the Lorenz attractor http://en.wikipedia.org/wiki/Lorenz_attractor which is an example of an ordinary differential equation in which the long term qualitative behavior can only be known if the initial data is known to an infinite precision. And in many cases, we don't yet know if equations that describe real life systems exhibit this kind of strange behavior (although we conjecture that they do).
While I really do think the OP (Norm) is not a crank, I do think that his position does open up lots more problems than it purportedly solves. Nevertheless, it is interesting discussion.