On Fri, 14 Jul 2006 21:41:59 GMT, Stephen Montgomery-Smith <email@example.com> wrote:
>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.')
If we're to take Turing at his word I suspect computable numbers are nothing more than what can be produced using Turing mechanics.
>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. > >Stephen