Surely You're Joking, Mr. Zeilberger?
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From: Angus Rodgers on 16 Feb 2010 04:07 On Mon, 15 Feb 2010 23:49:54 -0800 (PST), James Dow Allen <jdallen2000(a)yahoo.com> wrote: >I don't see the problem. That must be nice. -- Angus Rodgers
From: Angus Rodgers on 16 Feb 2010 04:25 On Tue, 16 Feb 2010 02:19:08 -0500, David Bernier <david250(a)videotron.ca> wrote: >I can't understand the last part... A zillion marbles in a bag, >we had one marble to the bag, and then there are no marbles (?) Yes. One wonders what his numbers actually do for a living, and what part they play in counting, or in the definition of what it means for a set to be finite, which is so important to him. It would seem, on the face of it, that his universe contains sets 'larger' than his set of natural numbers (such as the set Z x Z that I mentioned), and a finite 'successor function' could easily be defined on such a set (e.g., for the set Z x Z, we could take (m, n)' = (m, n'), except when n' = 0, and then (m, n)' = (m', 0)), and then it's not easy to see why this larger cycle of entities, whose Platonic existence he presumably accepts, would not function just as well in the role of 'natural numbers' as the cycle Z that he started with. Of course, he has Z = Z_p, where p is (in some sense!) a very large prime (never mind that it is equal to 0 in his system!), so his Z is an integral domain, whereas the larger system would not be. In the larger system, it would presumably work out that p^2 = 0 - but I can't see how that's any worse than p = 0! Nor can I see how he is able to get to choose that his Z is an integral domain (therefore also a field) in the first place. -- Angus Rodgers
From: Angus Rodgers on 16 Feb 2010 04:44 On Tue, 16 Feb 2010 09:28:23 +0200, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: >David Bernier <david250(a)videotron.ca> writes: > >> I think he probably is a finistic Platonist (whatever that means), but >> it appears to me a bit of mysticism to say that the largest natural >> number, plus one, is equal to zero. > >Taking the view that statements about naturals should be understood with >reference to the physical (computational?) universe (in some idealised >sense), and assuming we take it to follow on this view that there is a >largest natural, what to make of the successor of the largest natural is >a matter decided essentially by stipulation. We may decree it's zero, >that it's undefined, that applying the successor function to the largest >natural leaves it unperturbed, or pretty much anything that strikes our >fancy, anything we find convenient; just as in ordinary mathematics >whether zero is a natural or not is just a matter of stipulation. If that's so, I can't understand how he can feel himself free to 'stipulate' any properties that he likes, when he is so insistent that the way numbers are is the way that they really are in the world. I'm not saying you're wrong, just that I still can't imagine, on this view, how to think like him. In support of your view, there is the fact that his natural numbers conveniently form a field, which presumably (or at least so it eventually occurred to me as being likely) results from him preferring to avoid the possible embarrassment or awkwardness of them failing to form an integral domain, which in turn seems to imply that he is 'stipulating' his Platonic numbers to have whatever properties he finds convenient (or indeed, 'natural'). But I still can't get my head around it. Perhaps what he's basically Platonist about is [some finitist version] of set theory? The exact definition of what numbers are, in any version of set theory, does always seem to involve some sort of arbitrary stipulation, and perhaps he feels that his arbitrary stipulation makes as much sense as any other? I'm just guessing; I can never get very Platonically worked up about set theory myself; so I still can't imagine what's going on in his mind. But if it's something along these lines, then at least the mystery is closer to the mystery of what people usually take numbers to be when they take set theory as being fundamental. Me, I don't have any idea of what's fundamental, and I haven't even been thinking about it recently, until this interview in a television programme forced me to wonder what on Earth this guy was thinking about! >(Lest there be any confusion, lest any innocent mind be led astray, let >us note here that it is not at all a necessary component of >ultra-finitism or ultra-intuitionism that there be a largest >natural. Coming clean, I must also admit that I didn't bother to consult >Zeilberger's essays before composing this reply.) He does mention that there are other forms of ultrafinitism (some of which, at least, involve some sort of 'fading out' property). -- Angus Rodgers
From: Aatu Koskensilta on 16 Feb 2010 04:42 Tonico <Tonicopm(a)yahoo.com> writes: > He also acknowledges and thankx (!!!) WM for something, and he also > receives a positive feedback from A. Zenkin from whom I read something > some years ago but I can't now remember what, though I clearly > remember that he was on the edge of crankhood, at least. Zenkin's splendid contributions were discussed here a while back. He has uncovered several fatal flaws in the diagonal argument, in particular that it proves nothing, based on vaguely Wittgensteinian reflections. Somewhat dreary and tedious, but great stuff nonetheless! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 16 Feb 2010 04:56
Angus Rodgers <twirlip2(a)yahoo.co.uk> writes: > If that's so, I can't understand how he can feel himself free to > 'stipulate' any properties that he likes, when he is so insistent that > the way numbers are is the way that they really are in the world. Whether there's a largest natural is of course not something we can decide by stipulation -- no, such matters turn on the fundamental physical nature of reality. Even so, there are many questions that do not turn on physical fact, such as whether we count zero as a natural, or what to say about the successor of the largest natural. We can say it's zero, the largest natural itself, five, or whatever we want. This is just a question of what convention to adopt. > Perhaps what he's basically Platonist about is [some finitist version] > of set theory? The exact definition of what numbers are, in any > version of set theory, does always seem to involve some sort of > arbitrary stipulation, and perhaps he feels that his arbitrary > stipulation makes as much sense as any other? There's no need to drag in any set theory. Putting ultra-finitism to one side, we can ask what is, in the natural numbers, the predecessor of zero? In some context it's convenient to say there is no such thing, the predecessor function is not defined at zero, in others it's more convenient to stipulate that zero is its own predecessor. > He does mention that there are other forms of ultrafinitism (some > of which, at least, involve some sort of 'fading out' property). Yes, I hope to have something to say about that later on. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |