Surely You're Joking, Mr. Zeilberger?
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From: Tonico on 16 Feb 2010 12:48 On Feb 16, 6:44 pm, Angus Rodgers <twirl...(a)yahoo.co.uk> wrote: > On Tue, 16 Feb 2010 07:38:54 -0500, David Bernier > > <david...(a)videotron.ca> wrote: > >In the standard real analysis of today, the Intermediate Value Theorem and > >the Mean Value Theorem are used a lot. For example, if f(x) is > >a polynomial function of odd degree and real coefficients, > >f: R -> R, then the IVT implies that f has at least one real root. > >And the IVT rests upon the completeness of the reals. > > >The proof at Wikipedia uses the least upper bound property for reals: > ><http://en.wikipedia.org/wiki/Intermediate_value_theorem> > > >Zenkin rejects cardinals above aleph_0, and also Cantor's Diagonal proof.. > > >I'm not sure how those who reject the orthodox treatment of real numbers > >would want to prove that f in IR[x] of odd degree has at least one real root ... > >Go back to geometrical intuition? Construct root-finding algorithms? > > I already seem to detect signs of the dreaded convergence towards > a discussion of the usual suspects and their heresies. Of course > I have no power to stop that, just because I started the thread, > but, unless either there is something new being said about other > mathematical heretics, or jokers (I haven't been around for a long > time, and I never followed all such threads - only occasionally the > JSH ones), or such a widened discussion sheds light on Zeilberger's > serious or humorous beliefs about the natural number system (by the > way, he seems to reject the entirety of mathematical analysis, root > and branch, as meaningless!), I would beg for the discussion not to > go too far in that direction. > > On a more positive note, what I would like is to be helped to see > (a) how it is possible for anyone to think like Zeilberger at all, > and (b) how thinking like Zeilberger is nevertheless impossible for > me in particular, because it is incompatible with some principles I > hold to and he doesn't. If I can understand how even a professional > mathematician can believe something so fundamental, which I believe > to be false, I might better understand why I believe it to be false. > > It might be a bit like coming to an understanding of the principle > of conservation of energy, by seeing in detail a failed attempt to > build a perpetual motion machine. That analogy might not work very > well - because I can't imagine coming to a better understanding of > field theory by following in detail some foolish person's attempt > to duplicate the cube! - but I think that is because it omits one > feature of this case, which is that what Zeilberger denies seems so > 'obviously' true (unlike energy conservation, or the impossibility > of duplicating the cube by Euclidean means), that being able to > understand DZ's wacky point of view ought to help me to see beyond > the 'obvious', which is always a good thing - so long as you don't > completely lose sight of what is, in the end, still 'obvious'! > > At least, that's why I think I'm interested, but any account of > reasons for being interested is less interesting than the thing > itself. > > -- > Angus Rodgers It was "obvious" that the heavier an object is the fastest it'll freely fall towards the floor, specially after Mr. Aristotle determined this, until Mr. Galileo proved otherwise. That our intuition can get damaged by some facts in science in general and in maths in particular is no news, though professional mathematicians, used to this, could be more aware of the problem and develop a "new mathematical intuition"...which, again, can be wrong all the way. I don't know why Doron believes what he does, and reading his stuff I can't find some really compelling reason to that: perhaps a craving for "originality", perhaps something in his personal educational background, perhaps just a (mathematical or whatever) huntch. What I find hard to digest from him is the patronizing and belittling attitude he undertakes against those accepting "the paradise of fools" that Cantors bestowed us. Let us not forget that not only the huge majority of mathematicians, but at least a great deal (most of them...?) of logicians have no problems with that paradise. I'm gonna read some of Doron's writings once again... Tonio
From: Angus Rodgers on 16 Feb 2010 15:42 On Tue, 16 Feb 2010 09:48:59 -0800 (PST), Tonico <Tonicopm(a)yahoo.com> wrote: >I don't know why Doron believes what he does, and reading his stuff I >can't find some really compelling reason to that: perhaps a craving >for "originality", perhaps something in his personal educational >background, perhaps just a (mathematical or whatever) huntch. What I >find hard to digest from him is the patronizing and belittling >attitude he undertakes against those accepting "the paradise of fools" >that Cantors bestowed us. Let us not forget that not only the huge >majority of mathematicians, but at least a great deal (most of >them...?) of logicians have no problems with that paradise. > >I'm gonna read some of Doron's writings once again... Ditto. The first statement of his philosophy that I read was this: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html (extract, page 8) ``Myself, I don't care so much about the natural world. I am a platonist, and I believe that integers, finite sets of finite integers, and all finite combinatorial structures have an existence their own, regardless of humans (or computers). I also believe that symbols have an independent existence. What is completely meaningless is any kind of infinite, actual or potential. So I deny even the existence of the Peano axiom that every integer has a successor. Eventually we would get an overflow error in the big computer in the sky, and the sum and product [of] two integers is well-defined only if the result is less than p, or if one wishes, one can compute modulo p. Since p is so large, this is not a practical problem, since the overflow in our earthly computers comes so much sooner than the overflow errors in the big computer in the sky.'' I haven't found anything that clarifies this. But he also keeps saying things like this: http://www.math.rutgers.edu/~zeilberg/Opinion69.html ``Since all knowable math is ipso facto trivial, why bother? So only do /fun/ problems, that you really enjoy doing. It would be a shame to waste our short lives doing "important" math, since whatever /you/ can do, would be done, very soon (if not already) faster and better (and more elegantly!) by computers. So we may just as well enjoy our humble trivial work.'' This seems (at least to my depressed mind) to have an implicitly depressive tone to it, and it might give some sort of clue to his motivation, if not to his actual beliefs. Some sort of inferiority complex in relation to the computer, not only on his own behalf, but on behalf of humanity as a whole? He seems to /enjoy/ being tied down to the finite, the concrete, the physical, and the computable. I can't imagine enjoying that, nor can I imagine understanding mathematics on such a basis, any more than I can imagine understanding it on any other basis! But it's what he actually believes that matters, not speculation as to why he believes it. Even if I'm going to disbelieve something, I like at least to have a clear idea of what I'm disbelieving - and so far, I can't get my head around his ideas at all. There's another manifesto here: http://www.math.rutgers.edu/~zeilberg/Opinion43.html "It Is Time to Move On to NON-EUCLIDEAN MATHEMATICS" And (in spite of the overall optimistic tone) there's that sad refrain again: "Let's face it, anything we humans can know for sure is trivial, since we are creatures of such low complexity." I haven't had a look at this yet (referenced from the above): http://www.math.rutgers.edu/~zeilberg/GT.html "Plane Geometry by Shalosh B. Ekhad XIV" There's more about computers and programming here (I haven't read this one right through yet): http://www.math.rutgers.edu/~zeilberg/GT.html "Don't Ask: What Can The Computer do for ME?, But Rather: What CAN I do for the COMPUTER?" That article contains some feedback, including his own reply to Greg Kuperberg in sci.math.research, thus (edited down by me): `` >Doron's argument for his thesis is a little crazy, but it's not >completely crazy. You shouldn't interpret it literally, >even though Doron himself might. Of course, you should not interpret it literally, but neither should you interpret anybody's text literally. As Derrida, Rorty and several others have shown, we are slaves to our own final# vocabularies and we always have hidden agendas, and our `objective views' are just an instrument to bolster our ego, and to justify to ourselves our miserable existence. Now that plain Racism and Sexism is out of style, we cling to Human Chauvinism. '' Derrida and Rorty? Oh dear. Oh dearie me. That exchange (all of it, not just the bit I've quoted) seems quite revealing, at least as to his motives and his general philosophy, but it still leaves me in the dark as to how he can actually believe what he believes about the natural numbers. -- Angus Rodgers
From: Angus Rodgers on 16 Feb 2010 15:47 On Tue, 16 Feb 2010 20:42:43 +0000, I mistyped: >http://www.math.rutgers.edu/~zeilberg/GT.html >"Don't Ask: What Can The Computer do for ME?, >But Rather: What CAN I do for the COMPUTER?" Correction: that URL should be: http://www.math.rutgers.edu/~zeilberg/Opinion36.html -- Angus Rodgers
From: Angus Rodgers on 16 Feb 2010 16:08 On Tue, 16 Feb 2010 20:47:15 +0000, I wrote: >>"Don't Ask: What Can The Computer do for ME?, >>But Rather: What CAN I do for the COMPUTER?" > >Correction: that URL should be: > >http://www.math.rutgers.edu/~zeilberg/Opinion36.html Now that I actually start to read through it, I see that it even mentions the book title on which I based the title of this thread: 'Surely You're Joking Mr. Feynman'! I hadn't seen it, I promise! -- Angus Rodgers
From: David Bernier on 16 Feb 2010 16:22
Angus Rodgers wrote: [...] > On a more positive note, what I would like is to be helped to see > (a) how it is possible for anyone to think like Zeilberger at all, > and (b) how thinking like Zeilberger is nevertheless impossible for > me in particular, because it is incompatible with some principles I > hold to and he doesn't. If I can understand how even a professional > mathematician can believe something so fundamental, which I believe > to be false, I might better understand why I believe it to be false. [...] How can one know extensively about what exists in the physical universe? There could be particles or things besides neutrinos, etc. Generally speaking, aren't many philosophies close to systems of beliefs? About (a), I'd say "computer-centric" ultrafinitism is DZ's philosophy of mathematics, a kind of system of beliefs. It seems to me that many philosophical views are very hard to prove wrong, maybe because things can't be tested, etc. I'd say that Zeilberger is quite settled in his finitistic views of the universes of physics and maths. I had a look at a PDF file of a talk he gave at a conference: < http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf > |