Obections to Cantor's Theory (Wikipedia article)
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From: Han de Bruijn on 19 Jul 2005 06:52 David Kastrup wrote: > Cantor's work leads to quite unintuitive results, while still > being quite accessible to the layman. It has met opposition from > mathematically competent opponents at its time but has, partly > connected with changes in set theory, been made an integral part > of today's mathematics. While "anti-Cantorians" make themselves > quite visible on Usenet groups, they are actually few but > prolific, with a non-mathematical background, and unable to put > forward a coherent argument. Remarkably prevalent among them is > the inability to understand nested quantifiers. Non-mathematical background ? Look at yourself ! You claim that you know something about Numerical Analysis, while it is quite clear from your postings that you don't even have a clue. Han de Bruijn
From: Dave Seaman on 19 Jul 2005 07:48 On Tue, 19 Jul 2005 12:52:39 +0200, Han de Bruijn wrote: > David Kastrup wrote: >> Cantor's work leads to quite unintuitive results, while still >> being quite accessible to the layman. It has met opposition from >> mathematically competent opponents at its time but has, partly >> connected with changes in set theory, been made an integral part >> of today's mathematics. While "anti-Cantorians" make themselves >> quite visible on Usenet groups, they are actually few but >> prolific, with a non-mathematical background, and unable to put >> forward a coherent argument. Remarkably prevalent among them is >> the inability to understand nested quantifiers. > Non-mathematical background ? Look at yourself ! You claim that you know > something about Numerical Analysis, while it is quite clear from your > postings that you don't even have a clue. You, on the other hand, have shown that you do not understand the difference between numerical analysis and mere numerical methods. Hint: the former includes error analysis. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
From: David Kastrup on 19 Jul 2005 07:56 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > David Kastrup wrote: > >> Cantor's work leads to quite unintuitive results, while still >> being quite accessible to the layman. It has met opposition from >> mathematically competent opponents at its time but has, partly >> connected with changes in set theory, been made an integral part >> of today's mathematics. While "anti-Cantorians" make themselves >> quite visible on Usenet groups, they are actually few but >> prolific, with a non-mathematical background, and unable to put >> forward a coherent argument. Remarkably prevalent among them is >> the inability to understand nested quantifiers. > > Non-mathematical background ? Look at yourself ! You claim that you know > something about Numerical Analysis, while it is quite clear from your > postings that you don't even have a clue. Actually, I did some course work during my diploma studies. And helped out on the diploma thesis of a befriended mathematician that was trying to approximate some recursively defined probability distributions of piecewise exponential characteristics, proving to her that her naive approach of Simpson's rule was leading to large cascading errors (as well as exponentially increasing runtime). She was not alone with that ill-considered approach: the distribution in question had been handled that way in literature, taking weeks of computation time and coming up basically with junk. And yes, numerical analysis would not have just provided the method (which was employed here), but also the error estimates. And ignoring them was what turned this from mathematics into hand-waving. So we figured out how to do this semi-symbolically (this was before the widespread advent of symbolic calculation) by using combined exponentials and polynomials, and _useful_ results (namely with controllable errors) dropped out after few minutes of runtime (the symbolic expressions had a few hundred terms, reasonably fast to evaluate, but infeasible for manual calculation). In my engineering studies and diploma thesis, I also had to work a lot with error propagation in sliding-window Fourier transforms and synthesis, and I also did quite a bit of fixed and floating point arithmetic applications as a programmer where error estimates were important. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: sradhakr on 19 Jul 2005 08:08 Dear Prof. Montgomery-Smith, The Cantorian viewpoint has been seriously challenged by my newly proposed logic NAFL (Int. J. Quant. Inf., vol. 3, No. 1 (2005), pp. 263-267; see also <http://philsci-archive.pitt.edu/archive/00001923/>, math.LO/0506475, cs.LO/0411094, quant-ph/0504115. Please see my responses below to your message. Stephen Montgomery-Smith wrote: > david petry wrote: > > I'm in the process of writing an article about > > objections to Cantor's Theory, which I plan to contribute > > to the Wikipedia. I would be interested in having > > some intelligent feedback. Here' the article so far. > > > I have to admit that I don't follow the anti-Cantorian arguments very > much, but when I do, I get the sense that they lack coherence, and > perhaps they lack even intellectual honesty. I submit that it is the Cantorians who are intellectually dishonest, for they have failed to respond to my arguments presented in the above REFEREED, PUBLISHED work. For that matter, I don't think that refereed publications are essential in this day and age of instant electronic communication. If significant claims are posted to respectable electronic archives like the arXiv and the PhilSci Archive, then the academic community has a duty to respond, if only to correct any misunderstandings that may arise due to the wide reach of these archives. > > I can see Kronecker's point of view, which I guess is that Cantor's > theories depends upon the existence of mathematical objects that don't > seem to exist in real life (e.g. what is a real number, really?). If > the anti-Cantorians argued at this level, I think that I would > essentially be in agreement with them. I also think that the > pro-Cantorians and anti-Cantorians could co-exist side by side, holding > different philosophies as to what mathematics represents, but agreeing > upon its practical consequences. > > But I find that anti-Cantorians try to say something quite different, > which is that the Cantorian position is logically wrong. This is > clearly absurd, unless you change the laws of logic, and since they are > currently working well, and no-one is able to come up with something > different and sane, why change them? > > I had this experience when I tried to enter into a discussion with an > anti-Cantorian about how perhaps the Cantor approach is helpful in > telling us that we don't need to be searching for a halting function, > since a Cantor/Turing style argument shows that they don't exist. But > the response I got from this person wasn't even wrong - it was shear > nonsense, and I quickly gave up. > > Honestly, I feel that your article about anti-Cantorians is too generous > towards them, and in the final analysis I would not be supportive of > Wikipedia accepting such an article. I don't think that > anti-Cantorianism as I have experienced it is simply a different point > of view, rather I genuinely believe that those who propose such a > viewpoint are crackpots. > > I hope that you are not yourself an anti-Cantorian whom I have > inadvertently offended, or if you are I would certainly be interested in > hearing a non-crackpot approach against Cantor's arguments. > > Best, Stephen > The logic NAFL is the correct approach, and not just against Cantor's arguments; NAFL has important positive aspects, as is clear from my papers. So far the international academic community has pretended that NAFL does not exist, rather than answering me point by point. How does one deal with people who fail to acknowledge the existence of something as important as a new logic, a new philosophy of mathematical truth, a new way of doing theoretical science (physics, mathematics, computer science), etc.? I used to think that maybe it is my fault, maybe NAFL isn't that important, maybe I haven't explained my ideas clearly, etc. But not any more. Now that my work has been published, the onus is on the academic community to honestly evaluate/criticize it and give NAFL its due. I am assuming, of course, that the academic community consists of honest, sincere people who are genuinely interested in taking science forward, rather than merely "protecting turf". Regards, R. Srinivasan
From: David C. Ullrich on 19 Jul 2005 08:26
On Tue, 19 Jul 2005 09:34:45 +0100, Alec McKenzie <mckenzie(a)despammed.com> wrote: > "Stephen J. Herschkorn" <sjherschko(a)netscape.net> wrote: > >> Can anti-Cantorians identify correctly a flaw in the proof that there >> exists no enumeration of the subsets of the natural numbers? > >In my view the answer to that question a definite "No, they >can't". > >However, the fact that no flaw has yet been correctly identified >does not lead to a certainty that such a flaw cannot exist. Yet >that is just what pro-Cantorians appear to be asserting, with no >justification that I can see. I once had a person tell me the following, with a straight face: (*) "You can't say for sure there's no such thing as a square circle! I mean just because they haven't found one yet doesn't mean they won't discover one tomorrow." Please choose one of the following replies: (i) No, (*) is nonsense. If it's square then _by definition_ it's not a circle. So they will _never_ find a square circle. (ii) Hmm, good point. You really should choose one of (i) or (ii), so people know how to reply to your post. The point: If you say (ii) then we know that there's no point worrying about anything you say. Otoh if you say (i) then there's hope - you agree that we're _certain_ they will never find a square circle, now we just have to convince you that our assertions about enumerations of subsets of N are just as certain, for entirely similar (although slightly more complicated) reasons. So which is it, (i) or (ii? ************************ David C. Ullrich |