Obections to Cantor's Theory (Wikipedia article)
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From: MoeBlee on 19 Jul 2005 01:21 "david petry" <david_lawrence_petry(a)yahoo.com> wrote in message news:1121727755.158001.288300(a)g44g2000cwa.googlegroups.com... > > 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. > <SNIPPED> I think this article needs a lot of work. It really rubs me wrong. And not because of my bias in the subject, but because the article is barely informative and it is very poorly conceived. These are my criticisms and suggestions (not necessarily in order of importance): 1. This article sets up an opposition between pure mathematicians and a fragment of people interested more in applied mathematics. We may have an impression that pure mathematicians tend to be friendlier to Cantor than applied mathematicians tend to be, and this would make sense. But we don't have in front of us the hard numbers showing that this is a fact. You seem to use this supposed alignment more as a dramatic device for your story than as a well considered observation. 2. You don't qualify that mathematicians who accept the Cantorian basis, usually don't accept the original, inconsistent theory but only reformations made by Zermelo and others. 3. You've described anti-Cantorists as favoring computation and you continue to give this snapshot of the supposed typical anti-Cantorist. But that is only your impression. In general, what is the basis of your research for these kinds of claims? Mostly anecdotal from your own experience reading on-line, etc, I suspect. You have more a basis for an editorial or an essay than for an encyclopedia article. 4. You wrote: "Cantor's Theory, if taken seriously, would lead us to believe that while the collection of all objects in the world of computation is a countable set, and while the collection of all identifiable abstractions derived from the world of computation is a countable set, there nevertheless "exist" uncountable sets, implying (again, according to Cantor's logic) the "existence" of a super-infinite fantasy world having no connection to the underlying reality of mathematics. But you only qualified that that is an anti-Cantorist view in the following sentences. That might be okay for magazine writing, but for your article I would make very explicit, before you launch into that paragraph, that this is the view of an anti-Cantorist. 5. You wrote: "The pure mathematicians tend to view mathematics as an art form." Maybe so, but it's not substantiated, even with the vague qualifier 'tend to'. Also, you seem again to be setting up pictures in the reader's mind as if to make a "story" out of this with a "cast of characters." That's how it comes across to me, and I very much dislike it. This is my reaction to, admittedly, subtleties, but I find the approach gimmicky and that it detracts from the intellectual content of the subject. 6. You wrote: "Those who apply mathematics, tend to view mathematics as a science which explores an objective reality (the world of computation)." Again, what are your hard numbers showing that relatively fewer non-applied mathematicians view their work as science? Any surveys? Anything other than what you subjectively surmise? 7. You wrote: "In science, truth must have observable implications, and such a "reality check" would reveal Cantor's Theory to be a pseudoscience; many of the formal theorems in Cantor's Theory have no observable implications." But this is from the paragraph that seemed perhaps to start to give voice to a pro-Cantorist view. Already, after giving a paragraph or more to anti-Cantorists, you're back to adding more of their view. In general, the paper lacks structure. You should have some structure, such as: Intro What Cantor set theory is History of the controversy Pro Cantor view Anti Cantor view Dialectic on both views Conclusion But I agree with most posters here, that even this might be unfair, since it puts crackpots on equal intellectual footing with real mathematicians. Digressions As some posters have pointed out: You should distinguish between, on the one hand, rejecting axioms, and, on the other hand, denying that proofs from axioms are indeed proofs. Also, you should give examples of some of the supposed refutations of the uncountability proofs and point out why the supposed refutations are invalid. Also, you should mention that if an anti-Cantorist actually refuted uncountability, then that would be a discovery of an inconsistency in Zermelo set theory itself. And you might mention how little Zermelo set theory needs to assume to prove uncountability. Also, perhaps mention that some anti-Cantorists think that the source of supposed error in proofs of uncountability is some kind of problem with reductio ad absurdum, while the proofs don't even need reductio ad absurdum, as well as reductio ad absurdum is just a derived ruled of propositional logic anyway. Also, interesting questions are not even raised by you. For example, what axioms do anti-Cantorists propose? 8. You wrote: "there seems to be a never-ending heated debate about Cantor's Theory in the Usenet newsgroups sci.math and sci.logic. Typically, the anti-Cantorians accuse the pure mathematicians of living in a dream world, and the mathematicians respond by accusing the anti-Cantorians of being imbeciles, idiots and crackpots." Again, you're playing up the rivalry, as if it's a kind of culture-clash. These vignettes are not helpful. And when you mention the derision, the name calling, that anti-Cantorists take, this makes pro-Cantorists seem unreasonable and unsympathetic. Notice, the anti-Cantorist rebuke is said by you to be that the Cantorists are living in a dream world, which is a fairly mild rebuke. But the pro-Cantorists are made to look like grade schoolers for responding with "idiot" et. al. You omit mentioning the patience pro-Cantorists take in over and over and over again explaining the mistakes to anti-Cantorists who deny the proofs. The anti-Cantorists who deny the proofs are actually indulged. 9. You wrote: "It is plausible that in the future, mathematics will be split into two disciplines - scientific mathematics (i.e. the science of phenomena observable in the world of computation), and philosophical mathematics, wherein Cantor's Theory is merely one of the many possible "theories" of the infinite." Many things are plausible in the future. This paragraph is particularly not informative, and seems to me just a dashed off thought to be a spiffy closer. Oooh, you're peering into the future! Cut that silliness. You don't even mention current work that might be pointing to something in the future. Your closer, as your article, is fluff. Anyway, the Cantorian approach ALREADY IS one out of a selection of available theories. Your article is not at all scholarly. 10. You blur the distinction between contemporary crackpots and historical critics. 11. You wrote about anti-Cantorists: "they are people who have who have found mathematics to be of great practical value in science and technology" In other words, they believe a virtual truism. Who does NOT beleive that mathematics has had great practical value in science and technology.I don't know who this makes sound dopier, anti-Cantorists or the writer, for even mentioning. I would scrap even writiing this article. The very basis of this as an encyclopedia entry is extremely shaky. If other already existing articles on Cantor, set theory, foundations, constructionism, and intuitionism don't already mention the actually interesting intellectual controversies, then those articles should be bolstered to do so. There's no need for a separate article that barely separates intellectual history from Internet crackpotism. As well as, unless the intellectual organization of this article were vastly improved, it would stand out as well below the general level of writing on the proposed publication site. P.S. Perhaps not material for this article, but I am very curious what truly motivates some of the more dogged proof deniers. Is there something other than a mathematical issue? Are any people motivated by theological reasons? I so, what are they?
From: David Kastrup on 19 Jul 2005 01:53 Virgil <ITSnetNOTcom#virgil(a)COMCAST.com> writes: > In article <1121727755.158001.288300(a)g44g2000cwa.googlegroups.com>, > "david petry" <david_lawrence_petry(a)yahoo.com> 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. > > Seems fair enough to me, though it overlooks that what many > anti-Cantorians propose is as effectively counter-scientific as it > is counter-Cantor. It seems completely unfair and absurd to me since it gives the "Anti-Cantorians" equal consideration/hearing. That overlooks that a) it is a minuscule ratio of people b) they are visible only in non-professional circles like Usenet c) none of those that are visible can sustain a coherent argument or present a more or less consistent case. In fact, the ratio of people among them that are unable to understand the meaning of the order of quantifiers is almost 100%. d) there is no consistent theory that is put forward instead of what they aim to replace. Every crank has his own, different version of lala land that he presents. I won't say that serious mathematical philosophy that would be incompatible with findings from Cantor is impossible, and it would likely have to be based on different axioms. But if somebody came up with it, he has found no capable voice presenting him, at least here. Here we just have a small number of mathematically incapable but very verbose cranks that all have their own pet theories that are self-inconsistent at the most basic levels, let alone incompatible with each other. Giving them such consideration as in your "article" is completely lunatic. Their criticism boils down to "math must not be hard for me." We don't a Handicapper General in math: <URL:http://www.tnellen.com/cybereng/harrison.html> It is very much inappropriate to give them the exposure they don't get elsewhere. The article is much too long. It can be subsumed into 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. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Helene.Boucher on 19 Jul 2005 02:07 david petry wrote: <snip> There is no mention of one historical or living figure who is anti-Cantorian, what their objections were, and what logical and mathematical systems they advocated. It's a rant. Maybe Wikipedia accepts rants, but if it does it will quickly lose any status it might have (does it have any?).
From: Helene.Boucher on 19 Jul 2005 02:10 David Kastrup wrote: > It seems completely unfair and absurd to me since it gives the > "Anti-Cantorians" equal consideration/hearing. That overlooks that > > a) it is a minuscule ratio of people > b) they are visible only in non-professional circles like Usenet > c) none of those that are visible can sustain a coherent argument or > present a more or less consistent case. In fact, the ratio of > people among them that are unable to understand the meaning of the > order of quantifiers is almost 100%. > d) there is no consistent theory that is put forward instead of what > they aim to replace. Every crank has his own, different version of > lala land that he presents. > > I won't say that serious mathematical philosophy that would be > incompatible with findings from Cantor is impossible, and it would > likely have to be based on different axioms. But if somebody came up > with it, he has found no capable voice presenting him, at least here. > Solomon Feferman is probably the best known living anti-Cantorian.
From: Chan-Ho Suh on 19 Jul 2005 02:24
In article <1121727755.158001.288300(a)g44g2000cwa.googlegroups.com>, david petry <david_lawrence_petry(a)yahoo.com> 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. > The truth of the matter is that the article you wrote constitutes *original research* on your part, despite your attempt to ascribe your views to the "anti-Cantorians", which is, as quasi pointed out, not a well-defined group. Thus, it is not acceptable for inclusion into Wikipedia. You should read more on Wikipedia policy before wasting everyone's time including your own. Your article will either be VFD'd or successively modified in such a manner that you will be greatly unhappy with the result. But before any of that happens, you will have wasted a lot of other people's time and energy. |