Obections to Cantor's Theory (Wikipedia article)
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From: david petry on 18 Jul 2005 19:02 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. *** While the pure mathematicians almost unanimously accept Cantor's Theory (with the exception of a small group of constructivists), there are lots of intelligent people who believe it to be an absurdity. Typically, these people are non-experts in pure mathematics, but they are people who have who have found mathematics to be of great practical value in science and technology, and who like to view mathematics itself as a science. These "anti-Cantorians" see an underlying reality to mathematics, namely, computation. They tend to accept the idea that the computer can be thought of as a microscope into the world of computation, and mathematics is the science which studies the phenomena observed through that microscope. They claim that that paradigm includes all of the mathematics which has the potential to be applied to the task of understanding phenomena in the real world (e.g. in science and engineering). 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. The anti-Cantorians see such a belief as an absurdity (in the sense of being disconnected from reality, rather than merely counter-intuitive). The pure mathematicians tend to view mathematics as an art form. They seek to create beautiful theories, which may happen to be connected to reality, but only by accident. Those who apply mathematics, tend to view mathematics as a science which explores an objective reality (the world of computation). 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. The artists see the requirement that mathematical statements must have observable implications as a restriction on their intellectual freedom. The "anti-Cantorian" view has been around ever since Cantor introduced his ideas. Witness the following quote from a contemporary of Cantor: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there" (Kronecker) In the contemporary mainstream mathematical literature, there is almost no debate over the validity of Cantor's Theory. It was the advent of the internet which revealed just how prevalent the anti-Cantorian view still is; 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. 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.
From: Stephen Montgomery-Smith on 18 Jul 2005 19:26 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 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 argumnt 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 inadvertantly offended, or if you are I would certainly be interested in hearing a non-crackpot approach against Cantor's arguments. Best, Stephen > > *** > > While the pure mathematicians almost unanimously accept > Cantor's Theory (with the exception of a small group of > constructivists), there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. > > These "anti-Cantorians" see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). > > 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. The anti-Cantorians see > such a belief as an absurdity (in the sense of being > disconnected from reality, rather than merely counter-intuitive). > > The pure mathematicians tend to view mathematics as an art > form. They seek to create beautiful theories, which may happen > to be connected to reality, but only by accident. Those who apply > mathematics, tend to view mathematics as a science which explores > an objective reality (the world of computation). 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. > The artists see the requirement that mathematical statements must > have observable implications as a restriction on their intellectual > freedom. > > The "anti-Cantorian" view has been around ever since Cantor > introduced his ideas. Witness the following quote from a > contemporary of Cantor: > > > "I don't know what predominates in Cantor's > theory - philosophy or theology, but I am sure > that there is no mathematics there" (Kronecker) > > > In the contemporary mainstream mathematical literature, there > is almost no debate over the validity of Cantor's Theory. > It was the advent of the internet which revealed just how > prevalent the anti-Cantorian view still is; 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. > > > 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. >
From: Virgil on 18 Jul 2005 19:30 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. > > *** > > While the pure mathematicians almost unanimously accept > Cantor's Theory (with the exception of a small group of > constructivists), there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. > > These "anti-Cantorians" see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). > > 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. The anti-Cantorians see > such a belief as an absurdity (in the sense of being > disconnected from reality, rather than merely counter-intuitive). > > The pure mathematicians tend to view mathematics as an art > form. They seek to create beautiful theories, which may happen > to be connected to reality, but only by accident. Those who apply > mathematics, tend to view mathematics as a science which explores > an objective reality (the world of computation). 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. > The artists see the requirement that mathematical statements must > have observable implications as a restriction on their intellectual > freedom. > > The "anti-Cantorian" view has been around ever since Cantor > introduced his ideas. Witness the following quote from a > contemporary of Cantor: > > > "I don't know what predominates in Cantor's > theory - philosophy or theology, but I am sure > that there is no mathematics there" (Kronecker) > > > In the contemporary mainstream mathematical literature, there > is almost no debate over the validity of Cantor's Theory. > It was the advent of the internet which revealed just how > prevalent the anti-Cantorian view still is; 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. > > > 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. Seems fair enough to me, though it overlooks that what many anti-Cantorians propose is as effectively counter-scientific as it is counter-Cantor. Are most anti-Cantorians actual scientists or are they, like WM, neither mathematicians nor scientists.
From: Stephen J. Herschkorn on 18 Jul 2005 19:46 How does one justify the statement that "reality = computability"? To which of the axioms of Zermelo-Frankel do anti-Cantorians object? Can anti-Cantorians identify correctly a flaw in the proof that there exists no enumeration of the subsets of the natural numbers? Do anti-Cantorians accept that sum(i=1..infty, d_i / 10^i) "exists" for each collection (d_i) of decimal digits (as i ranges over the positive integers)? If so, how do they correctly justify the collection of these real number are countable? If one is to insert an entry on anti-Cantorians in any encylopedia, it must include the answers or lack thereof to these questions. -- Stephen J. Herschkorn sjherschko(a)netscape.net Math Tutor in Central New Jersey and Manhattan
From: Stephen Montgomery-Smith on 18 Jul 2005 20:03
david petry 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. The anti-Cantorians see > such a belief as an absurdity (in the sense of being > disconnected from reality, rather than merely counter-intuitive). > > The pure mathematicians tend to view mathematics as an art > form. They seek to create beautiful theories, which may happen > to be connected to reality, but only by accident. Those who apply > mathematics, tend to view mathematics as a science which explores > an objective reality (the world of computation). 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. > "I don't know what predominates in Cantor's > theory - philosophy or theology, but I am sure > that there is no mathematics there" (Kronecker) If you are going to argue at this level, it seems to me that the problem is not with the hierarchy of different super-infinities, but with the very concept of infinity itself. So, for example, I am told that Kronecker said something to the effect that "God invented the integers, man invented the rest." But I think that even Kronecker's statement is a huge statement of faith. I would contend that if you stick to mathematics that is actually observable in the real world, that even notions such as the set of integers, or the principle of induction, are dreams invented by mathematicians. For example, consider the collection of all numbers between 1 and googolplex. There are quite a large number of them (indeed most of them) that one could never write down on a piece of paper - even if that piece of paper was as large as the solar system. This means that a bunch of these integers, for all practical purposes, simply don't exist. And googolplex is small - what about googolplex taken to the power of itself googolplex times? Now you may counter that "in principle" we can write all these integers down. But as I said, this is essentially an act of faith, which has no visible proof. And if you can take this leap of faith, why not go a step further, and believe in the set of all real numbers, which, if you accept its existence, cannot logically be placed into one-to-one correspondence with the integers. Indeed why not go the whole way, and believe in the whole von-Neuman universe of sets? The only thing that will stop you in this belief is if you find some mathematical inconsistency. But as Goedel proved, this mathematical inconsistency might even exist amongst regular old number theory - you will never know unless you find it. You could also argue that the set of integers is a convenient abstraction to represent the notion of counting, which is definitely a real life activity. (Of course, we never actually count up to googolplex, so it really is an abstraction.) But then I would counter that the real numbers are merely a convenient abstraction to represent lengths and times and such like - definitely useful, because the resulting theories like calculus have clear real world applications. And if you accept this abstraction, even merely hypothetically, then you must accept the correctness of the Cantor's diagonal argument. Seriously, if you think that you have found a way to construct a one-one correspondence between the integers and the real numbers, I strongly advise you to spend a lot of time proof-reading your work. Because you will not have only contradicted the Cantorians, but you will have contradicted the whole way in which modern mathematicians think. I'm not saying that it cannot be done, but so many people have tried unsuccessfully that I am not going to spend a lot of time checking yet another attempt. Best, Stephen |