Obections to Cantor's Theory (Wikipedia article)
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From: Chris Menzel on 24 Jul 2005 09:05 On Sun, 24 Jul 2005 00:12:38 -0700, Vaughan Pratt <pratt(a)cs.stanford.edu> said: > Daryl McCullough wrote: >> Han de Bruijn says... >> >>>No contradiction. That's perhaps the only good thing about it [set >>>theory]. If that is the only thing you care about, let me tell you >>>that most of us care about other things, such as a physics that has a >>>reliable mathematical machinery at its disposal. >> >> Give an example of a calculation in physics in which using modern >> ("Cantorian") mathematics gives you the wrong answer, and using >> some other kind of mathematics gives you the right answer. > > That's pretty easy, isn't it? The Dirac delta function pervades > physics, but if you used "Cantorian mathematics", which defines a > function as a set of ordered pairs, you'd get either the wrong answer, > an overflow error of some kind, or a type check error. Could you elaborate on that?
From: David C. Ullrich on 24 Jul 2005 11:11 On Sun, 24 Jul 2005 00:12:38 -0700, Vaughan Pratt <pratt(a)cs.stanford.edu> wrote: >Daryl McCullough wrote: >> Han de Bruijn says... >> >> >>>No contradiction. That's perhaps the only good thing about it [set theory]. >>>If that is the only thing you care about, let me tell you that most of >>>us care about other things, such as a physics that has a reliable >>>mathematical machinery at its disposal. >> >> >> Give an example of a calculation in physics in which using modern >> ("Cantorian") mathematics gives you the wrong answer, and using >> some other kind of mathematics gives you the right answer. >> >> -- >> Daryl McCullough >> Ithaca, NY >> > >That's pretty easy, isn't it? The Dirac delta function pervades >physics, but if you used "Cantorian mathematics", which defines a >function as a set of ordered pairs, you'd get either the wrong answer, >an overflow error of some kind, or a type check error. Uh, no. The thing that's commonly called the "Dirac delta function" is actually a _distribution_, not a function. There's nothing anti-Cantorian about the theory of distributions, it can be (and is!) set up very nicely in terms of set theory, like more or less all the rest of mathematics. >Vaughan Pratt ************************ David C. Ullrich
From: Daryl McCullough on 24 Jul 2005 10:40 Vaughan Pratt says... >While I'm really enjoying this great thread, at the same time I'm >really bothered that no one has mentioned what to me is by far the >biggest problem with Cantor's theory, namely its claim to universality. >Cantor's paradise, as Hilbert fondly called it, is a mathematical >universe every object of which is a set. The basic objection to this >that I want to raise here can be summarized in the slogan "Groups are >not sets." [mucho deleted] The way I think about it is this: As you say, natural numbers are not finite bit-strings. However, bit-strings capture all the important relationships among natural numbers, in the sense that for any fact about natural numbers, there is an analogous fact about bit-strings. The reverse isn't really true, because bit-strings have extra structure that are irrelevant to the corresponding natural. But not all mathematical objects can be represented by finite bit-strings. For instance, to represent reals you need *infinite* bit-strings. To represent functions from reals to reals, you're out of luck. Sets are believed to be universal *representations*, in the sense that for just about any collection of mathematical objects you are likely to be interested in in "ordinary" mathematics, there is a corresponding collection of sets such that the properties you are interested in are mirrored as corresponding properties of the sets. (The set-theoretic representation may have additional structure that is irrelevant to your problem, but that's a universal problem with representations.) -- Daryl McCullough Ithaca, NY
From: Email me at CS not Boole on 24 Jul 2005 11:00 In article <85ll3w64n4.fsf(a)lola.goethe.zz>, David Kastrup <dak(a)gnu.org> wrote: >Vaughan Pratt <pratt(a)cs.stanford.edu> writes: > >> Daryl McCullough wrote: >>> Give an example of a calculation in physics in which using modern >>> ("Cantorian") mathematics gives you the wrong answer, and using >>> some other kind of mathematics gives you the right answer. >> >> That's pretty easy, isn't it? The Dirac delta function pervades >> physics, but if you used "Cantorian mathematics", which defines a >> function as a set of ordered pairs, you'd get either the wrong >> answer, an overflow error of some kind, or a type check error. > >So what? "Function" _is_ defined as a point-value mapping; that has >nothing to with Cantor. > >If you want to work with things like Dirac delta, the way to do this >is not to use functions (which can't represent them), but >distributions. Those are well-defined, too. > >If you want to talk about infinite cardinalities, you can't use the >naturals, and if you want to talk about integral sieves, you can't use >functions, even though you can get arbitrarily far on the way to >there. > >Just that you can't call a "function" which you use in contexts >usually reserved to functions is not reason enough to change the >concept. Rather, introduce a new one. Point taken. I guess my (narrower) point would have to be that it seems a shame for physicists to have to surrender the term "Dirac delta function" merely because the Cantorian perspective disallows it as a function. After all it is a perfectly good composable object (morphism as a category theorist would say), which gives it the "feel" of a function to its ordinary users. That said, perhaps "Dirac delta morphism" would be the more appropriate term in that case, since category theory is consistent with set theory regarding the usage of "function", where it is reserved for the morphisms of the category of sets, which we all agree the Dirac delta map is not among. As a distribution, Dirac delta is an instance of a morphism but not of a function. Vaughan Pratt -- Don't contact me at pratt(a)boole.stanford.edu, substitute cs for boole instead.
From: David McAnally on 24 Jul 2005 13:02
pratt(a)boole.Stanford.EDU (Email me at CS not Boole) writes: >In article <85ll3w64n4.fsf(a)lola.goethe.zz>, David Kastrup <dak(a)gnu.org> wrote: >>Vaughan Pratt <pratt(a)cs.stanford.edu> writes: >> >>> Daryl McCullough wrote: >>>> Give an example of a calculation in physics in which using modern >>>> ("Cantorian") mathematics gives you the wrong answer, and using >>>> some other kind of mathematics gives you the right answer. >>> >>> That's pretty easy, isn't it? The Dirac delta function pervades >>> physics, but if you used "Cantorian mathematics", which defines a >>> function as a set of ordered pairs, you'd get either the wrong >>> answer, an overflow error of some kind, or a type check error. >> >>So what? "Function" _is_ defined as a point-value mapping; that has >>nothing to with Cantor. >> >>If you want to work with things like Dirac delta, the way to do this >>is not to use functions (which can't represent them), but >>distributions. Those are well-defined, too. >> >>If you want to talk about infinite cardinalities, you can't use the >>naturals, and if you want to talk about integral sieves, you can't use >>functions, even though you can get arbitrarily far on the way to >>there. >> >>Just that you can't call a "function" which you use in contexts >>usually reserved to functions is not reason enough to change the >>concept. Rather, introduce a new one. >Point taken. I guess my (narrower) point would have to be that it >seems a shame for physicists to have to surrender the term "Dirac delta >function" merely because the Cantorian perspective disallows it as a >function. Not strictly accurate. The Dirac delta function can be treated as a function. The domain is S, the set of Schwartz test functions. The range is C. The Dirac delta function maps the Schwartz function f to f(0). ----- |