Obections to Cantor's Theory (Wikipedia article)
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From: Robert Kolker on 28 Jul 2005 08:12 Ross A. Finlayson wrote: > Virgil, why don't you reply to me anymore? > > Infinite sets are equivalent. In what sense? Not in the sense of having the same cardinality. The set of integers and the set of reals has different cardinalities. Bob Kolker
From: Robert Kolker on 28 Jul 2005 08:13 Han de Bruijn wrote: > > If I say to you that I can draw a straight line, will you then respond > to me that straight lines cannot be drawn, because they are "abstract"? Do not confuse a smudge on a writable surface with a curve. Your so-called lines are in fact three dimensional arrays of stuff. Bob Kolker
From: malbrain on 28 Jul 2005 09:40 Han de Bruijn wrote: > Martin Shobe wrote: > > > And the gist of our argument is that not being able to "go back to > > earth" is not a problem. Mathematics is not restricted to those > > things that can "go back to earth". > > Precisely. But perhaps "_useful_ mathematics" is restricted to those > things that can "go back to earth". If we make a distinction between > useful and useless mathematics, formalize it and present that to our > politicians, tax payers and bosses. Would that be a problem? Useful/useless for what? One important thread in the long history of mathematics has been to keep "politicians, taxpayers, and bosses" from getting stuck in ruts without resolving contradictions in their work. karl m
From: David Kastrup on 28 Jul 2005 09:53 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Martin Shobe wrote: > >> And the gist of our argument is that not being able to "go back to >> earth" is not a problem. Mathematics is not restricted to those >> things that can "go back to earth". > > Precisely. But perhaps "_useful_ mathematics" is restricted to those > things that can "go back to earth". But it will more often than not "go back" in completely unexpected ways. And that's why it is contraproductive to cling to the Earth like math's life depended on it. For example, number theoretic transforms have turned out to be a very good tool for fast convolutions, and convolutions happen to be one of the most common operation when modeling physical systems. And number theory certainly is not down to Earth. If one had kept down to Earth all the time, one would never have got where it actually becomes a compelling advantage to use this in a real-world application. > If we make a distinction between useful and useless mathematics, > formalize it and present that to our politicians, tax payers and > bosses. Would that be a problem? > > No? It would lead to the demise of the "useful" mathematics, since the "useful" mathematics are often offspring from the "useless" mathematics and could not be formed into one coherently working system without the theoretical underpinnings of "useless" mathematics. > Just as a matter of prevention, shouldn't we at least _try_ to find > a way of living together, then? If you want to do physics, do physics. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 28 Jul 2005 09:56
Martin Shobe wrote: > On Thu, 28 Jul 2005 11:48:09 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > >>Martin Shobe wrote: >> >> >>>Since sets aren't physical, it is not physically correct to assert >>>that a e A ==> a c A. From what I understand of the history of set >>>theory, physics wasn't the inspiration, and it certainly isn't its >>>purpose at the current time. >> >>Since sets aren't physical, since straight lines aren't physical, since >>numbers aren't physical. Of course they _are_ physical. Or better: they >>_were_ physical. Though some have been more physical than others. Those >>which have been more physical are the better ones, i.e. straight lines. > > > I'm sorry, but I can't make any sense out of this. Keep trying. Give up mathematics for a few moments and use what we call "common sense". Put your feet on the ground. Just once. Please ... >>>That doesn't mean that you can't explore theories where a e A ==> a c >>>A. Just don't confuse what you are doing with standard set theory. >>>(And it would be nice if you indicate that you aren't doing standard >>>set theory, if for no other reason than to avoid some confusion.) > > >>Heh, heh. That's precisely the first thing I want: throw _doubt_ upon >>everything that everybody finds so certain. And I thought that I have >>indicated that well enough, all over the place, ever since 1989. > > > But your objections don't cast any doubt, because they are irrelevant > to what you are objecting to. Set theory does not have physics as > it's inspiration, so failing to satisfy some physics viewpoint is > irrelevant. And what you do in non-standard theories doesn't > necessarily impact the standard theories. Set theory does not have physics as it's inspiration? I wouldn't be so certain of that! See my next poster in this thread, which is actually from 1991, Re: The Political Economy of Sets. I've said repeatedly that by "physics" I mean physics in a broad sense. Maybe "physical" provides a better clue of what I mean, I don't know. > If you want to make set theory an unimportant branch of mathematics, > your best bet is to find an alternative foundation that is acceptable > to mathematicians. (I.e. phsyics doesn't cut it). The only real > competitor for foundations of mathematics that I'm aware of is > category theory, and that does it by providing a foundation for set > theory, and using that to found other areas. Category Theory is just another form of Set Theory. That doesn't help. How about the Lambda Calculus by Church? But without the claim that it is the one and only foundation possible. Why not have _several_ pillars that provide a foundation, instead of just one? Han de Bruijn |