Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 28 Jul 2005 21:57 In article <u22je11t72jsdhk2dpjrrcu5ukeuk86eod(a)4ax.com>, Martin Shobe <mshobe(a)sbcglobal.net> wrote: > On Thu, 28 Jul 2005 13:50:09 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> 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". > > No. Some parts of mathematics were designed to bolter other parts of > mathematics, and do not need to "go back to earth" (At least not > directly). Other parts were designed purely for the gratification of > the designers. That is also "useful", and often enough, applications > that the original designers never considered are found later. Prime example of that is classical number theory. G. H. Hardy was sure that his work in that area could not ever be put to any practical use, but it has turned out to be the foundation for coding schemes imperative for the security of today's electronic commerce. > > > 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? > > >Just as a matter of prevention, shouldn't we at least _try_ to find > >a way of living together, then? > > Sure thing, let those who wish to concentrate on the mathematics that > "go back to earth", work on the mathematics that "go back to earth" > and allow them to define their terms as they see fit. Let those who > wish to work an mathematics that doesn't, work on mathematics that > doesn't, and let them define their terms as they see fit. And when > one borrows from the other, remember that those who are being borrowed > from do not have to change their theories to suit the borrowers. The > borrower is responsible for appropriately applying what was borrowed. Which is essentially the way it works now, however much the "back to earth" boys wish to censor the others. > > Martin
From: Martin Shobe on 28 Jul 2005 22:06 On Thu, 28 Jul 2005 15:06:47 +0100, Robert Low <mtx014(a)coventry.ac.uk> wrote: >Martin Shobe wrote: > >> 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, > >It's a common claim that one of Cantor's motivations in developing >sets and ordinal numbers was his attempt to understand Fourier series. >Whether you regard that as physics, applied maths, or just maths >may affect you opion of whether set theory had any physical >motivation :-) Maybe I should have said "Set theory does not have physics as it's immediate inspiration, so failing to satisfy some direct connection to physics is irrelevant." Martin
From: Martin Shobe on 28 Jul 2005 22:17 On Thu, 28 Jul 2005 15:56:32 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >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 ... Okay. Still doesn't make sense. How can abstractions and idealizations be physical? And why should "more physical" be better? >>>>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. No, it's not. > That doesn't help. >How about the Lambda Calculus by Church? Not even close. > 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? Actually, I don't have a problem with that. In a sense, we have that now with set theory and category theory (And lets not forget logic). But while physics will continue to provide inspirition to mathematics, it will not qualify as a foundation for methematics. Martin
From: Dik T. Winter on 28 Jul 2005 22:21 In article <dcaqg7$ti4$1(a)news.msu.edu> stephen(a)nomail.com writes: > In sci.math Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: .... > > So what? When someone talks about java with "unbounded" standard types, > > what is the point stating that C does not have "unbounded" standard types? > > He probably thinks that if you speak English loudly enough everyone > will understand. :) Reminds me about the British couple that was in Italy and asked for cold water. What they received, of course, was "aqua caldo". -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 28 Jul 2005 22:24
In article <1122567294.560463.170070(a)g49g2000cwa.googlegroups.com> malbrain(a)yahoo.com writes: > Dik T. Winter wrote: .... > > > Well, the OBVIOUS answer to your question is, "I'm talking about C" > > > However, I'm not that vulgar. I tend to translate discussions into C > > > because I find it to be more universally understood than java. karl m > > > > So what? When someone talks about java with "unbounded" standard types, > > what is the point stating that C does not have "unbounded" standard types? > > How is the length of an integer determined by a java program running on > the virtual-java-machine? karl m Yes, what are you trying to tell? I think you would be better off when you get back on your chair. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |