Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 23 Jul 2005 17:17 In article <1122150789.124648.311950(a)f14g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Jesse F. Hughes schreef: > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > > > Robert Low wrote: > > > > > >> stephen(a)nomail.com wrote: > > >> > > >>> Are you claiming that set theory is directly applied in General > > >>> Relativity? Do you actually have evidence of that? > > >> > > >> I have seen a paper in which transfinite induction was > > >> used. And in the existence of maximal solutions to the > > >> initial value problem, appeal is (at least sometimes) > > >> made to Zorn's lemma. > > > > > > I'm not surprised. Whether people like it or not, mathematics cannot be > > > separated from its applications. > > > > That conclusion is not justified from what Low said. All he said was: > > this particular part of mathematics has been applied. > > So what? Am I not allowed to draw my own conclusions? > > Han de Bruijn Are we not allowed to disagree with them?
From: Han.deBruijn on 23 Jul 2005 17:32 Jesse F. Hughes wrote: > "david petry" <david_lawrence_petry(a)yahoo.com> writes: > > > Han de Bruijn wrote: > > > >> Indeed, idealizations > >> don't necessarily obey the same laws of logic as the directly observable > >> objects. But, at least, there should exist a path from the abstractions > >> back to the observable objects. Directly observable sets are idealized - > >> and that's a good thing - but idealized sets must also be "materialized" > >> again, for the sake of _applications_. The latter tends to be forgotten. > > > > That hit's the nail on the head. Thanks Han. > > > > For those who missed it, the key sentence is: > > > > "But, at least, there should exist a path from the abstractions > > back to the observable objects." > > Seems like nonsense to me. > > You are still claiming that an abstraction is useful, but applying > deductive logic to the abstraction may yield false results unless we > check nature. But this has two obvious problems: I wish everybody should stop confusing idealizations and "abstractions" in the first place. Even a physical measurement is kind of abstraction: A measuring rod "abstracts" length from an object. But, after that, the abstraction of a length is "idealized" in mathematics. It is clear that an idealization of the _above_ kind, as it has originated from physical experience, will not likely yield false results and that we don't have to check nature every time again. The reason is that our "length" has been abstracted by a physical device (nature itself through the hands of men) and that it has been idealized "properly", meaning that there is a way back to "real" lengths. (A. Einstein has written nice things about this, BTW. It's not so simple after all, but it works.) > (1) The abstraction doesn't exist in nature. Now, maybe that means > (to you) that there is no "path from the abstractions back to the > observable objects", but then why did you call infinity a *useful* > abstraction? Or what path do you have in mind and how does it > contradict Cantor's theorem? An idealization _never_ exists in nature. As far as the idealization called 'infinity' is concerned, the classical concept of a limit has indeed a path back to earth, called: inaccuracy, errors, sloppyness. The gist is that the delta's and the epsilon's in the limit concept are idealizations from real measurement "errors". All we have to do with materialization (the way back) is to re-introduce these errors and accept them as "real". Or if N -> oo, replace it by "very big". But how about 'infinity' as introduced by Cantor's theories ? That's indeed a very good question. See below. > (2) How can any useful abstraction fail to satisfy the basic laws of > deductive logic? You have still not explained that. If our useful > abstraction justifies the presumption of A and A -> B, then it damn > well better justify the presumption of B and there's no need to check > nature to see. Once idealizations have been chosen carefully and properly, there should be no problem with logic whatsoever. If nature can prove its consistency, so can we. > But that is all that Cantor's theory uses, too. Basic axioms prove a > perfectly simple result. You agree that the proof is correct, but > want to claim that normal logic doesn't apply to infinite sets (which > are nonetheless "useful". The problem is with the "basic axioms", not with the logic. These axioms are definitely NOT a proper idealization from what physicists would call a "set" into what Cantor has defined as a set. See for example: http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cm > Of course, you don't give any principle to determine *what* > derivations involving infinite sets are acceptable, aside from "look > out the window and see how infinite sets really behave." I have said that derivations involving limits are acceptable. > Your position is really incoherent. You can't simply say that > infinite sets are legitimate mathematical objects but that normal > logic doesn't apply to them. You must have a principled objection > (not "Cantor violates my intuitions." or "Everything before Cantor was > about computation.") and you must have a solution. Which axioms are > wrong? Do you really suppose that deductive logic is not the right > logic for set theory? Then what is? I haven't said anything like that. > Anyway, viva la revolution and all that. Or that. Han de Bruijn
From: Han.deBruijn on 23 Jul 2005 17:40 Virgil wrote: > That is the difference between pure and industrial mathematics. Pure > mathematics does what is enjoyable, industrial mathematics only does > what is useful. Fortunately for both, they overlap to a remarkable > degree. What those who would restrict all mathematical studies to the > strictly industrial fail to realize is that it has often been pure > mathematical developments that have been of great use in expanding > industrial mathematics. You may be surprised, Virgil, but for once in my life I agree with you a great deal. Han de Bruijn
From: david petry on 23 Jul 2005 19:29 Jiri Lebl wrote: >Given all the results that modern >analysis (which is heavily based on Cantor's set theory) has brought >us, it seems that reality at least seems to work as in our models which >do use Cantor's ideas, rather then not. What we know of the real world >is based on models that were derived or proved from Cantor's theory. Cantor's set theory has been intertwined with modern analysis to such an extent that it sometimes appears that analysis would fall apart without the glue of set theory to hold it together. My claim is that all the results of analysis which have observable implications, an hence have the potential to be applied as models of the real world, necessarily do not need Cantor's Theory. >There is a difference in arguing that Cantor's theory has no basis in >reality on intuitive means and in arguing that Cantor's theory is >wrong. Have I every said that Cantor's Theory is "wrong"? >Perhaps it is that Cantor's theory is only useful in >approximate models of the reality. So far no one has come up with a >system of mathematics capable of solving real world problems that for >example doesn't use the completeness of the real numbers which is the >property that guarantees uncountability of the real numbers. Mathematicians were solving real world problems long before Cantor's Theory came along. Applied mathematicians never need to think about completeness, and I suspect many don't know what it is. >> In the contemporary mainstream mathematical literature, there >> is almost no debate over the validity of Cantor's Theory. >That is because mathematically it is a sound theory. You will NOT find >any further debate on this in mainstream mathematical literature. I certainly have not claimed that it is not a formally sound theory. My argument is that the mathematics that is relevant to real world problems necessarily has a property of "observability" - that is, mathematical statements must have observable implications, where we think of the computer as the mathematician's microscope which lets us make observations. Before Cantor came along, mathematicians would have listened to that argument. Now they don't. >> 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. >This paragraph is total nonsense put in purely because you were lacking >another paragraph to put into the article. Uh, I disagree.
From: Martin Shobe on 23 Jul 2005 19:36
On 23 Jul 2005 12:56:34 -0700, Han.deBruijn(a)DTO.TUDelft.NL wrote: >Martin Shobe wrote: > >> Your response to what is wrong with a definition of infinity was that >> "Most of us care about other things, such as a physics that has a >> reliable mathematical machinery at its disposal". The only connection >> between physics and that definition of infinity, is that the physicist >> choose to use that portion of mathematics that makes use of it. If >> that portion of mathematics does not reliably model reality, the >> *physicist* should be using a different portion of mathematics. This >> may result in the mathematicians having to develop a new branch (see >> Calculus, Dirac's delta, etc.), but that doesn't invalidate anything >> that was there before. >> >> Yet your response is that *mathematics* is what must change. Not the >> physicists. Most mathematicians do not believe that mathematics has >> more than an accidental connection to reality. If the physicists >> believe otherwise, then the phsycists need to have their training >> adjusted to emphisize that mathematics is not reality. > >That sounds reasonable. Alas. Seems that we have become such >separatists >that almost nobody of us still believes in that great ideal: > > One World or No World > >Doesn't anybody agree with me that a Unified Science would be >desirable? Not particularly. There certainly should be plenty of commmunication between them, but there is just too much unify and still expect people to be able to handle it. >If it only where for the sake of efficiency. I'm not convinced it would be efficient. Such a thing would be so big, that by the time one learned it, one would be dead. Martin |