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From: imaginatorium on 21 Apr 2005 12:20 W. Mueckenheim wrote: > "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote in message news:<1114002908.531397.20070(a)o13g2000cwo.googlegroups.com>... <skip rambling> > Even if accepting that, the means to store numbers are limited. The > infinitely many components and the virtual exchange particles of > forces, vacuum polarization and so on are not suitable to store > infinitely many bits. Mathematics is limited. Every reasonable set is > limited. Actual infinity in physics is nonsense, as every reasonable > physicist will confirm. The mathematics that mathematicians study is not limited in this way. Mathematics is not "about" the physical world. End of story. Brian Chandler http://imaginatorium.org
From: W. Mueckenheim on 22 Apr 2005 09:40 imaginatorium(a)despammed.com wrote in message news:<1114100440.140569.166900(a)o13g2000cwo.googlegroups.com>... > W. Mueckenheim wrote: > > "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote in message > news:<1114002908.531397.20070(a)o13g2000cwo.googlegroups.com>... > <skip rambling> > > > Even if accepting that, the means to store numbers are limited. The > > infinitely many components and the virtual exchange particles of > > forces, vacuum polarization and so on are not suitable to store > > infinitely many bits. Mathematics is limited. Every reasonable set is > > limited. Actual infinity in physics is nonsense, as every reasonable > > physicist will confirm. > > The mathematics that mathematicians study is not limited in this way. > Mathematics is not "about" the physical world. End of story. But you need physical means, if you do mathematics as an exact science (and even if you are only dreaming it). Regards, WM
From: imaginatorium on 22 Apr 2005 12:44 W. Mueckenheim wrote: > imaginatorium(a)despammed.com wrote in message news:<1114100440.140569.166900(a)o13g2000cwo.googlegroups.com>... > > W. Mueckenheim wrote: > > > "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote in message > > news:<1114002908.531397.20070(a)o13g2000cwo.googlegroups.com>... > > <skip rambling> > > > > > Even if accepting that, the means to store numbers are limited. The > > > infinitely many components and the virtual exchange particles of > > > forces, vacuum polarization and so on are not suitable to store > > > infinitely many bits. Mathematics is limited. Every reasonable set is > > > limited. Actual infinity in physics is nonsense, as every reasonable > > > physicist will confirm. > > > > The mathematics that mathematicians study is not limited in this way. > > Mathematics is not "about" the physical world. End of story. > > But you need physical means, if you do mathematics as an exact science > (and even if you are only dreaming it). You need pencils and paper*. You do not need physical models of the abstractions you are describing. Mathematics is about abstractions, not about mechanical calculation. * Sorry, add wastepaper baskets. (That was philosophy) Brian Chandler http://imaginatorium.org
From: Lester Zick on 22 Apr 2005 16:00 On Wed, 20 Apr 2005 16:12:27 -0400, Robert Kolker <nowhere(a)nowhere.com> in comp.ai.philosophy wrote: >Ross A. Finlayson wrote: >> Basically people point to Euclid and notice that geometric points and >> lines are defined in terms of each other. If you abstract that (make >> that abstract) in a slightly more enlightened way after thousands of >> years of research into geometry, the point, and line, and plane, and >> circle and sphere, and complex plane and hyperspheres, are all defined, >> at a very fundamental level of geometry and as well mathematical logic >> and numbers and various set relations. > >Check out Hilbert's Axioms for geometry. Points, planes and lines are >all undefined terms. > >Hilbert's Axioms >Undefined Terms > > * Points > * Lines > * Planes > * Lie on, contains > * Between > * Congruent > Yeah, a real milestone in the history of math. Regards - Lester
From: Ross A. Finlayson on 22 Apr 2005 16:12
Hi, I think that basically Chris asks to show that given integers and a rule for addition to prove the distributive law for the multiplication of integers. One method would seem to be through exhaustive induction: basically giving a definition for multiplication along the lines of for each integer, replacing the "3x" with (x+x+x), "4x" with (x+x+x+x), etcetera. Then it is asked why 3(x+1) = 3x + 3. With y = x+1, and the above shorthand notation for multiplication in terms of addition, that is explicitly described for each integer or via an inductive rule, x+1+x+1+x+1 = x+x+x+3 is seen to be true because associativity of addition is already a feature of the Presburger arithmetic with addition. Thus instead of using a "second variable" representing "nx" with an item indexed by n from an inductive rule, it appears that the associativity of addition shows the distributivity of multiplication, as a facet of addition, and multiplication of integers as a repetition of addition. Perhaps I'm misunderstanding what Chris wanted shown, in this case that distributivity of multiplication over associated addition (?) is a result of the associativity of addition and a definition of multiplication of integers in terms of addition. On another note, there is some discussion of natural language and its suitability for the representation of mathematical concepts. Considering that any symbolic language can be transcribed into a natural language, or "plain language", the opinion that some of the names of mathematical abstractions differ from a layman's understanding of the word is of no consequence. They're pieces of jargon, technical and specific, and "an inductive rule that parameterizes limitless quantities" is exactly that, and while vague and nonspecific in a particular interpretation is sufficiently exact for to be communicated. I want to address something about Chris seeming to think that I claimed that all infinite sets are the same size. That is not exactly so, where I do think that infinite sets are equivalent, equipollent, that bijections exist between them, I also accept a variety of set- and particularly number-theoretic ways of gauging the relative sizes of infinite sets, or the propensity of their subsets' elements to be in various distributions ranging over those sets. About arithmetic, I think that from defining zero and successor that all the familiar laws of addition and multiplication are a imple consequence of that, as is the existence of the infinite set of natural integers, and from that more or less directly are subtraction and division, and from zero and alternation the continuum, and from that more or less directly the Euclidean space, and from that and alternation more or less directly the Minkowski space. These "diversions" are flip sides of a coin, the system of numbers is plainly the most obvious thing it could be. Now, about the existence of non-logical axioms in a theory, in terms of a theory's possibility of completeness, that is something to consider. The set of all sets is its own powerset, and quantification over sets implies one. The universe is infinite, and, that's empirical proof that infinite sets are equivalent. Ross |