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From: Chris Menzel on 22 Apr 2005 19:11 On 22 Apr 2005 13:12:40 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > 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. No. I was trying to get you to see that you cannot define multiplication in terms of addition. > 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. Ok, try it out with the following statement: (x)(y)[y*(x+1) = (y*x)+y] (i.e., "For any natural numbers x and y, the product of y and x+1 is equal to the sum of y*x and y). See the problem? > 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. There's nothing to consider, there is only something to know. Goedel did not prove that all (consistent) theories with nonlogical axioms are incomplete. You should learn what he did prove. Chris Menzel
From: Ross A. Finlayson on 22 Apr 2005 22:11 Chris Menzel wrote: > On 22 Apr 2005 13:12:40 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > > 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. > > No. I was trying to get you to see that you cannot define > multiplication in terms of addition. > > > 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. > > Ok, try it out with the following statement: (x)(y)[y*(x+1) = (y*x)+y] > (i.e., "For any natural numbers x and y, the product of y and x+1 is > equal to the sum of y*x and y). See the problem? > > > 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. > > There's nothing to consider, there is only something to know. Goedel > did not prove that all (consistent) theories with nonlogical axioms are > incomplete. You should learn what he did prove. > > Chris Menzel Not really, no, because that's true. It doesn't matter which natural integers x and y are, if for each natural integer n there is the identity nx = x+x+... (n many times), for natural integer x there is natural integer z = x+1, and y+y+... (z many times) = y+y+... (x many times) + y through associativity y+y+... (z many times) = y+y+... (x+1 many times) and via z = x+1 0 = 0 So, in accepting that for each natural integer n there is a definition of integer multiplication, and what it is, then associativity of addition, and replacement of like terms, leads to distributivity of this definition of multiplication over addition. What's the problem? In terms of Goedel's incompleteness, I am under the impression that the conditions of Goedelian incompleteness include the existence of non-logical axioms in the theory, although perhaps not infinitely many. I'm talking about some things that Goedel perhaps did not preclude to exist. What do you think about the possibility of valid inferences from only the logical axioms? Does that change given the existence of any set or number? I arrived at the notion of the ur-element and excluded middle for various reasons. When I see that it meshes well with technical concerns about consistency and completeness, and that it reflects the thinking of Kant, Hegel, and other eminent philosophers in terms of a mathematical logic, I'm happy about that. Ross
From: W. Mueckenheim on 23 Apr 2005 11:51 imaginatorium(a)despammed.com wrote in message news:<1114188284.512430.25180(a)z14g2000cwz.googlegroups.com>... > > > > > > 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) Of course numbers are not mathematics, but numbers are an important part of mathematics. Further pencils, paper and wastepaper baskets (or recycling facilities) and your brain: All are made of matter. Regards, WM
From: Ross A. Finlayson on 23 Apr 2005 22:59 I do mathematics for my own personal gratification. There can be, only one, theory. Ross
From: Robert Kolker on 23 Apr 2005 23:37
Ross A. Finlayson wrote: > I do mathematics for my own personal gratification. > > There can be, only one, theory. There are many theories, some of them pairwise contradictory or contrary. Bob Kolker |