Obections to Cantor's Theory (Wikipedia article)
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From: Lee Rudolph on 18 Jul 2005 20:36 Stephen Montgomery-Smith <stephen(a)math.missouri.edu> writes: >So, for example, I am told that >Kronecker said something to the effect that "God invented the integers, >man invented the rest." > >But I think that even Kronecker's statement is a huge statement of >faith. I would contend that if you stick to mathematics that is >actually observable in the real world, that even notions such as the set >of integers, or the principle of induction, are dreams invented by >mathematicians. "If the integers did not exist, Man would have had to invent God." --Peano applied to Kronecker Lee Rudolph
From: quasi on 18 Jul 2005 23:59 On 18 Jul 2005 20:36:17 -0400, lrudolph(a)panix.com (Lee Rudolph) wrote: >Stephen Montgomery-Smith <stephen(a)math.missouri.edu> writes: > >>So, for example, I am told that >>Kronecker said something to the effect that "God invented the integers, >>man invented the rest." >> >>But I think that even Kronecker's statement is a huge statement of >>faith. I would contend that if you stick to mathematics that is >>actually observable in the real world, that even notions such as the set >>of integers, or the principle of induction, are dreams invented by >>mathematicians. > >"If the integers did not exist, Man would have had to invent God." >--Peano applied to Kronecker > >Lee Rudolph And then there's this quote: If math is ever proved inconsistent, we have too much invested -- we'll just change logic. quasi
From: Robert Kolker on 18 Jul 2005 21:05 david petry wrote: > > 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. There is no call to put scare quotes around the word -theory-. The theory of transfinite numbers is developed the same way as the theory of algorithmically computable numbers. To wit, it is developed from postulates by means of standard logical arguments. There is nothing special about the logic used to prove theorems in the theory of transfinite cardinals and ordinals. The beef that the anti-cantorians have is with the axioms, not with the means of deducing theorems from the axioms. The only constraint on theories is that they be internally consistent. It is not necessary that a mathematical theory be applicable to the physical world. Bob Kolker
From: The World Wide Wade on 18 Jul 2005 23:21 In article <1121727755.158001.288300(a)g44g2000cwa.googlegroups.com>, "david petry" <david_lawrence_petry(a)yahoo.com> wrote: > These "anti-Cantorians" see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. Such people are ignorant of mathematics. Why should we care what they think?
From: quasi on 19 Jul 2005 03:10
On 18 Jul 2005 16:02:35 -0700, "david petry" <david_lawrence_petry(a)yahoo.com> wrote: > >I'm in the process of writing an article about >objections to Cantor's Theory, which I plan to contribute >to the Wikipedia. I would be interested in having >some intelligent feedback. Here' the article so far. > > .... > >These "anti-Cantorians" see an underlying reality to >mathematics, namely, computation. They tend to accept the >idea that the computer can be thought of as a microscope >into the world of computation, and mathematics is the >science which studies the phenomena observed through that >microscope. They claim that that paradigm includes all >of the mathematics which has the potential to be applied to >the task of understanding phenomena in the real world (e.g. >in science and engineering). To me, it sounds very presumptuous to talk about anti-Cantorians as if they were a well defined group. Much of what you say strikes me as your own opinion. essentially an editorial, but camouflaged by attributing the views to a group. If you could even assemble a group of anti-Cantorians -- try it, I dare you -- I'll bet they would disagree with each other on almost everything. quasi |