[Marxism] One more on Fermat's Last Theorem

Rod Holt rholt at planeteria.net
Fri Apr 30 00:15:32 MDT 2004

Having been a professional mathematician and a meta-mathmaticion at 
that, I will note that *you* can mystify anything you like but do not 
pretend that others do so.
Russell, Goedel, Kleene, Zermelo and Church demystifyed number theory, 
the statement calculus, the predicate calculus, and set theory. 
Princeton published a monograph years ago by Goedel (ca.1950?) 
entitled*Axiomatic set theory and the Consistancy of the Axiom of choice 
and the Continuum Hypothesis* which can be followed by the intellegent 
reader with persistance over a year or two.
The notion of infinity you carry around is an intuitive notion, sort of 
like the "world is flat." It is not part of mathematics. We do not use 
the symbol (an eight lying on its side) except in analysis, which, as 
you know, is very sloppy. We do have another idea which is equivalent to 
your *infinity* and has the advantage of being precise. If you want me 
to expand on this topic I will be glad to later.
Number theory deals first with the integers, negative, zero, and 
positive. We do not care how big a particular number is. We introduce 
the negative integers as inverses. The + sign is used as shorthand for 
repeated counting; the minus sign as shorthand for counting backwards; 
multiplication is repeated adding, etc. We create the rational numbers 
simply by stating that for every integer a and b, ax=b (a?0) has a 
solution amongst the collection of rational numbers.
Throughout, we are very careful to distinguish between a formal theorem 
of number theory and a meta-theorem of number theory.
There are many definitions of *proof.* A convincing argument is 
considered by some people a proof. If a mathematical system is enlarged 
by adding an axiom (not inconsistent), any proof in the smaller system 
can be carried forward. The accepted notion of a proof can be derived 
from the observation that a theorem is the bottom line of a proof. 
Therefore, every line of a proof is either an instance of an axiom or is 
derived from the previous lines by the stated rule of inference (usually 
modu ponens).
--Rod Holt

Jurriaan Bendien wrote:

>Domnhall suggests mathematicians can talk rubbish and fool around with
>numbers, which is true in my experience and cost me a lot. But saying that
>doesn't prove or illuminate anything.
>I asked for a proof that infinity is not a member of the set of integers,
>but Domnhall just says that the set of integers is defined to exclude
>infinity and consequently also the set (inf-1, inf-2, inf-3....  n).
>In other words he fails to answer the question and provide proof, but
>nevertheless insists that infinity is not an integer. Then he talks about
>being "primitive" and "difficulty" and so on, but he is only talking about
>himself and pre-Galilean mathematics.
>As a consequence, he sophistically mystifies the process by which
>mathematics is created and the central problems which fuel its development.
>Galileo already observed that there are as many integers as there are
>perfect squares.
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