[Marxism] One more on Fermat's Last Theorem
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
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
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