[Marxism] Is Fermats Last Theorem False ?

["Jurriaan Bendien" <andromeda246@xxxxxxxxx>]

In 1867, Pierre de Fermat, a lawyer and amateur mathematician, noted in his
copy (now lost) of Bachet's translation of Diophantus' Arithmetika that
there are no positive integers where that x^n + y^n = z^n for n>2. Another
way of saying this, as Bruce E. Meserve notes is that if n is an integer
greater than 2, there do not exist any integers x, y, and z such that their
product xyz would equal a non-zero number and that xn + yn = zn
(Fundamental Concepts of Algebra, 1983, p. 96).

Fermat implied he could prove that relation was impossible ("I have
discovered a truly remarkable proof which this margin is too small to
contain") focusing on cases where n=3 or 4 or 5 but he never published on
it. In fact Fermat's last theorem proves at best only that the area of a
right triangle cannot be a square, and therefore that a rational triangle
cannot be a rational square. In symbols, there do not exist integers x, y, z
with x2 + y2 = z2 so that xy/2 is a square.

In mathematical constructivism, however, as contrasted with a more primal
intuitionist mathematics, it is necessary to actually find or construct a
mathematical object to prove that it really exists. If we assume that an
object does not exist, and derive a contradiction or inference from that
assumption, we have not established the contradiction.

Bruce Meserve notes that large prizes were offered for a proof of Fermat's
Last Theorem, and although the theorem was proved for all values of n
greater or equal to 617, no general proof was found (op. cit. p. 96-97;
Meserve cites H.S. Vandiver, "Fermat's Last Theorem, Its History and the
Nature of the Know Results Concerning It", American Mathematical Monthly,
vol. 53, p. 555-578, 1946).

In recent times, Princeton researcher Andrew Wiles claimed to have found an
aesthetically satisfying proof, referring to Euler and Hecke, but by 1994 it
was quite clear that there was a hole in the argument. (see e.g. Simon
Singh, Fermat's Last Theorem, 1998).

The question then arises whether number theory and set theory could shed
light on Fermat's last theorem, and possibly provide a general proof, and if
so, whether any solution is compatible with categorical set theory. The
properties of integers are studied in number theory, which defines integers
as a set of natural numbers (0, 1, 2, ...) and their negatives (-1, -2, -3,
...  This is usually denoted by Z and treated as a totally ordered set
without upper or lower limits. The idea can be expressed as ... < -2 < -1 <
0 < 1 < 2 < ... Normally it is assumed that -0 = +0 = 0, and so -0 is
ignored as a separate integer, even although the true mathematician would
include it as a separate integer. Rational numbers, defined as a ratio of
two integers a and b (such that a/b where b?0) comprise a subset of Z.

>From all this, you can deduce theorems of the type that, for the whole set
of integers, if a < b and c < d, then a + c < b + c. Likewise if a < b and 0
< c, then it must be the case that ac < bc and so on. Introducing negative
integers, you can solve x  for the more simple arithmetic problems of the
type a + x = b, provided that a and b are constant natural numbers. If
however x is a natural number, then only some of these equations, but not
all, can be solved.

Using integers, you can also do division with remainder. So then if a and b
are integers where b is a non-zero number, then you can always find a
quotient q as well as a remainder r  equal to a/b for the formula a = b.q +
r  if r  equals zero or is greater than zero, and if r is smaller than b.

In set theory, integers are defined as a "countable infinite set". A
countable set is a set which is either finite or countably infinite. Any set
which is not countable is uncountable. All uncountable sets are infinite,
but not all countable sets are infinite, because some are finite. A set is
called finite if there exists a one-to-one correspondence (or bijective
mapping) between that set and another set of the type {1, 2, 3, ..., n} for
a natural number n  including the empty set which results if n = 0.

With the aid of the concepts of ordinals, isomorphism and bijection, it can
be proved logically that, if you assume the axiom of choice, a set is
infinite if and only if it includes a denumerable subset (Azriel Levy, Basic
Set Theory, 1979, p. 79). This was first intuited by Archimedes who took
three pairs of magnitudes infinite in number, and pronounced that they were,
respectively, "equal in number", implying for the first time in the history
of Greek mathematical theory that objects infinite in number could be equal
in magnitude although simultaneously it was implied that not all objects,
infinite in number, are so equal

A set is called "countably infinite" if there exists a bijective mapping
between it and the set N comprising all natural numbers. So then if A
denotes the set of positive integers  {1,2,3,...}  and B denotes the set of
even positive integers {2,4,6,...} only, then A and B must have an identical
size, and therefore, set B is countably infinite. The mathematical concept
of infinity was developed from the late 19th century by Bolzano, Cantor,
Frege, Dedekind and others in set theory, a branch of endeavour which the
constructivist mathematician Errett Bishop dismissed irritably as "God's
mathematics" which we should just leave for God to do. Likewise the
encyclopedic but near-blind mathematician Henri Poincaré, a founder of
modern chaos theory, claimed that "set theory is a disease from which
mathematics will one day recover" (he had no use for God either).

Nevertheless, set theory involved not only the important concept of
one-to-one correspondence as a standard for comparing set size; it also
rejects the old Galilean idea that "the whole can never be the same size as
the part". So, infinite sets can have the same size as at least one of their
"proper" parts; they can have different sizes; and they could be countably
infinite or uncountable.  This is basic to the theory of cardinal numbers.
Extended number systems, like the surreal numbers, incorporate both ordinary
(finite) numbers and infinite numbers of different sizes. Thus,
interestingly, Science Daily Encyclopedia notes how "our intuition gained
from finite sets breaks down when dealing with infinite sets".

If we return to Archimedes and apply basic set theory to Fermat's Last
Theorem, we could take the case where x=1 and y=2 and z=3 and n=? . In that
case, we obtain x? + y? = z? and then we have the simplest proof that there
do exist positive integers which satisfy Fermat's conditions, and therefore
Fermat's theorem about their non-existence can be considered falsified.
Voltaire remarked, "The only way to comprehend what mathematicians mean by
infinity, is to contemplate the extent of human stupidity." Mathematicians
can be stupid in their antics. But we do need them.

Jurriaan




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