[Marxism] One more on Fermat's Last Theorem, or, Give Peace a Chance
Jurriaan Bendien
andromeda246 at hetnet.nl
Fri Apr 30 05:39:48 MDT 2004
I had actually logged off MM but now I have to respond to this... Thanks to
Rod for comment. Rod wrote:
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.
I don't believe the world is flat at all, I believe it is round (not a
perfect sphere but slightly conical, i.e. slightly egg-shaped), just as
Gerhardus Mercator did in 1569 in confronting the problem of expressing
three-dimensional space on a two-dimensional plane (a three-space open set
if you like). I am keen to hear your other idea "which is equivalent to your
*infinity* and has the advantage of being precise".
I agree that the set of integers, as a countably infinite set, could be
defined as "infinite extension" without any specific reference to infinity
as a number which is an element of the set, through an expression of
bijection or sequence, but the specific question raised, is whether or not
infinity itself is a separate element of the set of integers, and if not,
why not. It's a simple problem of number theory. I don't see why we could
not define infinity precisely with a formula which relates infinity to every
other member of the set, which seems consistent with the proposition that
the number one divided by infinity equals zero.
In 1993, ... [Fermat's Last Theorem] was partially proven by Andrew Wiles
(Cipra 1993, Stewart 1993) by proving the semistable case of the
Taniyama-Shimura conjecture. Unfortunately, several holes were discovered in
the proof shortly thereafter [in 1994] when Wiles' approach via the
Taniyama-Shimura conjecture became hung up on properties of the Selmer group
using a tool called an Euler system. However, the difficulty was
circumvented by Wiles and R. Taylor in late 1994 (Cipra 1994, 1995ab) and
published in Taylor and Wiles (1995) and Wiles (1995). Wiles' proof succeeds
by (1) replacing elliptic curves with Galois representations, (2) reducing
the problem to a class number formula, (3) proving that formula, and (4)
tying up loose ends that arise because the formalisms fail in the simplest
degenerate cases (Cipra 1995a). The proof of Fermat's Last Theorem marks the
end of a mathematical era.
http://mathworld.wolfram.com/FermatsLastTheorem.html
According to PlanetMath, given a global Galois representation with
representation space unramified at all but finitely many places, it is a
goal of number theory to prove that it arises naturally in arithmetic
geometry (namely, as a subrepresentation of an étale co-homology group of a
motive), and also to prove that it arises from an automorphic form. This can
only be shown in certain special cases.
Les Schaffer suggests that if an infinite countable set N did include
infinity as an element, then it would be a "qualitatively different set" N*
but why exactly would that be ? The question is not one of proof that Louis
Proyect "is not an element of the set Cap", but how we would establish
whether he is or isn't.
For the story of Sophie Germain ("Math's Hidden Woman") see for example
http://www.pbs.org/wgbh/nova/proof/germain.html Ms. Germain discovered that
if n and 2n+1 are primes then Fermat's equation implies that one of x, y, z
is divisible by n.
Platonism distinguishes between an intelligible world of "forms" (similar to
Kantian noumena) and the perceived external world, such that the perceived
external world and what it contains comprises "imperfect copies" (similar to
Kantian phenomena) of the intelligible pure forms or ideas which are
absolutely constant, timeless and perfect, and are cognized or intuited only
by the mind. This idea has merits, but is irreducibly dualistic rather than
dialectical. It ignores specifically:
(1) that the capacity for idealisation itself, based on the practical human
processes of stimulus identification, stimulus discrimination and stimulus
generalisation, is a historically emergent human characteristic developed
through human practice, (2) ideas exist only in the medium of time, and that
abstraction from time in order to permit logical continuity is a cognitive
operation, which proves only that the mind can abstract from time (or that
time, strictly speaking, does not exist for the brain); as far as I know, we
can express time only spatially or kinesthetically
(3) that humans can understand the world only in a way which is intelligible
to humans,
(4) that the cognition of the relationship between a Platonic form and its
imperfect copy is mediated by human practice,
(5) that meaning is constituted precisely by the RELATION between the form
and its so-called imperfect copy or object, i.e. that all meaning is
ultimately relational, and dependent on the cognitive processes I have
mentioned as well as on human practical activity.
Marx's concept of praxis contains both mind and world, in the sense that
praxis is dependent upon and constitutes a finite relation between mind and
world. In dialectical theory, infinity is constituted by the relation
between being and nothingness (not-being) defined as qualities.
"From a scientific perspective, there is no way to know whether there are
objectively existing, external mathematical entities or mathematical truths.
Human mathematics is embodied; it is grounded in bodily experience in the
world. Human mathematics is not about objectively existing, external
mathematical entities or mathematical truths. Human mathematics is primarily
a matter of mathematical ideas, which are significantly metaphorical in
nature. (...) There is not one notion of infinity but many, and not one
formal
logic but tens of thousands, not one set theory or geometry or statistics
but a wide range of them - all mathematics ! (...) The portrait of
mathematics has a human face." (George Lakoff and Rafeal Nunez, Where
Mathematics Comes From; How the Embodied Mind brings Mathematics into Being,
NY: Basic Books, 2000, p. 365, 379).
I am interested in Plato to some extent with regard to the problem of
universals, which has some relevance to the problem of the logical
specification of the concepts of economic value and price. But Platonism is
indefensible because it severs the relationship between the human mind and
the physical and social world in which it exists, and confuses social
entities with natural entities. The "myth of the cave" is a myth, and this
myth is produced by bad (imbalanced) personal development. Imbalanced
personal development is itself caused ultimately by socio-economic
inequality and status differences characteristic of hierarchical societies
divided into social classes, which attach deletrious consequences to
individual indifferences. The best way to reply to Platonic philosophy is
with a pop song lyric by the band Queen:
There's no time for us
There's no place for us
What is this thing that builds our dreams,
Yet slips away from us ?
Who wants to live forever?
Who wants to live forever.....?
There's no chance for us
It's all decided for us
This world has only one sweet moment
Set aside for us
Who wants to live forever?
Who wants to live forever ?
Who dares to love forever
When love must die?
But touch my tears with your lips
Touch my world with your fingertips
And we can have forever
And we can love forever
Forever is our today
Who wants to live forever?
Who wants to live forever?
Forever is our today
Who waits forever anyway?
Jurriaan
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