Das Kalenderblatt 100413

12/04/2010 - 09:49 von WM | Report spam
"You can keep counting forever. The answer is infinity. But, quite
frankly, I don't think I ever liked it. I always found something
repulsive about it.
I prefer finite mathematics much more than infinite mathematics. I
think that it is much more natural, much more appealing and the theory
is much more beautiful. It is very concrete. It is something that you
can touch and something you can feel and something to relate to.
Infinity mathematics, to me, is something that is meaningless, because
it is abstract nonsense."
[...] The great infinitarian Hugh Woodin (of Woodin "cardinal" fame)
[...] had the following reaction (my emphasis)

"To the person who does deny infinity and says that it doesn't exist,
*I feel sorry for them*, I don't see how such view enriches the world.
Infinity may be does not exist, but it is a beautiful subject. I can
say that the stars do not exist and always look down, but then I don't
see the beauty of the stars. Until one has a real reason to doubt the
existence of mathematical infinity, I just don't see the point."
[...] I can reciprocate and express my extreme pity and heartfelt
condolences to Professor Woodin for needing the fictional opiate (as
Marx would put it) of the so-called infinity to keep him going.

Let me first pause and enlist an unlikely sympathizer with finitistm,
the greatest set-theorist of our time, Paul Cohen. In the second-to-
last paragraph of "The Discovery of Forcing" (Rocky Mountain Journal
of Mathematics, v.32 (2002), 1071-1100) he said (p. 1099)

"The only reality we truly comprehend is that of our own
experience ... The laws of the infinite are extrapolations of our
experiences with the finite" {{Tatsàchlich? Extrapolationen der
Erfahrungen mit dem Endlichen? Hört man gewöhnliche Mengenlehrer, man
sollte es nicht für möglich halten!}}
So even the great "infinitarian" Paul Cohen was a devout finitist. But
even though he strongly disagreed with Woodin (and Kurt Gödel and
(practically!) infinitely many other people) about the ontology of the
infinite, he totally agreed with him about the aesthetics, when he
said (p. 1100):

"For me, it is the aesthetics which may well be the final
arbiter. For me it is rather a paradise of beautiful results, in
the end only dealing with the finite but living in the infinity of our
own minds." [...]

In my ultrafinitist weltanschauung, the great significance of both
Gödel's famous undecidability meta-theorem, and Paul Cohen's
independence proof is historical (or as Cohen would put it,
"sociological"). Both are reductio proofs that anything to do with
infinity is a priori utter nonsense, debunking the age-old erroneous
belief of human-kind in the actual (and even potential) infinity.
Granted, many statements: like "m+n=n+m for all (i.e. "infinitely"
many) integers m and n" could be made a posteriori sensible, by
replacing the phrase "for all" (when it ranges over "infinite" sets)
by the phrase for "symbolic (commuting) variables (or rather letters)
m and n". We have to kick the misleading word "undecidable" from the
mathematical lingo, since it tacitly assumes that infinity is real. We
should rather replace it by the phrase "not even wrong" (in other
words utter nonsense), that cannot even be resurrected by talking
about symbolic variables. Likewise, Cohen's celebrated meta-theorem
that the continuum hypothesis is "independent" of ZFC is a great proof
that none of Cantor's alephs make any (ontological) sense. [...]

[Doron Zeilberger: Opinion 108: ...The feeling is mutual: I Feel Sorry
for Infinitarian Hugh Woodin for Feeling Sorry for Finitists Like
Myself! (And the "Lowly" Finite is MUCH more Beautiful than any
"Infinite"), Written: March 16, 2010]

http://www.math.rutgers.edu/~zeilberg/Opinion108.html

Gruß, WM
 

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#1 Anonimo
13/04/2010 - 21:46 | Warnen spam
You can keep counting forever. The answer is infinity. But, quite



And what is the question, please?


"The only reality we truly comprehend is that of our own
experience ...



Das ist Positivismus. Eine Weltanschauung, die im Individuum nur eine auszubeutende und vollstàndig ersetzbare Arbeitskraft sieht, was es in den von den Positivisten regierten Staaten bis zum Zusammenbruch deren kapitalistischer Grundlage auch "tatsàchlich" "ist".


In my ultrafinitist weltanschauung, the great significance of both
Gödel's famous undecidability meta-theorem, and Paul Cohen's
independence proof is historical (or as Cohen would put it,
"sociological"). Both



Lassen sich nicht zueinanderbringen. Gödel hat erkannt, und "ebenbürtig" formuliert, dass Mathematik nur ein Werkzeug ist, sie *befàhigt* nur zum Lösen *abstrakter* Problemstellungen, welche nàmlich *auch rein beliebig* sein können.
Vor allem das "nàmlich" ist im letzten Satze wichtig, auch wenn dir das nicht passt, Universitàtshochschulenidiot!

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