Science > Physics > Nonrenormalization vs Renormalization 14.2 Fermat's Last Theorem vs Special Relativity
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Science > Physics |
| User: |
"OsherD" |
| Date: |
16 Apr 2006 01:17:27 AM |
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Nonrenormalization vs Renormalization 14.2 Fermat's Last Theorem vs Special Relativity |
From Osher Doctorow
Fermat's Last Theorem arguably has an "anomaly" or "paradox" in that
its proof is so difficult and long and its direct applications are so
few. But neither was characteristic of Pierre De Fermat of early to
mid-1600s France, who by the way as many of us know wrote of a simple
proof as being in the margin of his notebook or notes but which has
never been found.
Let's look at the equation which has no integer solutions for x, y, z,
n > 2:
1) x^n + y^n = z^n
according to the statement of Fermat's Last Theorem.
Now let's think of the psychology of Fermat himself, who was
professionally a Magistrate and only an "amateur" mathematician
although he co-discovered probability with Pascal and discovered modern
number theory that led to cryptography and discovered
analytic/Cartesian geometry before the "professional" Descartes and
discovered that light slows down in water (contrary to Descartes'
claim) and discovered several equations of calculus before either Sir
Isaac Newton or Leibniz.
To make it more interesting, let's think of the psychology of Albert
Einstein, who has been heavily attacked by Pentcho Valev here on
sci.physics for claimed errors in Special Relativity and possibly some
in General Relativity to my recollection. As a person who has the
opinion that the speed of light is not a finite constant upper bound to
all possible speeds, I only partly agree with Valev and do not
presently consider that Einstein was a deliberate liar or crook. In
fact, I think that he was a Creative Genius, though not necessarily as
great as Fermat or Sir Isaac Newton.
Creative Geniuses tend to be of two types: Beethoven's type, who have
to work painstakingly and long before they discover something, and
Chopin's type to whom a composition occurred immediately and suddenly
"full-blown". But all of them insofar as I can determine had and have
remarkable intuitions. So what is there "intuitive" in Fermat's Last
Theorem?
Look at this:
2) D^n(x^n) = n!, n > = 1 integer
The right hand side of (2) is a factorial (n times n-1 times n-2 times
.... times 3 times 2 times 1) which is a Gamma function. Fermat was an
expert in number theory. He also knew several key equations of
calculus.
I'll try to continue this shortly.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Nonrenormalization vs Renormalization 14.2 Fermat's Last Theorem vs Special Relativity |
16 Apr 2006 01:33:20 AM |
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From Osher Doctorow
I mentioned Beethoven and Chopin because Creative Geniuses tend
strongly to have "intuitive contact with the 'pulse' of the Universe."
But the "anomaly" or "paradox" about this is the fact, I think, that
they don't necessarily have intuitive contact with the details of the
Universe. This is why, I think, Beethoven took so long (with details)
and Chopin took so little time. And, by the way, Beethoven was better
than Chopin in the opinion of many "musical experts" even allowing for
the different types of music which they composed.
Fermat in my opinion had noticed the omnipresence of Gamma functions in
number theory and the extreme importance of derivatives which according
to Garrett Birkhoff are central to Causation.
Einstein noticed curious aspects of light (remember his discovery of
the photoelectric effect?) but in my opinion made a slight error in
details: he assumed that the speed of light was an upper bound rather
than a phase transition boundary or alternatively infinity. Actually,
phase transition boundaries can "look like infinity" to objects
approaching them, even theoreticlaly. There really is arguably a
phase difference in perceptual phases between subluminal motion (which
is perceived by human beings in general), luminal or light motion
(which is perceived by human beings but differently from the subluminal
types) and superluminal motion as occurred at least in the geometry of
the Universe in inflation and occurs at least as wave front or "phase"
(in a different sense) or group velocities as we now know.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Nonrenormalization vs Renormalization 14.2 Fermat's Last Theorem vs Special Relativity |
16 Apr 2006 01:53:30 AM |
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From Osher Doctorow
But I left out the "topper" or "punch line" or "key part" of the story
as I see it.
If Fermat's Last Theorem is true, and according to Wiles' proof it is,
then integer solutions of the equation x^n + y^n = z^n under the
indicated circumstances for n > 2 do not exist, but that also means
that under rather general conditions the "non-existent solutions,"
somewhat analogously to black holes in some respects if not others,
divide the solutions of the equation into number theory analogs of
"phases". For example, there are solutions of the above equation
which are not integers, so such solutions are bounded above and below
by non-solutions (an integer and its successor, the next higher
integer). This is reminiscent of a connected line segment, or in other
words an open interval here, bounded above and below by one-dimensional
"holes" as the endpoints of the interval (for example, (0, 1) versus
[0, 1] or versus the set {x: x = 0 or x = 1}). Even if the solution
set is not a whole connected interval or a union or class of such
intervals, the analogy can be extended.
So here we have Pierre De Fermat studying subluminal motion (light in
water), developing a theorem whose equation is reminiscent of phase
differences which would arguably include superluminal and subluminal
and luminal phases, and discovering modern number theory. Flying
saucers? No, I don't think so. I think that if you have your
intuition on the "pulse" of the Universe, then you get surprising
connections between facts no matter what era you live in. You may not
interpret them correctly, or like Fermat you might just "see" more
about their connections than later generations! Of course, Fermat
didn't like to publish (you can verify that by looking up his biography
for example under the keywords Fermat biography on the internet).
Neither did Sir Isaac Newton until relatively late in his career. I
will not over-emphasize this point, but it has occurred to me. I
wonder if it means anything :>)
Osher Doctorow
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