On Thu, 22 Mar 2007, Peter St. John wrote:
That result brought Robert's concern out in the open: do we believe a proof
that is too long for us to personally verify ourselves? With Wiles and the
group classification thing and some other such work, we've realized that
machines aren't required. It is a practical impossibility for me to verify
Wiles' work; in principle, maybe I could in, say, a decade. To verify the 4
color theorem would take me centuries. But really that's just a matter of
degree.
Or even more apropos (since a proof could IN PRINCIPLE be verified by
us) do we believe what we're told by scientists and mathematicians in
general? Take Usher vs Cepheid variables. I know "of" much of the work
that is done to date the Universe, from parallax observations to the
Hubble constant to cepheid variables to...
Those results currently peg the Universe at roughly 13 to 14 billion
years old (or so I read). Could I verify them? Only if I could gain
access to the only instrument near earth that generated a lot of the
data, or if I believe the data that was purportedly generated by this
instrument.
Compare to Usher's estimate of 6010.5 years (to all five significant
figures:-). It too has a "basis" of sorts.
How does one argue with a young earth creationist? They are no more
capable of actually validating the science than I am and reject most of
the science anyway just as I reject Lightfoot's work on establishing a
month-by-month chronology from the Bible.
Even most of what I teach in physics, I have not personally observed.
And then, inductive reasoning in general is not valid! It leads to
degree of belief, not certainty, and that can only be quantified if one
accepts a certain set of axioms laid out e.g. in Jaynes (and in a really
nifty gangbusters book that should be delivered to my door any day now)
which cannot THEMSELVES be accorded a degree of belief, only blind
acceptance.
But hey, when I finish my book entitled Axioms (a snapshot linked to my
website) I'll let y'all know.
It made a lot of us very uncomfortable, but we left behind the day when
everyone could read everything for himself, sometime between Guttenburg and
Leibnitz's Monadology.
We've never had it to leave behind. Each of us attempts to disspell out
own personal darkness, to peer out from Plato's cave, and make sense of
the pattern of light and darkness cast upon its walls. To borrow a bit
from one of George Robert's lectures once again:-).
rgb
Peter
On 3/22/07, Robert G. Brown <[EMAIL PROTECTED]> wrote:
On Thu, 22 Mar 2007, Peter St. John wrote:
> Well to me that's the point. My brain is too small for 500Kx500K
matrices
> over a ring of 22 degree polynomials, too. So we throw a 16-node
computer at
> it and crush it under the hobnailed jack-boots of Higher Mathematics.
> I wish I know more about the SAGE (machine) that hosts the SAGE
(software)
> that was used for this, but apparently washington.edu's web server can't
> handle the CNN exposure as well as their number cruncher can crunch
numbers.
> They are still down.
In the case of E8, using a computer is probably necessary, although one
would require a wetware interface to make the slightest bit of "sense"
out of the results anyway.
I do find this trend depressing, though. Fermat's lost theorem proven
using computing -- nothing elegant, just crush it underfoot, as you say.
E8. Next we'll hear that the Goldbach Conjecture is finally proven by
virtue of solving 10^17 specific cases and exploiting a proof that once
you have all of those cases proven you can iterate the result to
infinity somehow, or we'll hear of the Riemann Hypothesis being solved
this way -- nothing elegant, nothing that is (actually) of any USE. We
all know that these are true anyway, at least as well as we know that
the theory of gravitation is true. In neither case can they be proven
(yet, in the case of the math, never in the case of gravity), in both
cases they are known beyond any reasonable doubt via induction (see
Polya's lovely books on induction and mathematical reasoning). Proving
these things by computer adds nothing to this -- in addition to the near
impossibility of actually judging the computational results (deep bugs
remaining a ubiquitous possibility in ALL complex computer code) which
always leaves a sliver of doubt even then, that doubt (expressed as
Jaynesian/Bayesian "degree of belief" on an information
theoretic/entropic basis) is already so small that it hardly changes on
a log scale from having done an exhaustive computation.
In the SPECIFIC case of E8 that isn't quite true. Since string theory
as a theory of everything (TOE) may be covered by E8, and since string
theory is reportedly insanely complex and so big that exploring it to
find the RIGHT decomposition into whatever \times SU(whatever) by hand
might take lifetimes, it is barely possible that being able to enumerate
it even electronically will permit a systematic search to be performed
that can eliminate huge blocks of the possibilities and home in on what
we can at least HOPE is a small set of decompositions. Ideally a
single, unique decomposition.
That would actually be pretty cool. For the first time in pretty much
forever, we'd have an actual CANDIDATE TOE, and yet another important
step in "the end of physics" will have occurred. (And note well the
quotes, please -- I'm not suggesting that physics research will come
close to ending with a TOE, only that it will finally have a firm known
basic foundation.
rgb
> Peter
>
>
> On 3/22/07, Robert G. Brown <[EMAIL PROTECTED]> wrote:
>>
>> On Wed, 21 Mar 2007, Peter St. John wrote:
>>
>> > Times have sure changed; with Wiles and Fermat's Last Theorm in
>> newspapers
>> > for over a year, then "A Beautiful Mind" from Hollywood; it's almost
not
>> > surprising that the solution of a difficult math problem is mentioned
at
>> > CNN.com.
>> >
>> > The Exceptional Lie Group E8 computation just got done (some info at
>> > http://www.aimath.org/E8/computerdetails.html about the details of
the
>> > computation itself). Reference to the system SAGE is a bit ambiguous;
>> it's
>> > the name of a symbolic mathematics package and apparently also a
16-node
>> > system at the same University of Washington. Natually I was curious
>> about
>> > the computer, but ironically, it seems that while they can handle a
>> matrix
>> > with half a million rows and colums each (and each entry is a
polynomial
>> of
>> > degree up to 22, with 7 digit coeficients), their departmental web
>> server
>> > can't handle the load of all of CNN's readership browsing at once :-)
>> >
>> > The group E8 itself, together with some explanation of the recent
news,
>> is
>> > in wiki, http://en.wikipedia.org/wiki/E8_%28mathematics%29
>> >
>> > Dr Brown might explain better than I could how sometimes the best way
to
>> > understand a thing is to break it down into simple groups of
symmetries.
>>
>> Don't you be puttin' that off on me now. I get off of that particular
>> bus somewhere around the SU(N) stop, with rare excursions over into
>> point groups on the other side of the tracks. Unitary, yes.
>> Orthogonal, why not. SL(2,C) even. Strictly UNexceptional.
>>
>> > Apparently, one of the funky things about E8 is that the "easiest way
to
>> > understand it" is itself.
>>
>> Yeah, and like I have a brain that can manage ~500,000x500,000
>> complicated polynomial objects. Thanks, I think... but not.
>>
>> rgb
>>
>> >
>> > Peter
>> >
>>
>> --
>> Robert G. Brown http://www.phy.duke.edu/~rgb/
>> Duke University Dept. of Physics, Box 90305
>> Durham, N.C. 27708-0305
>> Phone: 1-919-660-2567 Fax: 919-660-2525 email:[EMAIL PROTECTED]
>>
>>
>>
>
--
Robert G. Brown http://www.phy.duke.edu/~rgb/
Duke University Dept. of Physics, Box 90305
Durham, N.C. 27708-0305
Phone: 1-919-660-2567 Fax: 919-660-2525 email:[EMAIL PROTECTED]
--
Robert G. Brown http://www.phy.duke.edu/~rgb/
Duke University Dept. of Physics, Box 90305
Durham, N.C. 27708-0305
Phone: 1-919-660-2567 Fax: 919-660-2525 email:[EMAIL PROTECTED]
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