From: Amara Graps (amara@amara.com)
Date: Sat Mar 02 2002 - 07:27:36 MST
Serafino:
>A topic from Historia-Matematica Discussion Group
>[HM] Archimedes Work Deciphered
(more on Archimedes' Palimpsest....)
The Physics Today article ("The Origins of Mathematical Physics: New
Light on an Old Question, June 2000) about this document goes into a
lot more detail of interpretations, as well as a historical adventure
of where this document travelled. In the following, I state some of
Reviel Netz's comments and conclusions from that nice article.
One of Archimedes original contributions to mathematical physics is
how he derived physical proofs based only on conceptual mathematical
considerations. He also worked backwards, using physical results to
derive results in pure mathematics. For example, thinking of
triangles and their symmetries and similarities, one finds the
center of the weight of any triangle; by thinking of centers of the
weight, one finds the area of a parabolic segment. His _Method_, now
seen in the Palimpsest, most explicitly connects the mathematics to
the physics. In it, he claims that he has invented a procedure that
allows him to use physics (in particular, mechanics) to derive
mathematical results. He also further anticipates integral calculus,
summing areas through lines, and of solids through areas, but it
possibly wasn't rigorous enough to his liking, and served merely as
a "method".
Because this manuscript was 'buried' from 212 B.C. until 1906 A.D.,
it is interesting to guess at how the history of science might have
looked if someone like Galileo or Newton was aware if its existence.
The author, Netz, also has some useful comments in the article
regarding regarding "proofs". He says:
=================================================================
"Proof was the real passion of Archimedes -- and that of his culture
in general. The Greeks were forever arguing, refuting, and
attempting to provide irrefutable arguments. Out of this
consummately argumentative society came that unique form of
expression that is characterized by its stress on argument, and on
argument alone: the Greek mathematical, deductive argument. [...] In
geometry, Archimedes could be irrefutable. My sense is that this is
the domain where he preferred to remain.
If this assessment is correct, we may also see why mathematical
physics is such a good idea. It embodies the principle that one
should aim for the best possible arguments, using the discipline in
which the highest standards of proof are available. Mathematics may
have little to say, directly, about the physical world, but it is
the only way to say anything at all with any certainty. The bet of
modern science -- following on Archimedes -- is that we are willing
to say very little, as long as what we say is well argued. Good
arguments are good starting points for truly productive discussion,
and so it is not surprising that the mathematical route has been so
productive in modern science.
But is this interpretation true? It is only a possibility, suggested
by the writings of Archimedes. He explicitly says very little about
his goals and conceptions. When -- as the legend goes -- he cried
"Eureka," sallying forth from the bath, this may have been because
he had discovered truths of physics. Or he may have discovered new
properties of geometrical solids. Like the citizens of Syracuse, we
cannot really tell, but can only gape at his discoveries with
amazement. We are extremely fortunate that now, thanks to the
Archimedes Palimpsest, we shall be able to gape from a bit closer."
=================================================================
BTW, here's more on that story of the sale of the manuscript. From
Greece to Palestine to Jerusalem to Constantinople it disappeared
after WWI, then appeared dramatically for sale at Christie's Auction
in October 1998. The sale was legally contested by the Greek
Orthodox Church, but the sale was allowed to proceed by a
last-minute court decision. The Greek government took the challenge
and sent a representative to the sale. At around one million
dollars, all the contenders dropped out, with two exceptions: the
Greek representative and the representative of a private collector.
The private collector held steady for two million dollars, which
Greece was unable to match. [The new owner wishes to keep his
anonymity but Archimedes' Palimpsest is currently being conserved
at the Walters Art Gallery in Baltimore, Maryland.]
-- ******************************************************************** Amara Graps, PhD email: amara@amara.com Computational Physics vita: ftp://ftp.amara.com/pub/resume.txt Multiplex Answers URL: http://www.amara.com/ ******************************************************************** "We don't see things as they are, we see them as we are." --Anais Nin
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