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Re: [HM] Magnitudes
Posted:
Nov 15, 2003 6:05 PM
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Dear all. Luigi Borzacchini, Laszlo Filep, Jochen Ziegenbalg, Gordon
Fisher, thanks for all the answers to my comments. I believe that
from the point of view of a mathematician I now understand what
magnitudes, or better, systems of magnitudes are (a partially ordered
Abelian semigroup; V.Def.4 suggests the Archimedean axiom). That's
not very deep, but I find satisfying, that one has to know nothing
about the individual magnitudes in the system. I think that
Eudoxos-Euclid didn't define magnitudes, because they thought it
would be applicable in more general situations. There is still a big
problem:
How did the Greeks think about addition (sum) in a system of
magnitudes? In my copy of Euclid V all illustrations are simply with
segments where addition is obvious and simple. In fact I have the
feeling that segments are symbolic for arbitrary magnitudes. But even
for the case of plane figures I am at a loss. What Mueller does in
his book, p. 122, I find unsatisfactory; it is just addition of
areas; no actual sum figure emerges. And Archimedes in the recently
deciphered part of the palimpsest (R. Netz et al) certainly has a
very different idea. . .
Luigi: I don't think the greek magnitudes were measured (i.e.,
numbers attached to them), at least not as a fixed necessity.I don't
find that in Book V. For plane figures the area is a suggestive
number to attach, but why not the perimeter? You are right that many
more kinds of magnitudes have appeared in the meantime; but it seems
that proportions now come from numerical attributes and not from
Eudoxos's definition.
Laszlo: At the moment I can't get to the Internet, so I can't open
your paper. I don't know much (or anything) about the greek attitude
to infinite sets, but they usually considered concrete sets like
numbers or extended lines, didn't they? According to R. Netz et al
Archimedes had ideas there that were way ahead of his time. (Using
these ideas he gave, what to me is a completely non-sensical proof of
Method, Proposition 14,on the palimpsest, as recently deciphered by
R. Netz et al.) The material about actually and potentially
increasing (or decreasing) sets is new to me.
Jochen: I'll try to find the book
Gordon: Assigning numbers to magnitudes is helpful for handling them
or to handle some aspects of them, but it doesn't seem essential to
me.(I find no suggestion for such numbers in Book V; but then I know
very little of the literature.)To assign coordinates to a vector is
helpful for computations, but doesn't go to the heart of the matter.
Or, to assign its area to a plane figure, is quite obviously the
thing to do, but it tells very little about the object at hand. I
wish Euclid or Eudoxos had said something about how they visualized
their magnitudes.Euclid in book V is completely blank, no hint of
anything.
Hans, from rainy (thank God) Stanford.
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