I have the highest admiration for those who achieved so much with
intuition alone. But, fascinating as "ethnomathematics" is, you can't
do quantum mechanics or relativity with it.
Roger Cooke
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Date: Mon, 9 Nov 1998 14:18:39 -0500 (EST)
From: Roger Cooke <cooke@emba.uvm.edu>
To: historia-matematica@chasque.apc.org
Subject: Re: [HM] Is Greek mathematics the *real* thing?
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Status: O
I would just like to add to Eric Schechter's note that one can also
see clearly in Euclid just where he was hampered in his quest for
generality. In proofs involving the Eudoxan theory of proportion, for
example, he is handicapped by not being able to say, "Let m and n be
positive integers...." The proof that the areas of triangles having
equal altitudes are proportional to their bases almost looks like a
"theorem schema" in that it is illustrated with just one example of
m and n from which the reader is expected to grasp the full idea.
It does seem to me that, despite their brilliance, Euclid and his
immediate
successors got geometry stuck in a rut with their "synthetic" methods,
from which it wasn't rescued until algebra and projective geometry came
along. An important question for pedagogy: how much of this formal proof
and axiomatization is still important in education today? Can the
symbolic
manipulation that constitutes modern algebra replace it in the study of
geometry? (HIstorical question: could algebra have been the source of
rigor in mathematics if geometry hadn't got there first? The Hindus
were well on the way to having the concept of a variable denoted by
a shorthand name (color names) around 1500 years ago, if not more.
Weierstrass was suspicious of any argument that wasn't essentially
algebraic. He trusted symbolic manipulation much more than he trusted
intuition.)