FOM: geometric proof (fwd)
Reuben Hersh
rhersh at math.unm.edu
Sun Feb 21 12:32:56 EST 1999
Is there such a thing as geometric proof as distinct from logical proof?
Is there or can there be any other kind of proof except logical?
Depends on what you mean by proof. If by proof you mean nothing
other than logical proof, then you win by petitio principi.
Another meaning is sometimes given to "proof: an unquestionable,
absolutely convincing argument, by whatever means.
An example: it would be a nice freshman exercise to prove logically
that a square has two diagonals. On the other hand, just drawing a
square and saying "Look!" might be more instructive.
Theorem 1, Book 1, of Euclid constructs an equilateral triangle
on a base AB by drawing two circles, centers A and B, radius AB.
Their intersection is the third vertex. Modern rigor says this
proof is defective
because Eclid doesn't provide an axiom to guarantee the two circles really
intersect. Hilbert supplies the deficiency with an appropriate axiom.
Nevertheless, Euclid's diagram is completely convincing. You see
the intersection, the third vertex. You use the visible
but "nonexistent" intersection to do further construction in
Euclidean geometry. Guaranteeing it's really there
by the completeness of the real numbers is nice and good but
really beside the point.
Back in the 30's and 40's there was a shibboleth in mathematics
outlawing reasoning from diagrams.
Nowadays we feel free to explain or argue by a
diagram--keeping aware of certain well known fallacies that can result
from bad diagrams or bad reasoning.
A proof by diagram is holistic, it gives an insight or a grasp into the
whole theorem. It is more likely to be understood, remembered, and used
in
further problem solving or research than a very detailed logical argument
that is impossible to keep in mind at once.
Logic and diagrams supplement each other. There's no competition unless
one
takes on an imperialistic attitude that says, my way or nothing.
Reuben Hersh
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