FOM: geometric proof (fwd)

Reuben Hersh rhersh at math.unm.edu
Sun Feb 21 12:32:56 EST 1999

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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

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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