FOM: Visual proofs -- two examples

Reuben Hersh rhersh at
Sat Feb 27 13:56:19 EST 1999

Thanks for your first example, which I understood.

I would like to understand the second.  CAn you tell me what
is an unknot, a wild knot, an isotopy?

It strikes me that this argument involves as usual two speakers
talking past each other.

The fom'ist, given a visual proof, demands proof that the visual
proof is not (in some sense) reducible to a logic proof.

The mathematician (typically)  could care less whether the
visual proof is reducible to a logic proof.  A  proof serves him
two purposes:  one, to certify the truth of the conclusion, given
the truth of the premises: and two, to convey insight, understanding,
a grip on what is going on.  In other words, the proof is serving
a communicative purpose between human mathematicians, not merely
an authenticating purpose to be stored in some buried archive.

A visual proof often gives immediate clarity, and the possibility of
further progress, more than a long complicated logic proof.

"Reducing" the visual proof to a long complicated logic proof is
going backwards from the point of view of the human mathematician.

I am aware that it is  possible to use logic to study the
question of shorter or more comprehensible proofs. I am
not "against" logic.  I am just pointing out that in general
on the question of visual proofs the fom'ist and the mathematician
are on opposite sides of the road, going to opposite destinations.

Reuben Hersh

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