FOM: Visual proofs -- two examples

[Moshe' Machover]

At 11:56 am -0700 27/2/99, Reuben Hersh wrote:

[...]
>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 proof, properly speaking, is a *logically conclusive* argument. By a
harmless extension of terminology, an outline from which it is clear how a
proof in the strict sense can be constructed is also called a "proof".

A proof--a rigorous proof (there is no other kind)--is not some kind of
logician's luxury. It is absolutely indispensible to the *mathematician*.
Perhaps Reuben Hersh does not agree, because he thinks that rigour is
negotiable. He seems to believe not only that humans make mathematics (a
thesis with which many would agree) but also that they make it more or less
as they please.

Of course, a mathematician needs *not only* a proof but also an intuitive
understanding why the proposition about to be proved is true (otherwise,
why try to find a proof?) and an intuitive idea of why and how a given
proof works (otherwise the proof cannot be understood, let alone invented).

Occasionally, an intuitive explanation why a proposition is true is such
that it can be easily converted into a proof. In such cases the intuitive
explanation can itslef be called a "proof"--again by a harmless extension
or abuse of terminology. Examples of such `visual proofs' were given by
Shipman. But often an explanation, no matter how *intuitively* persuasive,
cannot be so easily converted into a proof. In such a case a mathematician
needs both the explanation and a proof; both are indispensible, *and they
need not coincide*.

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

Yes; but if it is not obvious that the [so-called] visual proof can be
converted into a [rigorous] proof, then it is not sufficient, and a proper
proof must also be supplied. Modern mathematics insists on this. There have
been plenty of cases where an apparently persuasive visual `proof' turned
out to be fallacious.

Good teachers of mathematics know that in many cases the students have to
be given: first, an intuitive explanation why a proposition is true; and,
second, a rigorous proof of the proposition (and, if necessary, an
intuitive explanation of how the proof works). If one of these is omitted,
the teacher has not done his or her job properly.

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

No; doing so--or showing how it can be done--is absolutely essential for
doing the job properly. Without the assurance of what RH calls `logic
proof' there is no proof at all, therefore no theorem; at best there is an
intuitively appealing conjecture.

>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.

If RH is claiming (as I think he is) that the mathematician does not need a
logically correct proof, then I disagree, for the reasons explained above.


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