FOM: Re: Visual Proofs: Reply to Silver and Hayes

[Pat Hayes]

>
>Hayes:
>>Basically, your proofs aren't purely visual, and moreover they couldnt
>be,
>>in an important sense: they require an ability to generalise over
>classes
>>of pictures which cannot itself be represented pictorially.
>I'm NOT claiming these proofs are "purely" visual, just that they have a
>visual COMPONENT that is in some sense essential for understanding the
>proof and translating it into a "rigorous" proof in a formal system.

I suspect that you and I mean different things by "proof". It sounds like
that for you, a proof is something that suffices to convince the reader
that a proposition is true. That isnt what I mean by 'proof': for me, a
proof is a *rigorous demonstration* that a proposition *must be* true,
ultimately including all the details needed to ensure the rigor. Thus, your
notion of proof is ultimately a psychological notion, whereas for me it
most emphatically cannot rely on psychology.

Issues of pedagogy and ease of comprehension are quite irrelevant to the
nature of proofs, in my sense (though they may well be important matters,
of course.) It is entirely possible that a proof might exist which cannot
be understood by a human being, perhaps because it is too long or too
intricate. Several recent famous proofs almost qualify here, notably the
Fermat conjecture and the 4-color theorem.

Heres an observation to justify my position. To a sufficiently
sophisticated reader, some theorems might just be *obvious* on inspection.
On your view, then, it seems, these wouldnt need any other proof: is that
right?

>>Similar objections apply throught the rest of your proof. You constantly
>>suggest 'pictorial' demonstrations which are intended to establish
>>univerally quantified truths about all rectangles, all triangles, and so
>>forth; but you claim to demonstrate these by displaying single pictures.

>Wait a minute.  Are you now saying that my proof was not a "real
>proof"?

Yes.

>Did you find that it failed to persuade you that the theorem
>was true?

First, that's irrelevant to the question (see above); but in any case, yes,
it did fail: that is, I saw it as a kind of sketch narrative *about* how to
prove the theorem; it has the same kind of relationship to a real proof (in
my sense) as an artists sketch of a new building has to an architectural
builders plan. Given a certain degree of mathematical competence on the
part of the reader - enough to be able to fill in the details to get from
the sketch to the actual proof - then sketch might suffice to specify the
proof, but it does not by itself *constitute* a proof. (For example, I have
enough geometry to find your first proof-sketch convincing in this way, but
not enough topological competence to be able to fill out your second
proof-sketch. But even in the first case, I wouldnt feel sure that the
theorem was true until I had checked out the proof in detail. Diagrams can
be very misleading: I'm sure you know the diagrammatic "proof" that all
angles are zero, for example. )

>My point is that an appropriate picture often DOES suffice to
>establish some statement in that it persuades people of the truth of the
>statement (and probably also persuades them that a rigorous proof of the
>statement can be obtained in a straightforward way).  In fact my proofs
>didn't even draw pictures, just described pictures in words, but in such
>a way that people who were able to imagine the pictures I was describing
>would immediately assent to statements like "any triangle is equivalent
>to a rectangle by finite decomposition" or "the connected sum operation
>on knots is commutative".

I disagree: the diagram never suffices by itself. Such 'immediate asssent"
is only possible for those who have a specialist training in both the
relevant mathematics and in what might be called the 'received style' of
how to use diagrams in these various fields. (If one used topological
deformations on the pictures in your first proof, for example, all kinds of
nonsense would result.) In order to follow your proofs, one needs a lot
more than the ability to look at pictures: one needs to know a lot about
the subjectmatter these diagrams are supposed to refer to; and this
knowledge (even if it is not consciously articulated in the minds of those
who can follow your proofs) must be made explicit in a truly rigorous
demonstration of truth.

I could 'draw' all the pictures, but your second proof left me unconvinced.
Most of it has nothing to do with pictures, but with things like infinite
combinatorics, continuity and localness, none of which play any pictorial
role in your argument as given.

>>. ..[need for induction]....
>I'm not claiming my proof didn't leave out any details, just that it is
>sufficient to persuade some people of the truth of the theorem.  The
>"inductive" part of the argument is of course implicit here and could
>easily be made explicit; but in fact everyone here knows how to do that,
>it is making the PICTORIAL part of the argument explicit enough that is
>the interesting issue.

But the issue is whether or not there is a pictorial 'part' to the
*argument*, or whether (as I find much more plausible), the pictures are
simply a useful memory aid, and do not in fact play any direct role in the
actual proof at all. I agree this is controversial, but you havn't made
your side of the case yet: you seem to be simply *assuming* that proofs
have a visual component.

And by the way, its not at all obvious how to make induction explicit *over
pictures*. If you wish to claim that diagrams play a role in the proof
itself, then you need to explain how these 'visual' proofs hang together.
Induction over sets of diagrams is a tricky subject (so far the only
convincing example I have seen is in the work of Metaja Jamnik
(http://www.cs.bham.ac.uk/~mxj/) ).

>> If you
>>were to try to make these proofs explicit, I think that the pictures would
>>turn out to be more like convenient annotations acting as memory aids than
>>part of the proofs themselves.
>This doesn't follow -- you have shown that the pictures by themselves
>are not sufficient, not that they are not necessary.

True, which is why I said "I think...".

Best wishes

Pat Hayes

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