FOM: Reply to Hayes on proofs

[Joe Shipman]

> 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.

You are drawing a sharp line where none exists.  I am sure you do not
mean a
completely formalized proof with EVERY step filled in, because those are

incredibly tedious.  Any proof you accept will leave out details, but
there will
be enough there that you can fill in the gaps yourself.  My proofs were
rigorous
in the following sense: someone experienced in the fields of geometry
(1st
proof) or topology (2nd proof) would, after reading the proof reasonably

closely, be convinced, not only that the theorems were true, but that
all the
details he wanted could be easily filled in on demand.  Someone like
yourself,
less experienced in those fields, might not see how to fill all the gaps
and
could properly ask for more detail.  This means that the notion of
adequacy of a
proof is relative rather than absolute (unless you get all the way down
to a
formalized proof in the predicate calculus from the ZFC set-theoretical
axioms,
suitable for machine verification, which I am sure you do not do).  Put
another
way, there are some proofs *you* will accept that some other people will
not,
insisting on more detail -- that does not mean that either you or they
are wrong
about the adequacy of the proof.

> 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.

I'm afraid that if you are not willing to insist that proofs be totally
formalized this will always be an issue.  There will always be readers
of
greater and lesser degrees of sophistication, some of whom need more
details
filled in than others.

> 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?

This is an interesting question.  Consider the theorem "All but finitely
many
prime numbers are odd".  This is obvious on inspection, but it is still
possible
to logically reduce the theorem to even more obvious principles, so a
"proof"
still serves some purpose.  But if you only care about the theorem
itself and
not what it depends on, then there is no need for a proof.

> >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.

So precisely what level of detail would make this a proof?  You seem to
feel
that there is some level of detail which is "enough" in an absolute
sense, and I
don't see how you can draw a line short of getting to something
mechanically
verifiable.  A fair criticism would be that there is a gap in the proof
that a
mathematician experienced in the field could not *immediately* fill;
this is one
standard for publishability, and if I had been able to post the diagrams
with
the proofs my proofs would certainly meet it.

> (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. )

Well, the question is what level of attention you give to the proof when
reading
it.  You found it "convincing" on a cursory first reading, but not quite

convincing enough to be "sure", but you think that if you read it
carefully with
attention to detail you could reach this higher level of confidence.  It
is this
careful reading that I am claiming makes my proof a real proof (again, I
am
talking about a version of my proof with the diagrams actually drawn,
since many
people may be unable to follow the "text" version without drawing the
diagrams
for themselves; it would be fair to say that a proof which required the
reader
to actually write something down to fully satisfy himself wasn't really
detailed
enough for that reader).

> >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).

>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.
But in fact we all have SOME degree
of training; I have successfully presented my first proof to groups with
no
mathematical training beyond high school and they found it quite
sufficient
without any special difficulties.  Would you like to claim that the
articles in
the Transactions of the American Mathematical Society are not real
proofs
because each one can only be understood by a mathematician trained in
the
relevant area of mathematics?

>  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.

How explicit?  The difference between my proofs and the ones published
in the
journals is a matter of degree and not of kind (and not a large degree
either).

> 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.

Yes, but at least one part of the proof (the commutativity of the
connected sum
operation) is essentially pictorial in the sense that it can be made
rigorous
and picture-free only at the cost of clarity.

> 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.

Good point--this was in fact the reason I posted the proofs in the first
place,
because I wanted to see explanations of how the pictures could be
eliminated and
whether anything essential was lost when they were.

> 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/) ).

As always, the diagrams help make one's UNDERSTANDING of the proof so
clear that
it is easy to fill in all the details; I admit this is an informal
process and
the diagrams play no formal role, but the point is that somehow they
ensure that
the filling in of the formal details will be easy and straightforward.

-- Joe Shipman