FOM: Mathematical Certainty

[Joe Shipman]

I'd like to look at the three "big" theorems we have been discussing
(FLT, 4CT, CFSG) a little more to make some points about "mathematical
certainty".

Wiles's proof of FLT, which may in some sense be the "deepest" of the
three proofs, is also the most understandable.  There must be at least a
few dozen mathematicians by now who each "understand" the proof in the
sense that they "know why the theorem is true" and could fill in details
as far as necessary at any point in the proof, on demand.  But it is a
very difficult proof and in fact the original version of it had a
serious gap.  So Wiles, when he first announced a proof, thought he
understood the proof and was "certain" that the theorem was true (or he
would never have dared to announce the proof) but he was MISTAKEN.

I would like to suppose this was just the myopia that comes from being
too close to something you've worked on for a long time (the same reason
it's much easier to catch a spelling or grammar error in someone else's
writing than in your own), because in fact the gap became apparent
fairly quickly to people who put in a serious effort to understand
Wiles's proof, but many other theorems have had incorrect proofs that
were accepted for many years.  TWO separate proofs of the 4-color
theorem were wrong but not shown to be so for 11 years each!  (Kempe
1879 shown incorrect by Heawood 1890; Tait 1880 shown incorrect by
Peterson 1891).  Maybe the 4CT was not considered as important at the
time because it didn't have a notorious history (after 1891 it certainly
did!), and if a really serious effort had been put forth the gaps would
have been found much earlier (the existence of two independent "proofs"
may have reduced the urgency of checking each one really carefully).
But there were false proofs of FLT before then when it was already
notorious; can anyone provide information on how quickly those false
proofs were discovered?

I think the answer is "quite quickly", so my "myopia" theory holds up;
but there will be a serious problem if someone can show that an
incorrect proof of a "theorem" *considered very important at the time*
was *generally accepted* for *several* years.  Can anyone provide such
an example from the 20th Century?  Earlier examples aren't so pertinent
because standards of rigor became much higher (gradually) during the
19th century.  Such an example would have serious implications--it would
be very hard to assert that any theorem with a difficult proof was
"mathematically certain".  But for now I'll stand by the following
statement:

For the last century or so, the following sociological criterion for
"mathematical certainty" of a theorem has been sufficient:
1) Before the proof was announced, the conjecture was widely considered
to be very important
2) The proof can be read by a single person (rules out computer-aided
proofs like 4CT or proofs which required dozens of authors like CFSG)
3) The proof was *generally accepted* (no serious workers in the field
proclaimed that the proof failed to convince them) for at least 5 years.

If my statement is correct Reuben Hersh will be somewhat vindicated.

Turning to 4CT, we find that the "mathematical" portion of the proof is
quite well-established and many people understand it.  There is no
doubt, it really is "mathematically certain" that if the two algorithms
given produce the desired output then 4CT is true.  The "computer"
portion of the proof has also been multiply verified (this took longer
than most people realize, but it's been done enough by now).  So 4CT is
"scientifically certain" though maybe not "mathematically certain".  I
would actually be less surprised at this point if another gap is found
in Wiles's proof than if some mistake was found invalidating the proof
of 4CT.  Blass and Sazonov are quite right to point out the importance
of finding an "understandable" proof, but there may not be one; at some
point we may just have to say "we understand this phenomenon as well as
it can be understood, but it depends on so many finite combinatorial
'facts' that we still need to point to a computer output to persuade
someone else that it is true".

CFSG is less "mathematically certain" than FLT because the proof is not
yet digestible by any one person.  Depending on the word of a bunch of
other mathematicians for their parts of the proof isn't any better than
depending on a computer in my opinion.  It is not "scientifically
certain" either for the same reason.  If each of the dozens of pieces
has been really solidly verified, we may be justified in imputing some
sort of "certainty" to the theorem, but I'm not sure exactly what to
call it ("scientifically certain" is not quite right because since the
proof isn't small enough for a single person to understand there is a
sense in which it is "irreproducible").

Your comments, please!  I'd especially like to hear from number
theorists or group theorists who may have a better understanding of the
proofs of FLT and CFSG than I do and can confirm or correct my
characterizations (for 4CT I am confident I know what's going on).

-- Joe Shipman