FOM: Foundationalism and Secrecy

[Joe Shipman]

Shipman:

>No, I wasn't condemning them!  I said
>   IF the Fields medalists and the committees that selected them knew
that the
>*important* results of the medalists had incomplete proofs
>  THEN either
>1) You are correct about the dispensability of foundations
>OR
>2) The committees are to be condemned.

Friedman:

>I am not entirely convinced by this reasoning. You seem to be
implicitly
>assuming that the Fields committee is awarded the medal in SOLE
recognition
>of the mathematical results for which the candidate is credited as
having
>established.
>
>Other factors that may be properly taken into account are: the
influence on
>others, the importance of certain definitions, the importance of
incomplete
>proofs, etcetera.

I don't actually believe that the Fields medal is or should be awarded
solely for mathematical results which the candidate is credited as
having established.  I adopted this position provisionally for the sake
of argument, in order to clarify the differences I was having with
Reuben.  The idea was to give him a chance to back up his claim that
foundations are dispensable.  My position is that although the other
factors you mention are very important, they are ultimately important
because of the completely proved results they may eventually lead to.
The significant partial results that Thom and Thurston *did* completely
prove also helped confirm the importance of the claims and conjectures
they did not completely prove.

The case of Witten is more problematic.  My impression (which I hope
someone will correct if I am wrong) is that his Fields Medal was more
controversial because he had not conclusively proven anything and was
awarded the medal because of the inspiration his work had provided to
others.  I am not qualified to judge whether the degree of creativity
and correctness of his conjectures was so high that he deserved the
award; the question is whether his award was extraordinary and
singular.  He is the one of Hersh's three examples who really does tend
to support his thesis.

Reuben recognizes "consensus" as a hallmark of mathematics, but won't
admit that consensus ultimately depends on reducing a proposition to
basic propositions and definitions that everyone accepts.  Empirically,
we see that every result that is accepted by the mathematical community
as "having been proved" can be proved in ZFC.  This does not contradicts
Reuben's observation that mathematicians don't work "in ZFC"; but they
do work from common principles of reasoning and elementary propositions
that can be formalized in ZFC, and textbooks in a wide variety of
mathematical subjects recognize this by beginning with a chapter of
set-theoretical preliminaries.

There is a reason for this!  In the late 19th and early 20th centuries
there was a "crisis of consensus" that affected many areas of
mathematics (real analysis, algebraic geometry, abstract algebra).
Mathematicians were no longer unanimous in their recognition of claimed
results as actually proven.  This was because new principles of
reasoning (completed infinite sets, the axiom of choice, transfinite
induction, nonconstructive proofs, impredicative definitions, etc.) were
being used with various degrees of explicitness.  Attempts to formalize
the new principles led to contradictions (Russell paradox) or
repugnantly counterintuitive results (Banach-Tarski paradox).
Eventually, thanks to Frege, Cantor, Hilbert, and Zermelo (with assists
from others) a complete "foundation" for mathematical reasoning was
hammered out which appeared free of contradictions and allowed for the
restoration of consensus.  This restoration of consensus came at the
costs of accepting axioms with counterintuitive consequences for some
(but at least the role of AC could be explicitly identified in those
proofs), and rejecting certain bodies of mathematics as incapable of
being reduced to those axioms or other plausible ones (some of algebraic
geometry).

Shortly thereafter, Godel showed there was yet another cost to fixing a
foundation -- incompleteness.

But mathematics as a scientific profession was now healthy again.
(Please note that I have been talking about mathematics as what
mathematicians do; I am leaving questions of indubitability to the
philosophers and am trying to show Reuben that foundations are a
necessity not to restore a lost sense of indubitability, but simply for
the sake of making consensus possible!)  In the last 60-odd years, the
importance of foundations has faded from the consciousness of most
working mathematicians, because no further crises of consensus have
arisen.  In the cases where there has been disagreement about whether a
theorem has been really proven, either a clear mistake has been found
(Wiles's first proof), or the details have been satisfactorily filled in
(DeBranges's proof of the Bieberbach conjecture),
or gaps have remained and the theorem has been rejected (several cases I
will leave unmentioned).

What would be really interesting is a proof that was accepted as correct
by some mathematicians but not by others, where the two groups can't
change each others' minds.  For example, some of Thurston's claims may
have been established as far as he and some of his colleagues are
concerned by geometric reasoning that others are unable to visualize or
follow; but Thurston and those colleagues who are therefore persuaded of
the truth of those claims then have a duty to render the proof in terms
that those others can accept, and if that is not possible because of
technical difficulties then they should not assert that the claims have
really been proven.  (I'm not accusing anyone of such assertions--I'm
sure Thurston is quite honest about this, if indeed there are any
results which he feels he has established to his own satisfaction but
not proven to the satisfaction of the entire community.)

I will admit I am wrong about this if  some mathematical propositions
become broadly accepted as "proven" where it is not evident the proofs
can be formalized in ZFC or some obviously true extension of it.  (Many
propositions, e.g. Goldbach Conjecture, are broadly accepted as "true"
but this is a different issue [Goldbach would never win a Fields medal
just for making that conjecture]).

A crisis of consensus will come again someday, and then it will be clear
to everybody why foundations are indispensable.

-- Joe Shipman