FOM: naive or brainwashed?

[Martin Davis]

I will be giving an invited address at an Amer. Math. Soc. meeting
early in April. I chose as my title WHY FOUNDATIONS ARE IMPORTANT.
This was before I was aware of the fom list. Since joining the list,
I have watched in some amazement the various viewpoints swirl past
one another with perhaps some light, but certainly with lots of heat.
Matters that have seemed, and still seem, clear and straightforward
to me are the subject of seemingly endless disputes. I ask myself: Am
I naive or brainwashed? As part of the preparation for my talk, I'd
like to beg the indulgence of the fom list while I set out the
background from which my talk will emerge. (To avoid
misunderstanding: this is not at all the CONTENT of my proposed AMS
talk - but rather the predispositions out of which that exposition
will develop.) In any case, this is your last chance to convince me
how mistaken I am before my views are widely disseminated :-)

HISTORICAL BACKGROUND: Something I miss in fom postings is a sense of
the history out of which our current situation has developed. (This
despite talk about mathematics as a social process.) Frege, Cantor,
Dedekind: out of their work came on the one hand a great expansion of
the subject matter of mathematics into the infinite and, on the other
hand, a grand vision of mathematics as a unified subject with a
single simple logical foundation. Into this "paradise" the paradoxes
famously intruded giving rise to a sense of crisis. As G\"odel
already pointed out in the 1930s, the theory of types provided the
way that had been found to deal with the paradoxes. Russell's and
Zermelo's foundations, originally thought of as competitive visions,
were both fundamentally based on this same idea. Zermelo's version
was much simpler and more elegant being based on cumulative types.
With the realization that an object serving as ORDERED PAIR could be
defined in Zermelo's system, it became clear that all of ordinary
mathematics could be developed from a remarkably simple system of
axioms. This insight has been seamlessly absorbed into the body of
mathematical practice. The extent to which this is true is readily
apparent in comparing textbooks written in the early years of the
century to more recent works. Although the axioms are not ordinarily
made explicit, the set-theoretic foundations are to be clearly seen
in the typical introduction or first chapter on notation. Certainly,
the sense of crisis is gone. Mathematicians freely use set-theoretic
reasoning with confidence. They no longer feel the need to mention
uses of the axiom of choice explicitly. Analysts do not worry that
their subject is (Hermann Weyl), "a house built on sand". 

This is a hard-won victory for the Frege-Cantor-Dedekind grand
foundational vision.  There is no great mystery (as some have
suggested) in the reducibility of mathematics to set theory.
Mathematical patterns have to do with relations among arbitrary
objects. So what mathematics needs is enough objects and the
possibility of expressing arbitrary relations among them. With the
definability of ordered pair, set theory gives both. Simplistic
accounts of foundational work have suggested three opposing
"schools": logicism, formalism, and intuitionism. According to Reuben
Hersh all have "failed"; they have "dried up". He is certainly right
that the ideal of restoring a mythical "absolute certainty" to the
whole corpus of mathematics was never attained and never will be. But
the set-theoretic foundations of mathematics is a triumph for the
LOGICIST program. Hilbert`s "FORMALIST" program led directly to
G\"odel's discoveries, and important and interesting work on proof
theory continues. Even INTUITIONISM has spawned important work
including interesting applications to computer science (and yes, to
category theory).

When a crisis fades away, it is easy to forget the anxieties and
doubts to which it gave rise. Thus, instead of celebrating the great
success in incorporating infinitary methods into a unified
mathematics, we see attacks both on the notion of the unity of
mathematics and on the need and possibility of a unified foundation.
I believe the continued clash of arms over categorical foundations in
the fom reflects this. I don't believe the topos-folks believe in, or
care about, such a unified foundation. It suffices for them that
various PARTICULAR mathematical subjects can be elegantly treated
using categorical methods. This emerges quite clearly and explicitly
in Colin Mclarty's posting to fom today in which he lists as a
problem in categorical foundations: "on-going axiomatizations in
homology". I do not doubt that this problem is as important and
interesting as Colin says it is and even that it has a "foundational"
character: foundations of homology theory. But his is not a vision of
a unified F.O.M. I have the (probably vain) hope that this clear
demonstration that the friends and enemies of categorical foundations
are just talking about two different things when they use the word
"foundations", will end this interminable debate.

ANTI-FOUNDATIONALISM: Others, noting that mathematicians go about
their business unconcerned with foundational issues (and forgetting
the great efforts that have made this possible), reject the idea of
foundations on that ground. In a paper that has become something of a
classic, Hilary Putnam has informed us that mathematics doesn't
"need" foundations. More recently a whole "anti-foundationalism"
industry has sprung up. Leading the charge, Reuben Hersh makes the
remarkable discovery that mathematical activity is a social process.
He trivializes the work of the giants Frege, Hilbert, Brouwer, and
G\"odel, condensing their thought into slogans regarding the nature
of mathematical existence. His solution: the objects of mathematics
are neither to be located in mind nor in the physical world but exist
as social institutions like banks and governments. From my perhaps
simple-minded point of view his approach suffers form the twin
defects that while it sheds no light whatever on the real
foundational problems of today (of which more later), the problem it
does purport to solve is a non-problem. In the first place, to a
simple-minded materialist like me, the mind/world dichotomy makes no
sense: mental activity is just a function of the brain. Secondly it
is only a quirk of grammar that causes us to seek a locus where
"existence" can be instantiated. Lions exist: so we can ask "Where?"
and be told "In Africa". Unicorns do not exist; so they have no
location. The number 7 exists, but where? My simple-minded notion is
that the sense in which mathematical objects exist is only that their
properties are OBJECTIVE. (I think this is also what Bill Tait has
been saying.) We have only the power to discover (some of) these
properties, not to change them. A unanimous vote by all the
mathematicians in the world that 7 is composite would not suffice to
undo its property of being a prime.

G\"ODEL'S LEGACY: The famous cumulative hierarchy goes on and on. As
Harvey has pointed out repeatedly on fom, if we stop at \omega +
\omega we get a model of the Zermelo axioms and more than enough sets
for ordinary mathematical purposes. However, we know that no
computable set of axioms will be able to decide everything. Not only
propositions like CH that involve objects of higher order, but even
certain \Pi_1 sentences of arithmetic (by MRDP equivalent to some
Diophantine equation having no solution) remain undecided. Moreover,
adding axioms to ZF that can be thought of as requiring us to go
further out into the cumulative hierarchy (large cardinal axioms) is
guaranteed to enable more \Pi_1 sentences to be proved. Sol
Feferman's take on this situation is to recommend sticking to the kind of
mathematics that we know how to do. With a defiant NO NEW AXIOMS
NEEDED, he emphasizes that existing mathematical practice hardly
transcends Peano Arithmetic let alone ZF. In effect he is proposing
that problems that can't be solved in this framework are either of
little interest or are just forever beyond the reach of human
mathematics. He regards CH as simply incoherent. I believe that, on
the contrary, it is most unlikely that it is only propositions we
don't care about that ZF won't decide. G\"odel has left us the
challenge of learning how to bring to bear on problems we do care about
the higher order set-theoretic principles that they may require.
Harvey's recent combinatorial principles requiring large cardinal
axioms represents a very important step in this program. The great
success of descriptive set theory in the context of L(R) + AD can
already be regarded as a classical vindication of this point of view.

Of course, it must not be forgotten that venturing into the higher
transfinite is not without risk. There is no way to prove the
consistency of these large cardinal axioms. Inevitably this kind of
mathematics takes on an empirical dimension. A proof that the
existence of a measurable cardinal implies the Riemann Hypothesis or
GC would not today be accepted as a proof of these propositions.  (An
aside: I found recent postings about Hartry Field's ideas utterly
incomprehensible. We have stronger and stronger theories about whose
consistency we need to be more and more fearful. Should any of these
theories, alas, turn out to be inconsistent, a proof of GC from such a
theory would prove nothing.) So as we lose our innocence learning to
make use of the higher transfinite, we will give up any claim that
the mathematical truths obtained in this manner have any degree of
certainty greater than the truths of empirical science.

Martin Davis