FOM: wider cultural significance, part 1 (skepticism)

[john.kadvany@us.pwcglobal.com]

The following continues my replies to Steve Simpson vis a vis postmodernism
and skepticism:

SS:.  I hate the idea that
my field, f.o.m. (= foundations of mathematics), may be part and
parcel of the ongoing cultural break-up which postmodernism
epitomizes, and which I abhor.  I have spoken out many times, on the
FOM list and elsewhere, against the nihilism and lack of serious
intellectual standards in much current f.o.m. research.  I don't even
want this stuff to be called f.o.m.; I prefer to call it what it is:
technical work in mathematical logic, with no serious foundational
component.
JK:  L?art pour l?art?  Don?t talk to the NSF like this.

SS:You are making me think that I ought to learn more about Pyrrhonism.
JK: Popkin?s book The History of Skepticism (UC Press) is a great source.
Also, Myles Burnyeat, The Skeptical Tradition (UC Press) has many articles
by classicists and others explaining how clever and subtle the ancient
skeptics were.  As Pascal said, 'le pyrrrhonisme est le vrai.'

SS:Wait a minute.  If pomo = Pyrrhonism and Pyrrhonism = Lakatos, doesn't
it follow that pomo = Lakatos?  I tend to agree with both the premises
and the conclusion of this syllogism.
JK:  Pyrrhonism has reappeared a few times over the centuries; we are lucky
to have found out about it again. All versions are not =, in fact what is
diabolical is how closely related progressive ideas of critical history are
to postmodernism via Pyrrhonism. I like to point out these affinities and
watch folks run for the door.

SS: strikes me as offering no choice at all.  It sounds to me like "take
your pick, pomo or pomo".
JK: Ok, what?s the difference between all the versions of, say,
second-order logic? Seems like they are all the same to me.

SS: Sure, Godel's theorem has a history, but it is also a scientific fact.
In this
sense I see no option but to accept it for what it is and move on.
JK:   It?s a commonplace to recognize all facts as theory-laden, and open
to revision or criticism. Hard facts are the sign of ossification of
thought.

SS: Here is perhaps where we part company.  To me, foundations in general
and f.o.m. in particular (in the classical dogmatic sense?) are very
important.  Historical studies are no substitute for them.
JK:  Historical studies substitute for a philosophy of mathematics based on
a traditional search for foundations, and turn foundations into just more
mathematical objects, akin to the changes geometry underwent.  This process
between informal philosohical/empirical ideas and mathematical content has
occurred many times in the history of mathematics.

SS: I'm all in favor of historical understanding, but I don't see that
it's a substitute for foundational/philosophical understanding.
What is `the problem of foundations' if not the quest for an
appropriate foundation for a given field (mathematics in this case)?
You seem to take it for granted that this problem is unsolvable or
uninteresting, and all that's left is to study the history of failed
attempts to solve it.
JK:A model of historicism here is Ernst Mach, whose critical and historical
studies of Newtonian concepts of absolute space directly influenced
Einstein and the development of the special theory of relativity.   At its
best this type of history is critical and interventionist.  This is why
ahistoricist phil math is a bad influence on mathematical practice.

SS: What are some of these arguments [for anti-foundationalism]?  I hope
you will post them on FOM
and we can discuss them there.
JK: See for example Richard Rorty, Philosophy and the Mirror of Nature.
Then also Donald Davidson in analytic philosophy, Quine, Putnam, Popper,
Lakatos, Feyerabend, Kuhn, and others.  These are as abundant as proofs of
the fundamental theorem of algebra.

[Paraphrasing Simpson: ]What about Friedman?s combinatoric statements
proved via large cardinals, etc., vis a vis helping to give 'order' to the
'chaos' in foundations?
JK:  I have also heard of Woodin's results on determinacy and supercompacts
(is that right?).  This is an excellent starting point: i.e., let's assume
some kind of succes to the Godelian program of proving lower-order
statements via large cardinal or other axioms: What does that show us? A
crude approach follows.
First, I have to admit that I used the idea of postmodernism and the chaos
in fom years ago as an occasion to introduce ideas about skepticism in
mathematics; I don't really care much about the 'chaos' issue or pomo. I do
think the skepticism is there in fom and mathematics generally as implicit
methodological practice. It is the fallbile heaven which Godel gave us and
from which we will never be expelled. That means unlearning the bad ideas
of foundationalism.  Instead of focusing on new low-level statements as
'confirming' versions of Godel's program, I find the overall situation of
working with large cardinals or other strong axioms to be the interesting
story, since we know we can never prove relative consistency results for
these axioms even if they are consistent: I say look rather for the
fallibilism in mathematics, like the ancient Greek view of axioms and
postulates as hypothetical assumptions put forward in a dialectical debate,
and not Aristotelian 'first principles.'  Foundationalism and crude
inductivism are distorting lenses for contemporary mathematical theory, and
to the extent that there is 'order,' it is not indicative of a new-found
security in some level of large cardinal assumptions. Foundationalism has
lost its value as a positive heuristic for the growth of mathematical
theories and concepts: it is time to try attending to problems of
fallibilism appearing in fom, including the translation of informal
consistency into the formalized consistency statement, and exploit them for
mathematical content. The original Godelian program endorses a kind of
old-fashioned inductivism in its crude form and should be revised to take
account of these issues (translation intensionality in proof theory and
large cardinal fallibilism) and to devise heuristics for new problem areas.
You see the contradiction in Godel's trajectory:  he undoes the idea of
foundations from within, provides an initial mathematical program for
dealing with undecidability on a large scale and reverting to a
conservative epistemology, and then caps it off with his ridiculous
Platonism which only he can get away with because he's a great genius.
There was an article on Niels Bohr in Physics Today recently on how people
tried so hard to believe his every nebulous philosophical pronouncement;
similarly many mathematicians and philosophers also took KG a bit too
seriously on the Platonism. Then the Godelian program is still a good idea,
but minus the mysticism and crude inductivism, and plus Godel's own
skeptical heaven, whether in proof theory or set theory.
Cheers, JK.
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