FOM: wider cultural significance, part 1 (skepticism)

[Stephen G Simpson]

In my 1 Mar 1999 23:35:34 posting I commented on John Kadvany's 1989
paper in Philosophical Forum, in which he drew parallels between
Pyrrhonian skepticism and G"odel's incompleteness theorems.  Now
Kadvany 02 Mar 1999 15:38:10 has joined FOM and responded to my
comments.  Below, I continue the conversation.  This is beginning to
look like an interesting dialog between a Lakatos-inspired historicist
(Kadvany) and an old-fashioned, `naive', `dogmatic' fuddy-duddy (me).

[ Disclaimer: My comments below are actually in reply to an earlier,
off-line version of Kadvany's posting, so there may be some slight
discrepancies. ]

John, thanks for your FOM posting.  It's a wonderful contribution to
the discussion.  A few quick reactions:

 > I want to provoke people into thinking about this situation via the
 > "pomo" epithet.

Fair enough, and you have certainly provoked me.  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.

Your `pomo' epithet challenges me all the more to pursue what I regard
as genuine foundational themes.  In particular, I want to develop and
expand on my previous work on Hilbert's program, to which I referred
in my posting.  Also, I have some as yet unpublished ideas about an
Aristotelean approach to f.o.m.

 > what kind of "antitdote" is possible, but without invoking a
 > traditional dogmatic and naive search for foundations?

I'm gradually coming to realize that philosophers use the word
`dogmatic' in various senses.  Perhaps you would regard my work on
Hilbert's program etc as naive and dogmatic; I don't know.  Anyway, I
much prefer those epithets to the `pomo' epithet!

 > My answer, in short, is an historical understanding of mathematical
 > theorem-proving and concept-formation.

I'm all in favor of historical understanding, but I don't see that
it's a substitute for foundational/philosophical understanding.

 > Pyrrhonism, it should be recognized that this it is one of the
 > major, major influences in the development of modern science, as
 > discussed in Richard Popkin"s classic The History of Scepticism;
 > this is standard history of science and ideas, not a minor
 > offshoot.

You are making me think that I ought to learn more about Pyrrhonism.

 > In the 20th century Pierre Duhem was very influenced by Pyrrhonism;
 > so were Paul Feyerabend and Imre Lakatos.

Wait a minute.  If pomo equals Pyrrhonism and Pyrrhonism equals
Lakatos, doesn't it follow that pomo equals Lakatos?  I tend to agree
with both the premises and the conclusion of this syllogism.
Therefore, your dilemma

 > take your pick, either the "chaos" of postmodernism inducedby
 > skeptical practices implicit in the Godelian metamathematical
 > paradigm,OR mathematical historicism (a la Lakatos in Proofs and
 > Refutations).

strikes me as offering no choice at all.  It sounds to me like "take
your pick, pomo or pomo".  

Actually, I'm overstating this a little bit.  I don't want to claim
that Lakatos is a pomo-ist, and I rather enjoyed reading `Proofs and
Refutations' many years ago.  However, I now tend to think that some
of the intellectual heirs of Lakatos have turned Lakatosianism into
something like the pomo attitude: there is no truth or reality, only
historical analysis.

 > The little history I provide, from Godel to Rosser to Lob to
 > Kreisel and Feferman and then to Kripke and Solovay is just to show
 > that one of the greatest pieces of formal logicis itself a piece of
 > informal mathematics which had to go through its own historical
 > development.

This strikes me as a good historical point about the development of
G"odel's ideas, specifically the need to get clearer about the proof
predicate.  You are pointing out that G"odel's theorem, like almost
every other piece of mathematics, has a conceptual history.

However, I think you are putting too much weight on this point,
because you see historicism as an antidote, or even the only antidote,
to postmodernism/skepticism.  I don't see it that way.  Sure, G"odel's
theorem has a history, but it is also a scientific fact.  In this
sense I see no option but to accept G"odel's theorem for what it is
and move on.

 > There is no "foundation" in the classical dogmatic sense, but a
 > historical view of the problem of foundations in the history of
 > mathematics, just like we have for algebra, geometry, probability,
 > etc.

Here is perhaps where we disagree.  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.

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, so all that's left is to study the history of failed
attempts to solve it.  This attitude strikes me as defeatist.  My view
is that we should study the history to learn what we can from it, but
such historical studies are no substitute for the foundational
enterprise itself.

 > You can seek for a foundation for math outside of mathematics, as
 > you suggest, and I can"t prove that"s impossible to find, but there
 > sure are a heck of a lot of arguments about why about a zilllion
 > approaches will not work.

What are some of these arguments?  I hope you will post them here on
FOM so that we can discuss them here.

-- Steve