[FOM] Wittgenstein?

[Mark Steiner]

I appreciate Harvey's thoughtful answer.  Some of the points in the letter I
would like to think over before responding, but one clarification can
already be made:

I will use the capital letters FOM to denote the original meaning of the
phrase it abbreviates, which I believe had to do with epistemological issues
in mathematics.  Both Russell (1919) and Hilbert (1925) claimed to be
worried by such issues--FOM was supposed to SUPPLY foundations for
mathematics, which otherwise might be threatened in some way.  Russell
actually claimed that his favorite foundation, logicism, provided FOR THE
FIRST TIME, a rationale for 2+2=4.  Hilbert claimed to be justifying
classical proof procedures--including "infinitary" methods--which were
allegedly under attack by Brouwer and the Intuitionists.
LW objected to this obssession with Foundation: arithmetic and mathematics
in general, was in no need of a Foundation of any kind.  This point of view
is not limited to LW--I think Goedel himself held this, but his reason was
drastically different from that of LW (as he once explained to me on the
telephone).  LW's reason had to do with the relation of mathematics to its
applications (I won't go into details).  LW attributes the paranoiac fear
that "arithmetic totters" to Frege upon the latter's learning that Russell's
Paradox destroyed his logicist candidate for FOM.   LW felt that Frege's
(alleged) reaction was just as unjustified as the search for a Foundation
which engendered it.

    Much different from FOM is fom as Harvey (on the basis of his published
comments) practices it.  Harvey believes that the relation between fom and
philosophy is much looser than do Frege or Hilbert.  On the contrary, even
false philosophical theories can generate good fom research (Harvey has I
believe said as much).  Furthermore, fom (e.g. "reverse mathematics").  This
explains his willingness to dialogue with philosophers of all kinds,
"canabalizing" philosophical theories freely.  (I don't mind this--we
philosophers need all the friends we can get.).  The motivation of fom
(again, as I perceive it from reading his postings) is to get a nuanced
description of all kinds of mathematical reasoning, as well as finding the
limits of such reasoning (independence  and inconsistency proofs)--all this,
using mathematical techniques.  fom has no program to defend mathematics
against catastrophe.  Harvey's statement that he has even considered the
Wittgensteinian question of showing how even inconsistent systems could be
applied successfully, clinches the matter.  fom is thus mainly descriptive,
unlike FOM.  Incidentally, though Goedel had strong views on the philosophy
of mathematics, one could similarly say that this views were a source of
inspiration for his fom work which can be appreciated by anybody ("general
intellectual interest" if you insist).  LW's animosity to Goedel had to do
in part with his suspicion that Goedel was doing FOM, rather than fom--or
perhaps FOM in the guise of fom.

    So far, I see no grounds for antagonism between LW and fom.  LW in the
Investigations (PI) and in Remarks on the Foundations of Mathematics (RFM)
takes a similar, descriptive position.  He wants to describe the "motley" of
mathematics.  (Important note: I am speaking of the LW of the Remarks on the
Foundations of Mathematics period--I am skeptical about how much we can use
the "Philosophical Grammar" to project LW's later work.  In PG he still
regards FOM as relevant to philosophy, and comes close to adopting
Intuitionism--at least this is how I perceive the book now.)

    The main difference between fom (as practiced by Harvey and his school)
and LW as I see this, is: (a) to what extent can mathematics be used as a
tool to describe itself; and, especially (b) how significant is the
formalization of mathematics by logicians.  (LW of course agrees that
mathematical terminology need not be true to ordinary language, just as
physics need not be true to the "ordinary" meanings of "work" and "energy,"
but I think he held that "logicians" have overformalized mathematics beyond
what mathematics needs.)  He seems to have felt that the idealizations of
the logicians are sufficiently far from the practice of mathematics that we
do not get insight into mathematics and particularly its applications from
these idealizations.  Here, I believe he was just wrong--for example,
although Goedel's theorems and like independence results are proved for
formal systems, even the most hostile "core mathematician" would be ill
advised to try to prove the Continuum Hypothesis, Whitehead's hypothesis
(proved indepndent of ZFC, I believe, by my colleague Saharon Shelah), or
other statements proved by Harvey and others to be independent.  Also the
use by Harvey of formalizations to explain how nonrigorous reasoning can
given robust results (as in his reply to me) would be another possible reply
to LW's challenge.