[FOM] Wittgenstein?

[Mark Steiner]

    To ask whether Wittgenstein's LATER philosophy of mathematics has any
significance for FOM, in the sense that it can inspire research in the
field, may seem unfair--LW expresses hostility to the whole project of fom,
such as that of Russell and that of Hilbert.  I also read LW as arguing that
Goedel's theorems have no significance for the philosophy of mathematics--a
disastrous error even from LW's own point of view, as I understand it (cf.
my remarks on Juliet Floyd's views in Philosophia Mathematica, vol. 9,
257-279).

    Yet I believe that the kind of research that Harvey does (or some of it,
including "reverse mathematics"), and the motivations for it which he has
expressed from time to time on the fom list, are compatible with LW's
"descriptive" point of view (and even if LW would not have agreed to this
proposition, he should have) concerning mathematics.  (LW's insistence that
set theory, to be mathematics, must be relevant to applications
(mathematical and physical), is, as I understand it, exactly Harvey's view,
except that Harvey and other fom people have demonstrated such
connections--LW's prejudice to the contrary is, I feel, the same prejudice
that "core mathematicians" have about set theory, so LW should not be
singled out for blame here.)

    Let me give an example of this.  Wittgenstein, in a face to face
exchange with Turing (who attended his classes in 1939), argues that the
obsession with consistency on the part of logicians (Hilbert) is a
superstition.  Hidden consistencies he argues can never affect the
application of mathematics, and when an inconsistency crops up no engineer
would argue that "anything follows from this."  Turing, on the other hand,
argued that "bridges could fall down" if we apply an inconsistent system,
because there are ways of arguing for any conclusion q in an inconsistent
system without actually going through the step "p and not-p".  E.g., by
arguing from not-p to if p then q in one proof, and then proving p in
another.

    Now, in fact, "core" mathematicians inform me that the basic mathematics
used in quantum field theory, the Feynman Path Integral, has presently no
consistent formalization--unlike, say, the Dirac delta function, or the
Newtonian/Leibnizian differential calculus.  Feynman himself believed that
his formalism was in fact INconsistent (i.e. that there is no way to give it
a consistent foundation).  Nevertheless, physicists know how to extract the
most amazingly accurate "numbers" from this formalism, much as Euler was
able to extract mathematicial information from manipulating divergent
series.

    I would be grateful to the fom community if they could give me
references to literature, or create the literature themselves, on degrees of
inconsistency, "hidden" vs. "manifest" inconsistency, etc.  I'm not sure
myself how to formulate the problems mathematically, but an elementary
question raises itself: in the same way that Harvey has discussed the
completeness of set theory when limited to proofs of a certain magnitude or
"length" one could obviously ask the same question concerning inconsistency
(since the question is so obvious, there must already be literature on
this).

    A deeper question would be--is there any formal analogue to the notion
that a physicist knows how to "work around" an inconsistency (in the same
way that Euler did with divergent series), knowing where it is and where it
is not possible to draw conclusions (so that bridges don't fall)?

    More generally, we can ask concerning the whole question of rigor in
"core" mathematics.  Physicists tend to pooh pooh rigor in mathematics as
unnecessary in physics--they sometimes claim that rigor in a proof is
necessary only to rule out "monstrous" counterexamples never seen in nature.
Physicsts, for example, have no problem in considering a magnet of, say, 3+i
dimensions, as a formal trick to be able to apply complex analysis (analytic
continuation) to thermodynamics.  The physicists' attitude is similar to
LW's fulminations about the invasion of mathematics by formal logic.  It
would be ironic if the "logicians" were able to explain the efficacy of
nonrigorous mathematics as they give an account of "correct" mathematics.

    From the historic point of view I can testify that LW's problem of the
falling bridges has provoked at least one logician, Saul Kripke, to study
the problem.  He thinks LW is wrong and Turing is right, but in any case a
study of the problem would be fascinating.

Mark Steiner