[FOM] Wittgenstein?

[Mark Steiner]

In a previous posting, I suggested that Wittgenstein's work could stimulate research in fom in the direction of studying how physicists "work around" inconsistent systems.  I believe that Harvey, who originally requested examples of such stimulation, did not reject this idea out of hand.  The idea is, not necessary to defend LW's specific positions, but to gain insight and new research problems in the foundations of mathematics.

Here is another attempt at showing how LW could stimulate research.  LW, as is well known, rejected logical reductions, for example of arithmetic to logic (Frege, Russell).  His general objection to this had to do with the idea that arithmetic is a practice unto itself, and embedding it in something like set theory is like taking a block, wrapping it in string, and then claiming that it is "really" a sphere.  Some of the recent postings about real numbers have a Wittgensteinian atmosphere to them.  However, I think that core mathematicians would have little sympathy for the argument, however valid, that real numbers are not "really" Dedekind cuts.  (I'm not taking sides here, just stating a sociological fact.)

On the other hand, I think that core mathematicians WOULD have sympathy for a Wittgensteinian argument concerning the distinction between real analysis and complex analysis.  Looking at the ingenious ways that fom analysts show how much math you can do with systems far weaker than ZFC or even ZF, I am moved to comment (a la Wittgenstein) that something seems to be left out, though I don't know what.  

Every mathematician knows that there is a world of difference between real and complex analysis, despite the obvious objection from "fom" that they are really the same theory, where you inteterpret a complex number as an ordered pair of reals to get complex from real analysis.  An "elementary" proof of the Prime Number Theory is one which avoids characteristic concepts like analytic function, analytic continuation. etc., and just does "straight analysis of functions of a real variable."   Here the mathematicians, ironically, would take the "Wittgensteinian" position (I'm of course speaking of the later Wittgeinstein) that the logicians' hierarchies are too coarse to capture what is really going on in mathematics.

I'm not sure how to formulate the problem precisely, but: just as fom has so successfully studied the idea of mathematical PROOF, AXIOM, etc., is there a way to formalize and study the notion of mathematical REASONING, so that--even in a single theory, but surely in two different theories, even if of the same logical "strength"--we could discuss in a useful way concepts like "elementary," "explanatory," etc.  I'm aware of literature concerning complexity of proof, but am not sure that this literature is applicable.  At any rate, I would appreciate any enlightenment on this area.
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