[FOM] Godel and realism or platonism

mlink@math.bu.edu mlink at math.bu.edu
Sat Feb 18 12:23:41 EST 2006

Dear Professor Weaver,
  I read you were not interested in the WWGD questions,
but perhaps I might add some less speculative non-WWGD
details in passing.  Most of what I say is no doubt as
well-known to you as to other readers of this list.  I
am no expert on mathematics, nor on the history of
mathematics, nor on Goedel, as you must think already.
I also want to apologize since many of these details
have already been discussed on the f.o.m. list by real
experts but I have not provided cross-references.
  Of course, as you know, Goedel did state that he was
a `Platonist' or `Realist'.  You mentioned the analogy
between intuition of mathematical concepts and seeing
physical objects that is part of his `objectivistic
conception', which includes realism about mathematical
concepts as well as sets. Goedel writes that realism
about sets is not a problem for mathematics because of
the iterative conception of set (Collected Works,
vol. 2, p. 258; he appears to have changed his mind on
this: see Hao Wang, A Logical Journey, p. 270.  For
more on the philosophical problems associated with the
objectivistic conception see especially Tony Martin,
`Goedel's conceptual realism', Bulletin of Symbolic
Logic 11 (June 2005).).  From the correspondence with
Hao Wang comes the information that the objectivistic
conception was heuristically fundamental to Goedel's 
most important discoveries in logic (CW 5, p. 401n. d).  
  On the questionnaire from the sociologist Burke
Grandjean (CW 4, pp. 446-450) Goedel says that he had
persisted in his position onn conceptual realism and
platonism about mathematical objects such as sets since
1925. Goedel also mentions the influence of the Vienna
Circle, of Carnap, and of Hilbert and Ackermann.
Solomon Feferman and Martin Davis carefully pointed out
that this is not a static position for Goedel but
develops over time.  Goedel then has an unusually
complex form of platonism compared to what one might
say on that topic today.  Matters are complicated (maybe
Parsons wrote this somewhere) since the view of
platonism in the 1930s was not the ordinary view today.
  Some of the critical evidence for shifts in Goedel's
thought in the 1930s Davis presents in `What did Goedel
believe and when did he believe it?'  (Bulletin of
Symbolic Logic 11 (June 2005)).  One item on the list
of problems to reconcile is the lecture at Edgar
Zilsel's (1938a in CW 3, which is mentioned by Parsons
in the paper cited previously on p. 52), which
according to Parsons and Wilfried Sieg gives `an
overview of possibilities for a revised Hilbert
program', including the `crucial questions' as to what
`extensions of finitist methods will yield consistency
proofs, and what epistemological value such proofs will
have' (CW 3, p. 63).  This lecture contains a
reformulation of Gentzen's first consistency proof of
1936, for more on which see W. W. Tait, `Goedel's
reformulation of Gentzen's first consistency proof for
arithmetic: The no-counterexample interpretation',
Bulletin of Symbolic Logic 11 (June 2005).
  But, when one considers the major ongoing work of
1938-1940 on the hierarchy of constructible sets (see
CW 2, pp. 1-100, which includes Robert Solovay's note;
for what Goedel thought about the proof at that time,
see Feferman, `Goedel's life and work' (CW 1, p. 36n.
s), and p. 51 of the Parsons article previously cited
and n. 15 thereon), it is due to the impredicative
commitments, and the commitment I guess to omega_2,
connected to that project that would make a
generalization about a steady position for Goedel of
antirealism throughout the 1930's possibly awkward
without substantial qualification.  

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