FOM: Mathematicians' views of Goedel's incompleteness theorem(s)

[Solomon Feferman]

  The "splendid" biography, *Logical Dilemmas. The life and work of Kurt
Goedel* by John W. Dawson, Jr., has received a largely astute and
sympathetic review in the SIAM NEWS of Oct. 1997, pp.11-12, by Philip J.
Davis.  The following quotes from a small part of this review are relevant
to some of our discussions.

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 "Despite this development [of computationally undecidable problems], it is
an important fact that the brilliant, earth-shaking theorems of Goedel are
of absolute unimportance to 99.5% of research mathematicians in their
professional work.  This is a paradox that is rarely discussed.  I've
asked a number of informed mathematicians for their comments and have
received a wide spread of answers.  
 "One response, which I consider a bit flip is 'Well, the discipline of
mathematics is so large and so fragmented that most professionals hardly
know or care anything about what their colleagues at the next desk are
doing. We all do our own thing.' ...
 "While all this is true enough, considering that Goedel's work and that
of his followers lie at what are supposed to be the foundations of
mathematics, where does that leave the material at the higher levels of
the structure?
 "Another response is that Goedel's Incompleteness Theorem is like the
Bible, or perhaps the Ten Commandments. It states certain limitations.  We
all know what they are, honor them in principle, and then assume that they
do not pertain to our own work.  
 "Although Goedel's theorems are now a significant part of the unstated
metamathematical assumptions of research, they are relegated to a far back
burner. If a number theorist is working, say, on the famous and as yet
unsolved problem of whether there are an unlimited number of twin
primes..., then the strong assumption underlying the work, even in our
post-Goedelian period, is that the answer is either yes or no.  It is not
assumed that on the basis of the traditional axioms of arithmetic we
cannot decide for true or false.
 "Some have answered the paradox of irrelevance in this way: 'Wait, the
impact is yet to be felt.'  Thus philosopher of science David Berlinski's
'We have just begun--begun!--to assess the full importance of Goedel's
incompleteness theorem for philosophy, mathematics, and computer science."

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I shall, of course, inform Philip Davis of my own reasons for the
"paradox" of irrelevance, but perhaps others would like to communicate
their own views to him and/or to these pages.  (Davis is Prof. Emeritus of
Mathematics at Brown University.)

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Since Dawson is one of my co-editors on the Goedel *Collected Works* (and
has been from the word go), perhaps I won't be regarded as unbiased in my
own assessment of his biography, so let me repeat Davis' word "splendid"
for it.  I think it is a must read for all logicians [as befits the father
of our intellectual country], and should be recommended strongly to our
mathematical, informatical and philosophical colleagues, as well as those
farther afield.  It's also a good holiday present for relatives who want
to know what in the world you do.  

Sol Feferman