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Gödel, Nagel, minds and machines
Solomon Feferman
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1. The imbroglio between Nagel and Gödel.
Some fifty years ago (1957 to be exact),
Ernest Nagel and Kurt Gödel became involved in a contentious exchange about the
possible inclusion of Gödel’s original work on incompleteness in the book,
Gödel’s
Proof
, then being written by Nagel with James R. Newman.
What led to the conflict
were some unprecedented demands that Gödel made over the use of his material and his
involvement in the contents of the book
⎯
demands that resulted in an explosive reaction
on Nagel’s part.
In the end the proposal came to naught.
But the story is of interest
because of what was basically at issue, namely their provocative related but contrasting
views on the possible significance of Gödel’s theorems for minds vs. machines in the
development of mathematics. That is our point of departure for the attempts by Gödel,
and later Lucas and Penrose, to establish definitive consequences of those theorems,
attempts which
⎯
as we shall see
⎯
depend on highly idealized and problematic
assumptions about minds, machines and mathematics.
In particular, I shall argue that
there is a fundamental equivocation involved in those assumptions that needs to be
reexamined.
In conclusion, that will lead us to a new way of looking at how the mind
may work in deriving mathematics which straddles the mechanist and antimechanist
viewpoints.
The story of the conflict between Gödel and Nagel has been told in full in the
introductory note by Charles Parsons and Wilfried Sieg to the correspondence between
them in Volume V of the Gödel
Collected Works
(Gödel 2003a, pp. 135 ff), so I’ll
confine myself to the high points.
The first popular exposition of Gödel’s incompleteness theorems was published
by Nagel and Newman in 1956 in an article entitled “Goedel’s proof” for the
Scientific
American.
The article was reprinted soon after in the four volume anthology edited by
Newman,
The World of Mathematics: A small library of the literature of mathematics
1
Revised text of the Ernest Nagel Lecture given at Columbia University on September
27, 2007.
I wish to thank Hannes Leitgeb and Carol Rovane for their helpful comments
on an earlier version.
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from A'hmosé the Scribe to Albert Einstein, presented with commentaries and notes
.
That was an instant bestseller, and has since been reprinted many times.
Newman had
been trained as a mathematician but then became a lawyer and was in government service
during World War II.
Endlessly fascinated with mathematics, he became a member of
the editorial board of the
Scientific American
a few years after the war.
Ernest Nagel had long been recognized as one of the leading philosophers of
science in the United States, along with Rudolf Carnap, Carl Hempel and Hans
Reichenbach.
Like them, he was an immigrant to the US, but unlike them he had come
much earlier, in 1911 at the age of 10.
Later, while teaching in the public schools, Nagel
received his bachelor’s degree at CCNY in 1923 and his PhD in philosophy at Columbia
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 Spring '08
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