FOM: Goedel's alleged remark
Charles Parsons
parsons2 at fas.harvard.edu
Wed Nov 12 15:43:48 EST 1997
Several postings have referred to the following statement of Morris Kline:
>"Perhaps more surprising is Goedel's statement of 1950 that
the role of the alleged "foundations" is rather comparable to the function
discharged, in physical theory, by explanatory hypotheses... The so-called
logical or set-theoretical >foundation for number theory or of any other
well established mathematical theory is explanatory, rather than
foundational, exactly as in physics where the actual function of axioms is
to explain the phenomena described by the theorems of this system rather
than to provide a genuine foundation for such theorems.
What these leaders is [are?] acknowledging is that the attempt to establish
a universally acceptable, logically sound body of mathematics has failed."
I don't know of any statement to that effect by Goedel, and I've worked
pretty closely with Goedel's writings in this area. I suspect that Kline
had in mind remarks like Goedel's expression of approval for the view he
finds in Russell that axioms of logic and mathematics don't have to be
"evident in themselves, but rather their justification lies (exactly as in
physics) in the fact that they make it possible for these 'sense
perceptions' to be deduced" (Collected Works, II, 121). He refers to the
fact that the solutions to certain problems requires assumptions
"essentially transcending arithmetic, i. e. the domain of the kind of
elementary indisputable evidence that may be most fittingly compared to
sense perception."
Another, more widely quoted remark is the following:
It seems to me that the assumption of such objects [classes and concepts]
is quite as legitimate as the assumption of physical bodies and there is
quite as much reason to believe in their existence. They are in the same
sense necessary to obtain a satisfactory system of mathematics as physical
bodies are necessary for a satisfactory theory of our sense perceptions
..." (ibid., 128).
In neither of these places, nor so far as I know in other places where he
expresses a similar view, does Goedel even use the word "explanation",
still less contrast being "explanatory" with being "foundational."
I don't have Kline's book at hand and can't see what Goedel source he
gives, if any. But my conjecture is that Kline recalled these remarks,
misremembered their date, and read an interpretation of his own into them.
But one of you might show me wrong.
Charles Parsons
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