[FOM] alleged quote from Hilbert
Richard Heck
rgheck at bobjweil.com
Fri Apr 8 22:56:03 EDT 2005
On Thursday 07 April 2005 7:50 pm, Martin Davis wrote:
>>In Rebecca Goldstein's recent "Incompleteness" she quotes
>>Hilbert as follows:
>>
>> Mathematics is a game played according to certain simple
>>rules with meaningless marks on paper.
>>
>>I would appreciate any information about this quotation about
>>which I'm dubious.
>>
>>
Certainly, no such thing was Hilbert's view in the twenties. As I
understand it---and of course there are many people who know much more
about this matter than I do, such as Michael Hallett, Mic Detlefsen, and
Charles Parsons---Hilbert's view at that time was that primitive
recursive arithmetic could be grounded in some form of mathematical
intuition, and that extensions thereto were to be justified by proofs of
conservativity. And that, as opposed to some alleged commitment to
formalism, of course, is what has led most people to regard the second
imcompleteness theorem as a refutation of Hilbert's program (though
there are resisters, such as Detlefsen). Perhaps Hilbert could therefore
be interpreted as having a formalist attitude towards some parts of
mathematics---perhaps his view was that, say, set theory is a mere game
played with symbols, whose real content is given only by its
consequences for PRA---but not towards all of it.
Further to Roger Jones's quotation from one of Hilbert's letters to
Frege, it is worth noting (if memory serves) that Frege does NOT lump
Hilbert with Thomae and Heine, who, as Jeff Ketland noted, definitely
did hold the view Goldstein is attributing to Hilbert. Frege's
criticisms of Hilbert are quite different. Frege sees quite clearly that
Hilbert regards the axioms of geometry as DEFINING terms like "point"
and "line". But if they do that, then these terms are not meaningless
symbols but have whatever content the definitions give them, and the
issue Frege presses is what content a system of axiom can give the
symbols. Frege concludes that the definitions do succeed in assigning a
certain content to the terms, but not the kind of content Hilbert thinks
they assign.
Some of Frege's criticisms of Hilbert sound like some of his criticisms
of the formalists, but the reason is that the two positions have some
similar commmitments. For example, both positions regard the consistency
of a system as being all one can ask of it mathematically, there being
no further of the existence of its objects. (Some have wanted to add
categoricity as another requirement.) But Hilbert's reasons for this
view, around 1900, are very different from those of Thomae and Heine.
For Hilbert, it is proof that the system of axioms constitutes a
coherent set of implicit definitions. For Heine and Thomae, it is
because there is nothing beyond the symbols and the rules that govern
them, and a contradictory set of rules (as they see it) is no set of
rules at all.
Given Frege's aversion to formalism, one would have supposed he would
have tagged Hilbert with that label had he thought he deserved it. So,
even if Frege wasn't prepared to tag him with it....
Richard Heck
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