[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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