[FOM] alleged quote from Hilbert

Jeffrey Ketland ketland at ketland.fsnet.co.uk
Thu Apr 7 19:01:40 EDT 2005

Martin Davis,

> 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. Goldstein's source is: John de Pillis & Nick Rose, "Mathematical
> Maxims and Minims" (Raleigh, NC 1988).
> I've been unable to locate this book: it's listed neither in the unified U
> of Calif library catalog, nor in NYPL, nor (the ultimate authority) on 
> Amazon.

The quote sounds quite dodgy.
However, it is almost word for word the same as a direct quotation from a 
book by E.T. Bell (who claimed it was Hilbert's view):

    Mathematics, according to D. Hilbert (1862-1943), is
    nothing more than a game played according to certain
    simple rules with meaningless marks on paper.
    (E.T. Bell 1952, _Mathematics: Queen and Servant of
    Science_, G. Bell & Sons Ltd, London, p. 21.)

E.T. Bell was Professor of Mathematics at California Institute of 
Technology. He published _The Queen of the Sciences_ in 1931 and _The 
Handmaiden of the Sciences_ in 1937. The book cited above (Bell 1952) is "a 
thorough revision and a very considerable amplification" of these two 
earlier volumes.

Bell does not use quotation marks. Nor, so far as I can see from a quick 
glance, does he cite any of Hilbert's works (the book has no bibliography). 
Bell repeats his opinions about what "Hilbert asserted" on p. 38:

   If indeed, as Hilbert asserted, mathematics is a meaningless game
   played with meaningless marks on paper, the only mathematical
   experience to which we can refer is the making of marks on paper.
   (op. cit., p. 38).

So, perhaps this quote began its life in Bell's book.

The position it refers to sounds more like the formalist position of E. 
Heine and J. Thomae (a colleague of Frege's at Jena). Thomae's view was,

   For the formalist, arithmetic is a game with signs, which are
   called empty. That means they have no other content (in
   the calculating game) than they are assigned by their
   behaviour with respect to certain rules of combination (rules
   of the game).

The original is (somewhere) in J. Thomae 1898, _Elementare Theorie der 
analytische Functionen einer complexen Veraenderlichen_. It is quoted in 
Frege 1903, _Grundgesetze der Arithmetic_, Vol II, Section 88. It also 
appears in English in the long section "Frege Against the Formalists" in 
_Philosophical Writings of Gottlob Frege_ (1980, 3rd edition), translated by 
P. Geach and M. Black, p. 163.

Regards --- Jeff

Jeffrey Ketland
Department of Philosophy
University of Edinburgh
George Square
Edinburgh EH8 9JX
United Kingdom
jeffrey.ketland at ed.ac.uk

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