[FOM] alleged quote from Hilbert

Richard Zach rzach at ucalgary.ca
Fri Apr 8 12:56:21 EDT 2005

> 	Mathematics is a game played according to certain simple rules with 
> meaningless marks 	on paper.

As far as I know, Hilbert never said this.  There are two parts to this
claim: 1. that the content of mathematics is given by, or that
mathematical objects/statements are just "marks on paper" and 2. that
inference is just a game with these marks.

One can certainly read a formalistic view of mathematics into Hilbert's
writings of the 1920s.  E.g., in the 1921 Hamburg talk, translated as
"The New Grounding of Mathematics. First Communication" in
Mancosu's /From Brouwer to Hilbert/, you can find the original statement
of this formalism:

[For the purpose of developing analysis within the axiomatic method, and
in order to enable a metamathematical study of mathematical theories]
the usual contentual ideas of the mathematical theory must be replaced
by formulae and rules, and imitated by formalisms.  In other words, we
need to have a strict formalization of the entire mathematical theory,
inclusive of its proofs, so that--following the example of the logical
calculus--the mathematical inferences and definitions become a formal
part of the edifice of mathematics. (p. 204)

[Two points emerge in regards to Hilbert's proof theory:]
*First*: everything that hitherto made up mathematics proper is now to
be strictly formalized, so that *mathematics proper*, or mathematics in
the strict sense, becomes a stock of provable formulae. [...] (p. 211)

It is clear from the context that this replacement of "mathematics
proper" by a formalism is not an interpretation of what mathematics
actually is, rather, it is a reconstruction carried out for the
methodological purpose of proving consistency.

Part 2., that this formalism is a "mere game with symbols" is, to the
best of my knowledge, something that Hilbert's critics ascribed to him,
e.g., Weyl in "The Current Epistemological Situation in
Mathematics" (also in the Mancosu volume).  To wit:

The statements [of mathematics] become meaningless figures built up from
signs. Mathematics is no longer knowledge but a *game of formulae*.
ruled by certain conventions, which is very well comparable to the game
of chess. (p. 136)

I don't know where the "on paper" part of the quote comes from.  It is
pretty clear from Hilbert's and Bernays's writings that the signs
(stroke symbols, formulae) are not to be understood as physical objects
(but they do seem to change their mind on this; see Bernays's exchange
with Müller and Mancosu's discussion, esp. pp. 168--173).
Richard Zach ...... http://www.ucalgary.ca/~rzach/
Associate  Professor,   Department  of  Philosophy
University of Calgary, Calgary, AB T2N 1N4, Canada

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