[FOM] alleged quote from Hilbert

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Fri Apr 8 14:30:42 EDT 2005

Quoting Martin Davis <martin at eipye.com>:

> 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. 

Even if Hilbert said exactly this, I think this should be 
considered in the context of that time as quite reasonable 
opposition both to platonism and intuitionism with their 
pretension on having some kind of eternal, absolute 
truth/meaning in mathematics. There is also no crime in 
asserting an idea in somewhat exaggerated form. But even 
in this form it makes sense: 

the last and main argument in mathematics is rigorous
proof/construction/algorithm, that is something formal 
(at least to some degree). It would be against the 
mathematical nature to check correctness of a rigorous 
proof without abstraction (may be only partial abstraction) 
from the meaning (whatever it is) of symbols. This is an 
ideal which we should follow as soon as we are mathematicians. 
Whether and when this ideal is (or should or can be) really 
achieved is a different question. 

Such style of a phrase needs in some further interpretation, 
additions which could mild, clarify and ramify it and made 
it even more plausible. What if (just for example only) to 
reread this phrase as: 

Mathematics is a game played according to certain simple 
rules with marks on paper having no predefined, unique 

The above reformulated form is also some exaggeration 
(I mean "game" - chess as well?), but I am permanently 
expressing the opinion in FOM that formalist view can 
have quite reasonable reading.  It puts in the center 
exactly that feature which distinguishes mathematics 
from other human activities. This is mathematical rigour 
and the way how it is achieved - by formalizing (mechanizing, 
automating, thereby strengthening) the thought and intuition. 
In an appropriate reading there is *nothing* against intuition 
and meaning in formalist position.  

Roger Jones: 
> Hilbert's view seems to have been not that primitives are
> meaningless, but that their meaning should exclusively
> be given formally.

Very much agree! Formalism is a *tool* to express 
intuition (which otherwise would be even not 
expressible at all) and to strengthen our thought 
(see above). 

These are opponents of formalism who take its 
deliberately exaggerated formulations as the full 
and official version. As is often necessary, 
we should also read "between lines" and may be 
reconsider the initial great idea. 

Vladimir Sazonov

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