[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
meaning.
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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