[FOM] alleged Hilbert quote

[Klaus Frovin Joergensen]

> While Hilbert probably never said that math is a meaningless game with
> symbols,
> ths spirit of this phrase is somewhat similar to a famous actual citation
> from
> Hilbert's Intro to his Foundations of Geometry:
>
> "One must be able to say at all times -- instead of points, straight lines,
> and
> planes -- tables, beer mugs, and chairs."

As far as I know it is not settled whether Hilbert actually said that. I would
be interested in a precise reference. It is of course attributed to him, and it
could have been said by him, but I think it is one of the most understood
'quotes' of Hilbert. Let me give my interpretation of the 'spirit' it may be
said in.

One of the main concerns for Hilbert when developing the axiomatic method by the
end of the 19th century was not rigor, but discovery. Hilbert does not see
geometry as some formal and meaningless theory with no content; Hilbert's
geometry is an investigation of how we - as human beings - conceptualize space,
thus he writes (with implicit reference to Kant) "[t]his problem is tantamount
to the logical analysis of our intution of space" (Foundations, p. 1).

The main motivation for applying formal methods in _Foundations_ is that new
discoveries are made *possible* by the method. In his seminal book the
axioms are not treated as self-evident axioms which are true in some absolute
sense. Hilbert is interested in groups of axioms, their relations and
consequences. The five groups are I. Axioms of connection, II. Axioms of order,
III. Axioms of parallels, IV. Axioms of congruence and V. Axioms of continuity.
He shows that the groups are pairwise independent and this is done by
providing different models validating different groups of axioms. Thus the
groups give rise to different geometries such as Euclidean, non-Euclidean,
Archimedian, non-Archemedian, etc. Therefore, when Hilbert talks
about lines they can be lines as Euclid understands them, or they can be great
circles on a surface of a square, or they can be something quite different.

The core of Hilbert's method is that he detaches the geometrical concepts from
their semantics. Hilbert is not studying one, and only one, particular model.
He is interested in a whole variety of models. This is *not* for the sake of
rigor, this is for the sake of discovery. Thus, by the axiomatic formal method
he can discover relations between and properties of geometries, which are
not possible to discover unless language and semantics are separated. And this
is not done in the traditional (Euclidean) way of doing geometry,
which is about one particular model. As the tradional way of doing geometry
is about one model, the same need to separate language and semantics is not
present. With Hilbert the idea of an uninterpreted language is new, and the
purpose of it is to study varieties of models. The subject matter is, however,
the different models - the (semi) formal language just means.

/Klaus

***
Klaus Frovin Jørgensen
http://akira.ruc.dk/~frovin