Hilbert's Beer-mugs

なんとなく引用しておこうと思って。

• Ivor Grattan-Guinness, The Search for Mathematical Roots 1870-1940, Princeton University Press, 2000

から。

Concerning a lecture by Hermann Weiner to the DMV on proving Pascal's and Desargues's theorems, he had stated that `one must be able to say ``tables, chairs, berr-mugs'' each time in place of ``points, lines, planes'''; but this famous remark is normally misunderstood and Hilbert may not have thought it through at the time.
Firstly, Hilbert was advocating model theory for the axioms (intuitively at this early stage), not the mere use of words nor the marks-on-paper formalism that Frege detected in the symbol-loving arithmeticians; intuitive knowledge of Euclidean geometry motivated the axiomatising enterprise in the first place. [...]one obvious consequence is that intuitive knowledge of beer-mugs is different. Secondly, he treated concepts such as `point' as implicitly defined via axioms. Thirdly, the same versatility could not be demanded of the logical connectives used to form and connect his propositions; for example, `and' cannot become `wine-glass'. (p. 208f.)

Hilbert was perhaps too enchanted with presenting geometrical axioms by beer-mugs to think through the consequences for concepts and for logic, and did not sufficiently stress the place of intuitive thoery prior to its axiomatisation. (p. 213)