ONT Re: Excuses, Exercises, Exergues, Exorabilities
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| No more than any other science can mathematics be founded by logic alone;
| rather, as a condition for the use of logical inferences and the performance
| of logical operations, something must already be given to us in our faculty of
| representation [in der Vorstellung], certain extralogical concrete objects that
| are intuitively [anschaulich] present as immediate experience prior to all thought.
| If logical inference is to be reliable, it must be possible to survey these objects
| completely in all their parts, and the fact that they occur, that they differ from one
| another, and that they follow each other, or are concatenated, is immediately given
| intuitively, together with the objects, as something that neither can be reduced to
| anything else nor requires reduction. This is the basic philosophical position
| that I regard as requisite for mathematics and, in general, for all scientific
| thinking, understanding, and communication. And in mathematics, in particular,
| what we consider is the concrete signs themselves, whose shape, according to
| the conception we have adopted, is immediately clear and recognizable.
| This is the very least that must be presupposed; no scientific thinker
| can dispense with it, and therefore everyone must maintain it,
| consciously or not.
|
| DH, TFOM, pages 464-465.
|
| David Hilbert,
|"The Foundations Of Mathematics",
| address delivered to the Hamburg Mathematical Seminar, July 1927,
| reprinted in Jean van Heijenoort (ed.), 'From Frege To Gödel,
| A Source Book in Mathematical Logic', 1879-1931',
| Harvard University Press, Cambridge, MA, 1967.
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