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Full Idea
Just as in the limit processes of the infinitesimal calculus, the infinitely large and small proved to be a mere figure of speech, so too we must realise that the infinite in the sense of an infinite totality, used in deductive methods, is an illusion.
Gist of Idea
The idea of an infinite totality is an illusion
Source
David Hilbert (On the Infinite [1925], p.184)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.184
A Reaction
This is a very authoritative rearguard action. I no longer think the dispute matters much, it being just a dispute over a proposed new meaning for the word 'number'.
| 12456 | I aim to establish certainty for mathematical methods [Hilbert] |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| 9636 | My theory aims at the certitude of mathematical methods [Hilbert] |
| 12457 | There is no continuum in reality to realise the infinitely small [Hilbert] |
| 9633 | No one shall drive us out of the paradise the Cantor has created for us [Hilbert] |
| 12459 | The subject matter of mathematics is immediate and clear concrete symbols [Hilbert] |
| 12460 | We extend finite statements with ideal ones, in order to preserve our logic [Hilbert] |
| 18112 | Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert] |
| 12461 | We believe all mathematical problems are solvable [Hilbert] |
| 12462 | Only the finite can bring certainty to the infinite [Hilbert] |