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Full Idea
To preserve the simple formal rules of ordinary Aristotelian logic, we must supplement the finitary statements with ideal statements.
Gist of Idea
We extend finite statements with ideal ones, in order to preserve our logic
Source
David Hilbert (On the Infinite [1925], p.195)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.195
A Reaction
I find very appealing the picture of mathematics as rooted in the physical world, and then gradually extended by a series of 'idealisations', which should perhaps be thought of as fictions.
| 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] |