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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.