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Full Idea
Operating with the infinite can be made certain only by the finitary.
Gist of Idea
Only the finite can bring certainty to the infinite
Source
David Hilbert (On the Infinite [1925], p.201)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.201
A Reaction
See 'Compactness' for one aspect of this claim. I think Hilbert was fighting a rearguard action, and his idea now has few followers.