Quote: proof by induction allows us to generalize, to pass from the finite to the infinite

Quotation

The process is proof by recurrence. We first show that a theorem is true for n=1; we then show that if it is true for n-1 it is true for n, and we conclude that it is true for all integers. ... if we look carefully, we find this mode of reasoning at every step ... [p. 11] If he applies his faculties to Arithmetic, he cannot conceive its general truths by direct intuition alone; to prove even the smallest theorem he must use reasoning by recurrence, for that is the only instrument which enables us to pass from the finite to the infinite. ... without [the infinite] there would be no science at all, because there would be nothing general.   Google-1   Google-2

Published before 1923

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