Page:Philosophy and Fun of Algebra.djvu/30

 CHAPTER V MATHEMATICAL CERTAINTY AND REDUCTIO AD ABSURDUM

is very often said that we cannot have mathematical certainty about anything except a few special subjects, such as number, or quantity, or dimensions.

Mathematical certainty depends, not on the subject matter of our investigation, but upon three conditions. The first is a constant recognition of the limits of our own knowledge and the fact of our own ignorance. The second is reverence for the As-Yet-Unknown. The third is absolute fearlessness in meeting the reductio ad absurdum. In mathematics we are always delighted when we come to any such conclusion as 2 + 3 = 7. We feel that we have absolutely cleared out of the way one among the several possible hypotheses, and are ready to try another.

We may be still groping in the dark, but we know that one stumbling-block has been cleared 26