Page:Philosophy and Fun of Algebra.djvu/26

PHILOSOPHY AND FUN OF ALGEBRA Such suggestions as "suppose x were Unity" are called "working hypotheses," or "hypothetical data." In Algebra we are very careful to distinguish clearly between actual data and hypothetical data.

This is only part of the essence of Algebra, which, as I told you, consists in preserving a constant, reverent, and conscientious awareness of our own ignorance.

When we have exhausted all the possible hypotheses connected with Unity and Zero, we next begin to experiment with other values of x; e.g.—suppose x were 2, suppose x were 3, suppose it were 4. Then, suppose it were one half, or one and a half, and so on, registering among our data, each time, either "x may be so and so," or "x cannot be so and so."

The method of finding out what x cannot be, by showing that certain suppositions or hypotheses lead to a ridiculous statement, is called the method of reductio ad absurdum. It is largely used by Euclid. 22