Chapter 3, Simultaneous Problems, part 2

If the result of doing so is to bring out some such ridiculous answer as “2 and 3 make 7,” we then know that x cannot be 1. We now add to our column of data, “x cannot be 1.” But if we come to a truism, such as “2 and 3 make 5,” we add to our column of data, “x may be 1.” Some people add to their column of data, “x is 1,” but that again is not Algebra. Next we try the experiment of supposing x to be equal to zero (0), and go over the ground again. Then we go over the same ground, trying y as 1 and as 0. And then we try the same with z. Some people think that it is waste of time to go over all this ground so carefully, when all you get by it is either nonsense, such as “2 and 3 are 7”; or truisms, such as “2 and 3 are 5.” But it is not waste of time. For, even if we never arrive at finding out the value of x, or y, or z, every conscientious attempt such as I have described adds to our knowledge of the structure of Algebra, and assists us in solving other problems. 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.