Analogy of Logical and Numerical Terms.
If my assertion is correct that number arises out of logical conditions, we ought to find number obeying all the laws of logic. It is almost superfluous to point out that this is the case with the fundamental laws of identity and difference, and it only remains to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out (p. 32). Thus the Law of Commutativeness, is equally true of quality and quantity. As in logic we have
so in mathematics it is familiarly known that
The properties of space are as indifferent in multiplication as we found them in pure logical thought.
Similarly, as in logic
| triangle or square = | square or triangle, | |
|---|---|---|
| or generally | A ꖌ B = | B ꖌ A, |
| so in quantity | 2 + 3 = | 3 + 2, |
| or generally | x + y = | y + x . |
The symbol ꖌ is not identical with +, but it is thus far analogous.
How far, now, is it true that mathematical symbols obey the Law of Simplicity expressed in the form
or the example
Apparently there are but two numbers which obey this law; for it is certain that
is true only in the two cases when x = 1, or x = 0.
In reality all numbers obey the law, for 2 × 2 = 2 is not really analogous to AA = A. According to the definition of a unit already given, each unit is discriminated from each other in the same problem, so that in 2′ × 2″, the first two involves a different discrimination from the second two. I get four kinds of things, for instance, if I first discriminate “heavy and light” and then “cubical and spherical,” for we now have the following classes—
| heavy, cubical. | light, cubical. |
|---|---|
| heavy, spherical. | light, spherical. |
But suppose that my two classes are in both cases discriminated by the same difference of light and heavy, then we have
| heavy | heavy = | heavy, |
|---|---|---|
| heavy | light = | 0, |
| light | heavy = | 0, |
| light | light = | light. |