The principles of science

by William Stanley Jevons

I should make the third coin into two, and should imply the existence of difference where there is no difference.‍[90] C‴ and C‴ are but the names of one coin named twice over. But according to one of the conditions of logical symbols, which I have called the Law of Unity (p. [72]), the same name repeated has no effect, and

A ꖌ A = A.

We must apply the Law of Unity, and must reduce all identical alternatives before we can count with certainty and use the processes of numerical calculation. Identical alternatives are harmless in logic, but are wholly inadmissible in number. Thus logical science ascertains the nature of the mathematical unit, and the definition may be given in these terms—A unit is any object of thought which can be discriminated from every other object treated as a unit in the same problem.

It has often been said that units are units in respect of being perfectly similar to each other; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality. If three coins were so similar that they occupied the same space at the same time, they would not be three coins, but one coin. It is a property of space that every point is discriminable from every other point, and in time every moment is necessarily distinct from any other moment before or after. Hence we frequently count in space or time, and Locke, with some other philosophers, has held that number arises from repetition in time. Beats of a pendulum may be so perfectly similar that we can discover no difference except that one beat is before and another after. Time alone is here the ground of difference and is a sufficient foundation for the discrimination of plurality; but it is by no means the only foundation. Three coins are three coins, whether we count them successively or regard them all simultaneously. In many cases neither time nor space is the ground of difference, but pure quality alone enters. We can discriminate the weight, inertia, and hardness of gold as three qualities, though none of these is before nor after the other, neither in space nor time. Every means of discrimination may be a source of plurality.

Our logical notation may be used to express the rise of number. The symbol A stands for one thing or one class, and in itself must be regarded as a unit, because no difference is specified. But the combinations AB and Ab are necessarily two, because they cannot logically coalesce, and there is a mark B which distinguishes one from the other. A logical definition of the number four is given in the combinations ABC, ABc, AbC, Abc, where there is a double difference. As Puck says—

“Yet but three? Come one more;
Two of both kinds makes up four.”

I conceive that all numbers might be represented as arising out of the combinations of the Logical Alphabet, more or less of each series being struck out by various logical conditions. The number three, for instance, arises from the condition that A must be either B or C, so that the combinations are ABC, ABc, AbC.

Of Numerical Abstraction.

There will now be little difficulty in forming a clear notion of the nature of numerical abstraction. It consists in abstracting the character of the difference from which plurality arises, retaining merely the fact. When I speak of three men I need not at once specify the marks by which each may be known from each. Those marks must exist if they are really three men and not one and the same, and in speaking of them as many I imply the existence of the requisite differences. Abstract number, then, is the empty form of difference; the abstract number three asserts the existence of marks without specifying their kind.

Numerical abstraction is thus seen to be a different process from logical abstraction (p. [27]), for in the latter process we drop out of notice the very existence of difference and plurality. In forming the abstract notion hardness, we ignore entirely the diverse circumstances in which the quality may appear. It is the concrete notion three hard objects, which asserts the existence of hardness along with sufficient other undefined qualities, to mark out three such objects. Numerical thought is indeed closely interwoven with logical thought. We cannot use a concrete term in the plural, as men, without implying that there are marks of difference. But when we use an abstract term, we deal with unity.