Number is but another name for *diversity*. Exact identity is unity, and with difference arises plurality. An abstract notion, as was pointed out (p. 28), possesses a certain *oneness*. The quality of *justice*, for instance, is one and the same in whatever just acts it is manifested. In justice itself there are no marks of difference by which to discriminate justice from justice. But one just act can be discriminated from another just act by circumstances of time and place, and we can count many acts thus discriminated each from each. In like manner pure gold is simply pure gold, and is so far one and the same throughout. But besides its intrinsic qualities, gold occupies space and must have shape and size. Portions of gold are always mutually exclusive and capable of discrimination, in respect that they must be each without the other. Hence they may be numbered. Plurality arises when and only when we detect difference. For instance, in counting a number of gold coins I must count each coin once, and not more than once. Let C denote a coin, and the mark above it the order of counting. Then I must count the coins C′ + C″ + C‴ + C″″ + .... If I were to count them as follows C′ + C″ + C‴ + C‴ + C″″ + ..., 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. [90] *Pure Logic*, Appendix, p. 82, § 192. 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.*