Were this not the case and were the units really
1′ + 1″ + 1″ + 1‴ + 1″″,
the Law of Unity would, as before remarked, apply, and
1″ + 1″ = 1″.
Mathematical symbols then obey all the laws of logical symbols, but two of these laws seem to be inapplicable simply because they are presupposed in the definition of the mathematical unit. Logic thus lays down the conditions of number, and the science of arithmetic developed as it is into all the wondrous branches of mathematical calculus is but an outgrowth of logical discrimination.
Principle of Mathematical Inference.
The universal principle of all reasoning, as I have asserted, is that which allows us to substitute like for like. I have now to point out how in the mathematical sciences this principle is involved in each step of reasoning. It is in these sciences indeed that we meet with the clearest cases of substitution, and it is the simplicity with which the principle can be applied which probably led to the comparatively early perfection of the sciences of geometry and arithmetic. Euclid, and the Greek mathematicians from the first, recognised equality as the fundamental relation of quantitative thought, but Aristotle rejected the exactly analogous, but far more general relation of identity, and thus crippled the formal science of logic as it has descended to the present day.
Geometrical reasoning starts from the axiom that “things equal to the same thing are equal to each other.” Two equalities enable us to infer a third equality; and this is true not only of lines and angles, but of areas, volumes, numbers, intervals of time, forces, velocities, degrees of intensity, or, in short, anything which is capable of being equal or unequal. Two stars equally bright with the same star must be equally bright with each other, and two forces equally intense with a third force are equally intense with each other. It is remarkable that Euclid has not explicitly stated two other axioms, the truth of which is necessarily implied. The second axiom should be that “Two things of which one is equal and the other unequal to a third common thing, are unequal to each other.” An equality and inequality, in short, give an inequality, and this is equally true with the first axiom of all kinds of quantity. If Venus, for instance, agrees with Mars in density, but Mars differs from Jupiter, then Venus differs from Jupiter. A third axiom must exist to the effect that “Things unequal to the same thing may or may not be equal to each other.” Two inequalities give no ground of inference whatever. If we only know, for instance, that Mercury and Jupiter differ in density from Mars, we cannot say whether or not they agree between themselves. As a fact they do not agree; but Venus and Mars on the other hand both differ from Jupiter and yet closely agree with each other. The force of the axioms can be most clearly illustrated by drawing equal and unequal lines.[91]
The general conclusion then must be that where there is equality there may be inference, but where there is not equality there cannot be inference. A plain induction will lead us to believe that equality is the condition of inference concerning quantity. All the three axioms may in fact be summed up in one, to the effect, that “in whatever relation one quantity stands to another, it stands in the same relation to the equal of that other.”
The active power is always the substitution of equals, and it is an accident that in a pair of equalities we can make the substitution in two ways. From a = b = c we can infer a = c, either by substituting in a = b the value of b as given in b = c, or else by substituting in b = c the value of b as given in a = b. In a = b ~ d we can make but the one substitution of a for b. In e ~ f ~ g we can make no substitution and get no inference.