Thus, (heavy or light) × (heavy or light) = (heavy or light).
In short, twice two is two unless we take care that the second two has a different meaning from the first. But under similar circumstances logical terms give the like result, and it is not true that A′A″ = A′, when A″ is different in meaning from A′.
In a similar manner it may be shown that the Law of Unity
holds true alike of logical and mathematical terms. It is absurd indeed to say that
except in the one case when x = absolute zero. But this contradiction x + x = x arises from the fact that we have already defined the units in one x as differing from those in the other. Under such circumstances the Law of Unity does not apply. For if in
we mean that A″ is in any way different from A′ the assertion of identity is evidently false.
The contrast then which seems to exist between logical and mathematical symbols is only apparent. It is because the Laws of Simplicity and Unity must always be observed in the operation of counting that those laws seem no further to apply. This is the understood condition under which we use all numerical symbols. Whenever I write the symbol 5 I really mean
and it is perfectly understood that each of these units is distinct from each other. If requisite I might mark them thus
Were this not the case and were the units really
the Law of Unity would, as before remarked, apply, and
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,