Nevertheless, it will appear on consideration, that this apparently so
decisive instance is no instance at all; that there is in every step of an
arithmetical or algebraical calculation a real induction, a real inference
of facts from facts; and that what disguises the induction is simply its
comprehensive nature, and the consequent extreme generality of the
language. All numbers must be numbers of something: there are no such
things as numbers in the abstract. _Ten_ must mean ten bodies, or ten
sounds, or ten beatings of the pulse. But though numbers must be numbers
of something, they may be numbers of any thing. Propositions, therefore,
concerning numbers, have the remarkable peculiarity that they are
propositions concerning all things whatever; all objects, all existences
of every kind, known to our experience. All things possess quantity;
consist of parts which can be numbered; and in that character possess all
the properties which are called properties of numbers. That half of four
is two, must be true whatever the word four represents, whether four
hours, four miles, or four pounds weight. We need only conceive a thing
divided into four equal parts (and all things may be conceived as so
divided), to be able to predicate of it every property of the number four,
that is, every arithmetical proposition in which the number four stands on
one side of the equation. Algebra extends the generalization still
farther: every number represents that particular number of all things
without distinction, but every algebraical symbol does more, it represents
all numbers without distinction. As soon as we conceive a thing divided
into equal parts, without knowing into what number of parts, we may call
it _a_ or _x_, and apply to it, without danger of error, every algebraical
formula in the books. The proposition, 2 (_a_ + _b_)= 2 _a_ + 2 _b_, is a
truth co-extensive with all nature. Since then algebraical truths are true
of all things whatever, and not, like those of geometry, true of lines
only or of angles only, it is no wonder that the symbols should not excite
in our minds ideas of any things in particular. When we demonstrate the
forty-seventh proposition of Euclid, it is not necessary that the words
should raise in us an image of all right-angled triangles, but only of
some one right-angled triangle: so in algebra we need not, under the
symbol _a_, picture to ourselves all things whatever, but only some one
thing; why not, then, the letter itself? The mere written characters, _a_,
_b_, _x_, _y_, _z_, serve as well for representatives of Things in
general, as any more complex and apparently more concrete conception. That
we are conscious of them, however, in their character of things, and not
of mere signs, is evident from the fact that our whole process of
reasoning is carried on by predicating of them the properties of things.
In resolving an algebraic equation, by what rules do we proceed? By