of the language. For numbers, though they must be numbers of something,
may be numbers of anything; and therefore, as we need not, when using an
algebraical symbol (which represents all numbers without distinction),
or an arithmetical number, picture to ourselves all that it stands for,
we may picture to ourselves (and this not as a sign of things, but as
being itself a thing) the number or symbol itself as conveniently as any
other single thing. That we are conscious of the numbers or symbols, in
their character of things, and not of mere signs, is shown by the fact
that our whole process of reasoning is carried on by predicating of them
the properties of things.

Another reason why the propositions in arithmetic and algebra have been
thought merely verbal, is that they seem to be _identical_ propositions.
But in 'Two pebbles and one pebble are equal to three pebbles,' equality
but not identity is affirmed; the subject and predicate, though names of
the same objects, being names of them in different states, that is, as
producing different impressions on the senses. It is on such inductive
truths, resting on the evidence of sense, that the Science of Number is
based; and it is, therefore, like the other deductive sciences, an
inductive science. It is also, like them, hypothetical. Its inductions
are the definitions (which, as in geometry, assert a fact as well as
explain a name) of the numbers, and two axioms, viz. The sums of equals
are equal; the differences of equals are equal. These axioms, and
so-called definitions are themselves exactly, and not merely
hypothetically, true. Yet the conclusions are true only on the
assumption that, 1 = 1, i.e. that all the numbers are numbers of the
same or equal units. Otherwise, the certainty in arithmetical processes,
as in those of geometry or mechanics, is not _mathematical_, i.e.
unconditional certainty, but only certainty of inference. It is the
enquiry (which can be gone through once for all) into the inferences
which can be drawn from assumptions, which properly constitutes all
demonstrative science.

New conclusions may be got as well from fictitious as from real
inductions; and this is even consciously done, viz. in the _reductio ad
absurdum_, in order to show the falsity of an assumption. It has even
been argued that all ratiocination rests, in the last resort, on this
process. But as this is itself syllogistic, it is useless, as a proof of
a syllogism, against a man w