OF THE AXIOMS WHICH RELATE TO NUMBER.
1. THE foundations of our speculative knowledge of the
relations and properties of Number, as well as of Space, are
contained in the mode in which we represent to ourselves the
magnitudes which are the subjects of our reasonings. To
express these foundations in axioms in the case of number,
is a matter requiring some consideration, for the same
reason as in the case of geometry; that is, because these
axioms are principles which we assume as true, without being
aware that we have made any assumption; and we cannot,
without careful scrutiny, determine when we have stated, in
the form of axioms, all that is necessary for the formation
of the science, and no more than is necessary. We will,
however, attempt to detect the principles which really must
form the basis of theoretical arithmetic.
2. Why is it that three and two are equal to four and one?
Because if we look at five things of any kind, we see that
it is so. The five are four and one; they are also three and
two. The truth of our assertion is involved in our being
able to conceive the number five at all. We perceive this
truth by _intuition_, for we cannot see, or imagine we see,
five things, without perceiving also that the assertion
above stated is true.
But how do we state in words this fundamental principle of
the doctrine of numbers? Let us consider a very simple case.
If we wish to show that seven and two are equal to four and
five, we say that seven are four and three, _therefore_
seven and two are four and three and two; and because three
and two are {139} five, this is four and five. Mathematical
reasoners justify the first inference (marked by the
conjunctive word _therefore_), by saying that "When equals
are added to equals the wholes are equal," and that thus,
since seven is equal to three and four, if we add two to
both, seven and two are equal to four and three and two.
3. Such _axioms_ as this, that when equals are added to
equals the wholes are equal, are, in fact, expressions of
the general condition of intuition, by which a whole is
contemplated as made up of parts, and as identical with the
aggregate of the parts. And a yet more general form in which
we might more adequately express this condition of intuition
would be this; that 'Two magnitudes are equal when they can
be divided into parts which are equal, each to each.' Thus
in the above example, seven and two are equal to four and
five, because each of the two sums can be divided into the
parts, four, three, and two.
4. In all these cases, a person who had never seen such
axioms enunciated in a verbal form would employ the same
reasoning as a practised mathematician, in order to satisfy
himself that the proposition was true. The steps of the
reasoning, being seen to be true by intuition, would carry
an entire conviction, whether or not the argument were made
verbally complete. Hence the axioms may appear superfluous,
and on this account such axioms have often been spoken
contemptuously of, as empty and barren assertions. In fact,
however, although they cannot supply the deficiency of the
clear intuition of number and space in the reasoner himself,
and although when he possesses such a faculty, he will
reason rightly if he have never heard of such axioms, they
still have their place properly at the beginning of our
treatises on the science of quantity; since they express, as
simply as words can express, those conditions of the
intuition of magnitudes on which all reasoning concerning
quantity must be based; and are necessary when we want, not
only to see the truth of the elementary reasonings on these
subjects, but to put such reasonings in a formal and logical
shape. {140}
5. We have considered the above-mentioned axioms as the
basis of all arithmetical operations of the nature of
_addition_. But it is easily seen that the same principle
may be carried into other cases; as for instance,
_multiplication_, which is merely a repeated addition, and
admits of the same kind of evidence. Thus five times three
are equal to three times five; why is this? If we arrange
fifteen things in five rows of three, it is seen by looking,
or by imaginary looking, which is _intuition_, that they may
also be taken as three rows of five. And thus the principle
that those wholes are equal which can be resolved into the
same partial magnitudes, is immediately applicable in this
as in the other case.
6. We may proceed to higher numbers, and may find ourselves
obliged to use artificial nomenclature and notation in order
to represent and reckon them; but the reasoning in these
cases also is still the same. And the usual artifice by
which our reasoning in such instances is assisted is, that
the number which is the root of our scale of notation (which
is _ten_ in our usual system), is alternately separated into
parts and treated as a single thing. Thus 47 and 35 are 82;
for 47 is four tens and seven; 35 is three tens and five;
whence 47 and 35 are seven tens and twelve; that is, 7 tens,
1 ten, and 2; which is 8 tens and 2, or 82. The like
reasoning is applicable in other cases. And since the most
remote and complex properties of numbers are obtained by a
prolongation of a course of reasoning exactly similar to
that by which we thus establish the most elementary
propositions, we have, in the principles just noticed, the
foundation of the whole of Theoretical Arithmetic.
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