Arithmetic is first a manipulation of symbols called 'figures.' There
are ten of these, and they are capable of many species of combination,
and an indefinite number of individual operations under each species.
Certain rules govern each sort of operation, and when the rules are
properly understood and recollected the operations can be performed
with absolute certainty. Although the figures have names relating to
number, and the problems given for exercise make mention of acres,
pounds, tons, miles, and all sorts of concrete objects, the symbolic
calculations of books have no necessary relation to real things,
numbers, or quantities. They are a purely conventional treatment
of arbitrary marks that may mean anything or nothing. That is the
arithmetic of the 'schools.' There is no trace of reasoning or
argument in it--it is mere rule and recollection.
There is however real Number and there is real Quantity. Number is
that quality in which a group of three things (for instance) is seen
to differ from a group of four or seven, even when the things are
otherwise quite similar. We begin by distinguishing ten primary
degrees of this difference, and then consider other degrees as
multiples or parts of these primary degrees.
Quantity is degree in size, and is a property quite different from
number. But, for convenience, we assume that quantities are all units
or fractions of certain standard quantities, and we are thus enabled
to use the same terms for both number and quantity.
The names which written language provides for the numerical degrees
and their combinations are inconvenient to use, and so a set of
symbols was devised exclusively for numerical designation. These
are the figures of arithmetic. They are the technical vocabulary of
number, and of quantity considered as number.
Number and quantity admit of but two kinds of variation--increase and
diminution. These variations can be denoted so correctly by figures,
that any combination we first make in figures according to rule can be
reproduced in real objects, provided the objects are in other respects
possible. The result of this perfection of technical nomenclature is
that our study of number and quantity has been transferred from real
objects to figures. It has become symbolic and indirect, and most of
us never go beyond the symbols; that is, what we call arithmetic is
an affair of figures, not of true quantities and numbers. We talk
of miles, tons, and pounds sterling, but we do not _think_ of miles,
tons, and pounds sterling--we think of _figures_. A thousand shillings
is to us, when arithmetically stated, '1000_s._,' just as it is here
represented on paper; we do not think of silver coins, and we could
not if we tried imagine a thousand things of any sort. There is in
reality an enormous difference between '0001_s._' and '1000_s._,'
but to the arithmetician the only objective difference is one of
arrangement in figures.
From these considerations it follows that there are two sciences of
number. There is the true science which deals with quantities really
seen in objects and imagined in the mind, and an artificial science
dealing with figures which have only a historical connection with real
quantity. Of the latter, unfortunately, our arithmetical education
chiefly consists. We are never taught to distinguish number and size
in things by the 'eye,' that is, by reason. The symbolism that was
originally intended to assist real arithmetical thought has ended by
supplanting it. An ignorant shepherd, bricklayer, or carpenter, who is
accustomed to make a rapid estimate of the number of things in a
mass, or the area of planking in a log, has a better training in real
arithmetical science than some mathematicians. If we are obliged to
practise genuine arithmetical thought in engineering, astronomy, and
other professions, our scholastic symbolism gets realised to some
extent, and is a great assistance in arithmetical estimation. But
without this it has no more reference to number and quantity than a
musical education, based entirely on the printed or written notation,
has to the appreciation of musical sounds. A book arithmetician is in
the position of a person thoroughly acquainted with theoretical music,
and who can even compose music _according to rule_, but who is unable
to distinguish a high note from a low one or harmony from discord in
actual sound.
It will thus be seen that it is only in the real arithmetic that
reasoning can enter. The judgment in free arithmetical observation is
the counting of actual groups and the measurement of actual surfaces,
and the argument consists in estimating the number of individuals
in other groups, and the size of other surfaces, without counting or
measurement. But this exercise never enters into symbolic arithmetic.
All the apparent conclusions of book arithmetic are tautological; they
consist in repeating in one combination of symbols the whole or
part of what has been already given in another combination. It is an
exercise in expression--nothing more.
Arithmetical ratio has a resemblance to the rational parallel.
3:5::9:15 might be arranged thus--
5 | 15
--+---
3 | 9
This is not argument, for two reasons. (1) The apparent conclusion
is not an effort of rational imagination; it is a figure that can
be obtained with infallible certainty by treating the other figures
according to a rule, which has only to be recollected and applied. (2)
The relation between the left-hand figures and the right-hand figures
is not a categorical judgment; it is a form of resemblance, and so it
cannot yield a valid conclusion.