The origin of the great generality of number is now apparent. Three sounds differ from three colours, or three riders from three horses; but they agree in respect of the variety of marks by which they can be discriminated. The symbols 1 + 1 + 1 are thus the empty marks asserting the existence of discrimination. But in dropping out of sight the character of the differences we give rise to new agreements on which mathematical reasoning is founded. Numerical abstraction is so far from being incompatible with logical abstraction that it is the origin of our widest acts of generalization.
Concrete and Abstract Number.
The common distinction between concrete and abstract number can now be easily stated. In proportion as we specify the logical characters of the things numbered, we render them concrete. In the abstract number three there is no statement of the points in which the three objects agree; but in three coins, three men, or three horses, not only are the objects numbered but their nature is restricted. Concrete number thus implies the same consciousness of difference as abstract number, but it is mingled with a groundwork of similarity expressed in the logical terms. There is identity so far as logical terms enter; difference so far as the terms are merely numerical.
The reason of the important Law of Homogeneity will now be apparent. This law asserts that in every arithmetical calculation the logical nature of the things numbered must remain unaltered. The specified logical agreement of the things must not be affected by the unspecified numerical differences. A calculation would be palpably absurd which, after commencing with length, gave a result in hours. It is equally absurd, in a purely arithmetical point of view, to deduce areas from the calculation of lengths, masses from the combination of volume and density, or momenta from mass and velocity. It must remain for subsequent consideration to decide in what sense we may truly say that two linear feet multiplied by two linear feet give four superficial feet; arithmetically it is absurd, because there is a change of unit.
As a general rule we treat in each calculation only objects of one nature. We do not, and cannot properly add, in the same sum yards of cloth and pounds of sugar. We cannot even conceive the result of adding area to velocity, or length to density, or weight to value. The units added must have a basis of homogeneity, or must be reducible to some common denominator. Nevertheless it is possible, and in fact common, to treat in one complex calculation the most heterogeneous quantities, on the condition that each kind of object is kept distinct, and treated numerically only in conjunction with its own kind. Different units, so far as their logical differences are specified, must never be substituted one for the other. Chemists continually use equations which assert the equivalence of groups of atoms. Ordinary fermentation is represented by the formula
C6 H12 O6 = 2C2 H6 O + 2CO2.
Three kinds of units, the atoms respectively of carbon, hydrogen, and oxygen, are here intermingled, but there is really a separate equation in regard to each kind. Mathematicians also employ compound equations of the same kind; for in, a + b √ - 1 = c + d √ - 1, it is impossible by ordinary addition to add a to b √ -1. Hence we really have the separate equations a = b, and c √ - 1 = d √ - 1. Similarly an equation between two quaternions is equivalent to four equations between ordinary quantities, whence indeed the name quaternion.
Analogy of Logical and Numerical Terms.
If my assertion is correct that number arises out of logical conditions, we ought to find number obeying all the laws of logic. It is almost superfluous to point out that this is the case with the fundamental laws of identity and difference, and it only remains to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out (p. [32]). Thus the Law of Commutativeness, is equally true of quality and quantity. As in logic we have
AB = BA,