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derived from experience.” (Mansel, Prol. Log. p. 93, ed. 1860.)

204. Abstract terms are strongly distinguished from general terms by possessing only one kind of meaning ; for as they denote qualities there is nothing which they cannot in addition imply. The adjective 'red' is the name of red objects, but it implies the possession by them of the quality redness; but this latter term has one single meaning—the quality alone. Thus it arises that abstract terms are incapable of plurality.” (Jevons, Prin. Science, p. 27, ed. 1877.). “ Numerical Abstraction consists in abstracting the character of the difference from which plurality arises, retaining merely the fact. When I speak of three men I need not at once specify the marks by which each may be known from each. Those marks must exist if they are really three men and not one and the same, and in speaking of them as many I imply the existence of the requisite differences. Abstract number, then, is the empty form of difference ; the abstract number three asserts the existence of marks without specifying their kind. Numerical abstraction is thus seen to be a different process from logical abstraction, for in the latter process we drop out of notice the very existence of difference and plurality.

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.

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 one 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. (Ibid., pp. 158–60.)

"Abstractly considered, Number is the measure of the relation between quantities or things of the same kind. We can form no conception of the absolute magnitude of any quantity, and can only acquire a relative conception of it, by comparing it with some other quantity of the same kind, assumed as a standard of comparison.

The comparison is made by seeking how many times the standard is contained in the quantity measured. The result of this

comparison is number."

(Davies and Peck, Dict. of Math.) Groups of units are what we really treat in arithmetic. The number five is really 1+1+1


+1+1, but for the sake of conciseness we substitute the more compact sign 5, or the name five. These names being arbitrarily imposed in any one manner, an infinite variety of relations spring up between them which are not in the least arbitrary. If we define four as 1+1+1+1, and five as 1+1+1+1+1, then of course it follows that five=four +1; but it would be equally possible to take this latter equality as a definition, in which case one of the former equalities would become an inference. It is hardly requisite to decide how we define the names of numbers, provided we remember that out of the infinitely numerous relations of one number to others, some one relation expressed in an equality must be a definition of the number in question and the other relations immediately become necessary inferences.'

“ În the science of number the variety of classes which can be formed is altogether infinite, and statements of perfect generality may be made subject only to difficulty or exception at the lower end of the scale. Every existing number for instance belongs to the class M+7; that is, every number must be the sum of another number and seven, except of course the first six or seven numbers, negative quantities not being here taken into account. Every number is the half of some other, and so on. The subject of generalization, as exhibited in mathematical truths, is an infinitely wide one. In number we are only at the first step of an extensive series of generalizations. As number is general compared with the particular things numbered, so we have general symbols for numbers, and general symbols for relations between indetermine numbers. There is an unlimited hierarchy of successive generalizations." (Jevons, Pr. Sc., , pp. 167–168.)

A large proportion of the mathematical functions which are conceivable have no application to the circumstances of this world. Physicists certainly do investigate the nature and consequences of forces which nowhere exist. Newton's Principia is full of such investigations. In one chapter of his Mécanique, Céleste Laplace indulges in a remarkable speculation as to what the laws of motion would have been if momenturn, instead of varying simply as the velocity, had been a more complicated function of it.

Thought is not bound down to the limits of what is materially existent, but is circumscribed only by those Fundamental Laws of Identity, Contradiction, and Duality, which have ‘already been laid down.” (Ibid., pp. 70–45.)

205. “Mathematical Judgments may be divided into two kinds—indemonstrable or axiomatic judgments, whose necessity is self-evident ; and demonstrable judgments, whose necessity depends on some previous assumption.

The necessity of the latter is derived from that of the former, so that the indemonstrable judgments alone require a special examination. Under this class are comprehended the axioms of geometry, properly so-called—viz., the original assumptions concerning magnitudes in space as

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