Quantity is that attribute of things in virtue of which they are susceptible of exact mensuration. The question how much, or how many (quantus), implies the answer, so much, or so many (tantus); but the answer is possible only through reference to some standard of magnitude or multitude arbitrarily assumed. Every object, therefore, of which quantity, in the mathematical sense, is predicable, must be by its essential nature mensurable. Now mensurability implies the existence of actual, definite limits, since without them there could be no fixed relation between the given object and the standard of measurement, and, consequently, no possibility of exact mensuration. In fact, since quantification is the object of all mathematical operations, mathematics may be not inaptly defined as the science of the determinations of limits. It is evident, therefore, that the terms quantity and finitude express the same attribute, namely, limitation--the former relatively, the latter absolutely; for quantity is limitation considered with relation to some standard of measurement, and finitude is limitation considered simply in itself. The sphere of quantity, therefore, is absolutely identical with the sphere of the finite; and the phrase infinite quantity, if strictly construed, is a contradiction in terms.
The result thus attained by considering abstract quantity is corroborated by considering concrete and discrete quantities. Such expressions as infinite sphere, radius, parallelogram, line, and so forth, are self-contradictory. A sphere is limited by its own periphery, and a radius by the centre and circumference of its circle. A parallelogram of infinite altitude is impossible, because the limit of its altitude is assigned in the side which must be parallel to its base in order to constitute it a parallelogram. In brief, all figuration is limitation. The contradiction in the term infinite line is not quite so obvious, but can readily be made apparent. Objectively, a line is only the termination of a surface, and a surface the termination of a solid; hence a line can not exist apart from an extended quantity, nor an infinite line apart from an infinite quantity. But as this term has just been shown to be self-contradictory, an infinite line can not exist objectively at all. Again, every line is extension in one dimension; hence a mathematical quantity, hence mensurable, hence finite; you must therefore, deny that a line is a quantity, or else affirm that it is finite.
The same conclusion is forced upon us, if from geometry we turn to arithmetic. The phrases infinite number, infinite series, infinite process, and so forth, are all contradictory when literally construed. Number is a relation among separate unities or integers, which, considered objectively as independent of our cognitive powers, must constitute an exact sum; and this exactitude, or synthetic totality, is limitation. If considered subjectively in the mode of its cognition, a number is infinite only in the sense that it is beyond the power of our imagination or conception, which is an abuse of the term. In either case the totality is fixed; that is, finite. So, too, of series and process. Since every series involves a succession of terms or numbers, and every process a succession of steps or stages, the notion of series and process plainly involves that of number, and must be rigorously dissociated from the idea of infinity. At any one step, at any one term, the number attained is determinate, hence finite. The fact that, by the law of the series or of the process, we may continue the operation as long as we please, does not justify the application of the term infinite to the operation itself; if any thing is infinite, it is the will which continues the operation, which is absurd if said of human wills.
Consequently, the attribute of infinity is not predicable either of 'diminution without limit,' 'augmentation without limit,' or 'endless approximation to a fixed limit,' for these mathematical processes continue only as we continue them, consist of steps successively accomplished, and are limited by the very fact of this serial incompletion.
"We can not forbear pointing out an important application of these results to the Critical Philosophy. Kant bases each of his famous four antinomies on the demand of pure reason for unconditioned totality in a regressive series of conditions. This, he says, must be realized either in an absolute first of the series, conditioning all the other members, but itself unconditioned, or else in the absolute infinity of the series without a first; but reason is utterly unable, on account of mutual contradiction, to decide in which of the two alternatives the unconditioned is found. By the principles we have laid down, however, the problem is solved. The absolute infinity of a series is a contradiction in adjecto. As every number, although immeasurably and inconceivably great, is impossible unless unity is given as its basis, so every series, being itself a number, is impossible unless a first term is given as a commencement. Through a first term alone is the unconditioned possible; that is, if it does not exist in a first term, it can not exist at all; of the two alternatives, therefore, one altogether disappears, and reason is freed from the dilemma of a compulsory yet impossible decision. Even if it should be allowed that the series has no first term, but has originated ab œterno, it must always at each instant have a last term; the series, as a whole, can not be infinite, and hence can not, as Kant claims it can, realize in its wholeness unconditioned totality. Since countless terms forever remain unreached, the series is forever limited by them. Kant himself admits that it can never be completed, and is only potentially infinite; actually, therefore, by his own admission, it is finite. But a last term implies a first, as absolutely as one end of a string implies the other; the only possibility of an unconditioned lies in Kant's first alternative, and if, as he maintains Reason must demand it, she can not hesitate in her decisions. That number is a limitation is no new truth, and that every series involves number is self-evident; and it is surprising that so radical a criticism on Kant's system should never have suggested itself to his opponents. Even the so-called moments of time can not be regarded as constituting a real series, for a series can not be real except through its divisibility into members whereas time is indivisible, and its partition into moments is a conventional fiction. Exterior limitability and interior divisibility result equally from the possibility of discontinuity. Exterior illimitability and interior indivisibility are simple phases of the same attribute of necessary continuity contemplated under different aspects. From this principle flows another upon which it is impossible to lay too much stress, namely; illimitability and indivisibility, infinity and unity, reciprocally necessitate each other. Hence the Quantitative Infinites must be also Units, and the division of space and time, implying absolute contradiction, is not even cogitable as an hypothesis. [220]
"The word infinite, therefore, in mathematical usage, as applied to process and to quantity, has a two-fold signification. An infinite process is one which we can continue as long as we please, but which exists solely in our continuance of it. [221] An infinite quantity is one which exceeds our powers of mensuration or of conception, but which, nevertheless, has bounds and limits in itself. [222] Hence the possibility of relation among infinite quantities, and of different orders of infinities. If the words infinite, infinity, infinitesimal, should be banished from mathematical treatises and replaced by the words indefinite, indefinity, and indefinitesimal, mathematics would suffer no loss, while, by removing a perpetual source of confusion, metaphysics would get great gain."
[Footnote 220: ][ (return) ] By the application of these principles the writer in the "North American Review" completely dissolves the antinomies by which Hamilton seeks to sustain his "Philosophy of the Conditioned." See "North American Review," 1864, pp. 432-437.
[Footnote 221: ][ (return) ] De Morgan, "Diff. and Integ. Calc." p. 9.
[Footnote 222: ][ (return) ] Id., ib., p. 25.