One of the ablest mathematicians, and the most persevering Hamiltono-mastix of the day, maintains the applicability of the metaphysical notion of infinity to mathematical magnitudes; but with an assumption which unintentionally vindicates Hamilton’s position more fully than could have been done by a professed disciple. “I shall assume,” says Professor De Morgan, in a paper recently printed among the Transactions of the Cambridge Philosophical Society, “the notion of infinity and of its reciprocal infinitesimal: that a line can be conceived infinite, and therefore having points at an infinite distance. Image apart, which we cannot have, it seems to me clear that a line of infinite length, without points at an infinite distance, is a contradiction.” Now it is easy to show, by mere reasoning, without any image, that this assumption is equally a contradiction. For if space is finite, every line in space must be finite also; and if space is infinite, every point in space must have infinite space beyond it in every direction, and therefore cannot be at the greatest possible distance from another point. Or thus: Any two points in space are the extremities of the line connecting them; but an infinite line has no extremities; therefore no two points in space can be connected together by an infinite line.
In fact, it is the “concrete reality,” the “something infinite,” and not the mere abstraction of infinity, which is only conceivable as a negation. Every “something” that has ever been intuitively present to my consciousness is a something finite. When, therefore, I speak of a “something infinite,” I mean a something existing in a different manner from all the “somethings” of which I have had experience in intuition. Thus it is apprehended, not positively, but negatively—not directly by what it is, but indirectly by what it is not. A negative idea is not negative because it is expressed by a negative term, but because it has never been realised in intuition. If infinity, as applied to space, means the same thing as being greater than any finite space, both conceptions are equally positive or equally negative. If it does not mean the same thing, then, in conceiving a space greater than any finite space, we do not conceive an infinite space.
Mr. Mill’s next string of criticisms may be very briefly dismissed. First, Hamilton does not, as Mr. Mill asserts, say that “the Unconditioned is inconceivable, because it includes both the Infinite and the Absolute, and these are contradictory of one another.” His argument is a common disjunctive syllogism. The unconditioned, if conceivable at all, must be conceived either as the absolute or as the infinite; neither of these is possible; therefore the unconditioned is not conceivable at all. Nor, secondly, is Sir W. Hamilton guilty of the “strange confusion of ideas” which Mr. Mill ascribes to him, when he says that the Absolute, as being absolutely One, cannot be known under the conditions of plurality and difference. The absolute, as such, must be out of all relation, and consequently cannot be conceived in the relation of plurality. “The plurality required,” says Mr. Mill, “is not within the thing itself, but is made up between itself and other things.” It is, in fact, both; but even granting Mr. Mill’s assumption, what is a “plurality between a thing and other things” but a relation between them? There is undoubtedly a “strange confusion of ideas” in this paragraph; but the confusion is not on the part of Sir W. Hamilton. “Again,” continues Mr. Mill, “even if we concede that a thing cannot be known at all unless known as plural, does it follow that it cannot be known as plural because it is also One? Since when have the One and the Many been incompatible things, instead of different aspects of the same thing?... If there is any meaning in the words, must not Absolute Unity be Absolute Plurality likewise?” Mr. Mill’s “since when?” may be answered in the words of Plato:—“Οὐδὲν ἔμoιγε ἄτoπoν δoκεῖ εἶναι εἰ ἓν ἅπαντα ἀπoφαίνει τις τῷ μετέχειν τoῦ ἑνὸς καὶ ταὐτὰ ταῦτα πoλλὰ τῷ πλήθoυς αὖ μετέχειν· ἀλλ’ εἰ ὃ ἔστιν ἕν, αὐτὸ τoῦτo πoλλὰ ἀπoδείξει, καὶ αὖ τὰ πoλλὰ δὴ ἕν, τoῦτo ἤδη θαυμάσoμαι.”[AV] Here we are expressly told that “absolute unity” cannot be “absolute plurality.” Mr. Mill may say that Plato is wrong; but he will hardly go so far as to say that there is no meaning in his words. In point of fact, however, it is Mr. Mill who is in error, and not Plato. In different relations, no doubt, the same concrete object may be regarded as one or as many. The same measure is one foot or twelve inches; the same sum is one shilling or twelve pence; but it no more follows that “absolute unity must be absolute plurality likewise,” than it follows from the above instances that one is equal to twelve. And, thirdly, when Mr. Mill accuses Sir W. Hamilton of departing from his own meaning of the term absolute, in maintaining that the Absolute cannot be a Cause, he only shows that he does not himself know what Hamilton’s meaning is. “If Absolute,” he says, “means finished, perfected, completed, may there not be a finished, perfected, and completed Cause?” Hamilton’s Absolute is that which is “out of relation, as finished, perfect, complete;” and a Cause, as such, is both in relation and incomplete. It is in relation to its effect; and it is incomplete without its effect. Finally, when Mr. Mill charges Sir W. Hamilton with maintaining “that extension and figure are of the essence of matter, and perceived as such by intuition,” we must briefly reply that Hamilton does no such thing. He is not speaking of the essence of matter per se, but only of matter as apprehended in relation to us.
Parmenides, p. 129.
Mr. Mill concludes this chapter with an attempt to discover the meaning of Hamilton’s assertion, “to think is to condition.” We have already explained what Hamilton meant by this expression; and we recur to the subject now, only to show the easy manner in which Mr. Mill manages to miss the point of an argument with the clue lying straight before him. “Did any,” he says (of those who say that the Absolute is thinkable), “profess to think it in any other manner than by distinguishing it from other things?” Now this is the very thing which, according to Hamilton, Schelling actually did. Mr. Mill does not attempt to show that Hamilton is wrong in his interpretation of Schelling, nor, if he is right, what were the reasons which led Schelling to so paradoxical a position: he simply assumes that no man could hold Schelling’s view, and there is an end of it.[AW] Hamilton’s purpose is to reassert in substance the doctrine which Kant maintained, and which Schelling denied; and the natural way to ascertain his meaning would be by reference to these two philosophers. But this is not the method of Mr. Mill, here or elsewhere. He generally endeavours to ascertain Hamilton’s meaning by ranging the wide field of possibilities. He tells us what a phrase means in certain authors of whom Hamilton is not thinking, or in reference to certain matters which Hamilton is not discussing; but he hardly ever attempts to trace the history of Hamilton’s own view, or the train of thought by which it suggested itself to his mind. And the result of this is, that Mr. Mill’s interpretations are generally in the potential mood. He wastes a good deal of conjecture in discovering what Hamilton might have meant, when a little attention in the right quarter would have shown what he did mean.
Mr. Mill does not expressly name Schelling in this sentence: but he does so shortly afterwards; and his remark is of the same character with the previous one. “Even Schelling,” he says, “was not so gratuitously absurd as to deny that the Absolute must be known according to the capacities of that which knows it—though he was forced to invent a special capacity for the purpose.” But if this capacity is an “invention” of Schelling’s, and if he was “forced” to invent it, Hamilton’s point is proved. To think, according to all the real operations of thought which consciousness makes known to us, is to condition. And the faculty of the unconditioned is an invention of Schelling’s, not known to consciousness. In other words: all our real faculties bear witness to the truth of Hamilton’s statement; and the only way of controverting it is to invent an imaginary faculty for the purpose.
The third feature of Hamilton’s philosophy which we charged Mr. Mill with misunderstanding, is the distinction between Knowledge and Belief. In the early part of this article, we endeavoured to explain the true nature of this distinction; we have now only a very limited space to notice Mr. Mill’s criticisms on it. Hamilton, he says, admitted “a second source of intellectual conviction called Belief.” Now Belief is not a “source” of any conviction, but the conviction itself. No man would say that he is convinced of the truth of a proposition because he believes it; his belief in its truth is the same thing as his conviction of its truth. Belief, then, is not a source of conviction, but a conviction having sources of its own. The question is, have we legitimate sources of conviction, distinct from those which constitute Knowledge properly so called? Now here it should be remembered that the distinction is not one invented by Hamilton to meet the exigencies of his own system. He enumerates as many as twenty-two authors, of the most various schools of philosophy, who all acknowledged it before him. Such a concurrence is no slight argument in favour of the reality of the distinction. We do not say that these writers, or Hamilton himself, have always expressed this distinction in the best language, or applied it in the best manner; but we say that it is a true distinction, and that it is valid for the principal purpose to which Hamilton applied it.