Part 6

"Something infinite," says Mr. Mill, "is a conception which, like most of our complex ideas, contains a negative element, but which contains positive elements also. Infinite space, for instance; is there nothing positive in that? The negative part of this conception is the absence of bounds. The positive are, the idea of space, and of space greater than any finite space."--(P. 45.)

This definition of infinite space is exactly that which Descartes gives us of indefinite extension,--"Ita quia non possumus imaginari extensionem tam magnam, quin intelligamus adhuc majorem esse posse, dicemus magnitudinem rerum possibilium esse indefinitam." So too, Cudworth,--"There appeareth no sufficient ground for this positive infinity of space; we being certain of no more than this, that be the world or any figurative body never so great, it is not impossible but that it might be still greater and greater without end. Which indefinite increasableness of body and space seems to be mistaken for a positive infinity thereof." And Locke, a philosopher for whom Mr. Mill will probably have more respect than for Descartes or Cudworth, writes more plainly: "To have actually in the mind the idea of a space infinite, is to suppose the mind already passed over, and actually to have a view of all those repeated ideas of space, which an endless repetition can never totally represent to it,--which carries in it a plain contradiction." Mr. Mill thus unwittingly illustrates, in his own person, the truth of Hamilton's remark, "If we dream of effecting this [conceiving the infinite in time or space], we only deceive ourselves by substituting the indefinite for the infinite, than which no two notions can be more opposed." In fact, Mr. Mill does not seem to be aware that what the mathematician calls infinite, the metaphysician calls indefinite, and that arguments drawn from the mathematical use of the term infinite are wholly irrelevant to the metaphysical. How, indeed, could it be otherwise? Can any man suppose that, when the Divine attributes are spoken of as infinite, it is meant that they are indefinitely increasable?

Principia, i., 26.

Intellectual System, ed. Harrison, vol. iii., p. 131.

Essay, ii., 17, 7.

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:--"[Greek: Ouden emoige atopon dokei heinai ei hen hapanta apophainei tis tô metechein· tou henos kai tauta tauta polla tô plêthous au metechein; all' ei ho estin hen, auto touto polla apodeixei, kai au ta polla dê hen, touto êdê thaumasomai.]" 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. 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.

We do not agree with all the details of Hamilton's application. We do not agree with him, though he is supported by very eminent authorities, in classifying our conviction of axiomatic principles as belief, and not as knowledge. But this question does not directly bear on Mr. Mill's criticism. The point of that criticism is, that Hamilton, by admitting a belief in the infinite and unrelated, nullifies his own doctrine, that all knowledge is of the finite and relative. Let us see.

Hamilton's distinction is in principle the same as that which we have given in our previous remarks (pp. 18, 19). He says, "A conviction is incomprehensible when there is merely given to us in consciousness--That its object is ([Greek: hoti esti]), and when we are unable to comprehend through a higher notion or belief Why or How it is ([Greek: dioti esti])."--(Reid's Works, p. 754.) We would distinguish between why and how, between [Greek: dioti], and [Greek: pôs]. We can give no reason why two straight lines cannot enclose a space; but we can comprehend how they cannot. We have only to form the corresponding image, to see the manner in which the two attributes coexist in one object. But when I say that I believe in the existence of a spiritual being who sees without eyes, I cannot conceive the manner in which seeing coexists with the absence of the bodily organ of sight. We believe that the true distinction between knowledge and belief may ultimately be referred to the presence or absence of the corresponding intuition; but to show this in the various instances would require a longer dissertation than our present limits will allow.

We may believe that a thing is, without being able to conceive how it is. I believe that God is a person, and also that He is infinite; though I cannot conceive how the attributes of personality and infinity exist together. All my knowledge of personality is derived from my consciousness of my own finite personality. I therefore believe in the coexistence of attributes in God, in some manner different from that in which they coexist in me as limiting each other: and thus I believe in the fact, though I am unable to conceive the manner. So, again, Kant brings certain counter arguments, to prove, on the one side, that the world has a beginning in time, and, on the other side, that it has not a beginning. Now suppose I am unable to refute either of these courses of argument, am I therefore compelled to have no belief at all? May I not say, I believe, in spite of Kant, that the world has a beginning in time, though I am unable to conceive how it can have so begun? What is this, again, but a belief in an absolute reality beyond the sphere of my relative knowledge?

"I am not now considering," says Mr. Mill, "what it is that, in our author's opinion, we are bound to believe concerning the unknowable." Why, this was the very thing he ought to have considered, before pronouncing the position to be untenable, or to be irreconcilable with something else. Meanwhile, it is instructive to observe that Mr. Mill himself believes, or requires his readers to believe, something concerning the unknown. He does not know, or at any rate he does not tell his readers, what Hamilton requires them to believe concerning the unknowable; but he himself believes, and requires them to believe, that this unknown something is incompatible with the doctrine that knowledge is relative. We cannot regard this as a very satisfactory mode of refuting Hamilton's thesis.

In a subsequent chapter (p. 120), Mr. Mill endeavours to overthrow this distinction between Knowledge and Belief, by means of Hamilton's own theory of Consciousness. Hamilton maintains that we cannot be conscious of a mental operation without being conscious of its object. On this Mr. Mill retorts that if, as Hamilton admits, we are conscious of a belief in the Infinite and the Absolute, we must be conscious of the Infinite and the Absolute themselves; and such consciousness is Knowledge. The fallacy of this retort is transparent. The immediate object of Belief is a proposition which I hold to be true, not a thing apprehended in an act of conception. I believe in an infinite God; i.e., I believe that God is infinite: I believe that the attributes which I ascribe to God exist in Him in an infinite degree. Now, to believe this proposition, I must, of course, be conscious of its meaning; but I am not therefore conscious of the Infinite God as an object of conception; for this would require further an apprehension of the manner in which these infinite attributes coexist so as to form one object. The whole argument of this eighth chapter is confused, owing to Mr. Mill not having distinguished between those passages in which Sir W. Hamilton is merely using an argumentum ad hominem in relation to Reid, and those in which he is reasoning from general principles.

But if Mr. Mill is unjust towards the distinction between Knowledge and Belief, as held by Sir W. Hamilton, he makes ample amends to the injured theory in the next chapter, by enlarging the province of credibility far beyond any extent which Hamilton would have dreamed of claiming for it. Conceivability or inconceivability, he tells us, are usually dependent on association; and it is quite possible that, under other associations, we might be able to conceive, and therefore to believe, anything short of the direct contradiction that the same thing is and is not. It is not in itself incredible, that a square may at the same time be round, that two straight lines may enclose a space, or even that two and two may make five. But whatever concessions Mr. Mill may make on this point, he is at least fully determined that Sir W. Hamilton shall derive no benefit from them; for he forthwith proceeds to charge Sir William with confusing three distinct senses of the term conception--a confusion which exists solely in his own imagination,--and to assert that the Philosophy of the Conditioned is entirely founded on a mistake, inasmuch as infinite space on the one hand, and, on the other, both an absolute minimum and an infinite divisibility of space, are perfectly conceivable. With regard to the former of these two assertions, Mr. Mill's whole argument is vitiated, as we have already shown, by his confusion between infinite and indefinite; but it is worth while to quote one of his special instances in this chapter, as a specimen of the kind of reasoning which an eminent writer on logic can sometimes employ. In reference to Sir W. Hamilton's assertion, that infinite space would require infinite time to conceive it, he says, "Let us try the doctrine upon a complex whole, short of infinite, such as the number 695,788. Sir W. Hamilton would not, I suppose, have maintained that this number is inconceivable. How long did he think it would take to go over every separate unit of this whole, so as to obtain a perfect knowledge of the exact sum, as different from all other sums, either greater or less?"

In reference to this last paradox, Mr. Mill quotes from Essays by a Barrister: "There is a world in which, whenever two pairs of things are either placed in proximity or are contemplated together, a fifth thing is immediately created and brought within the contemplation of the mind engaged in putting two and two together.... In such a world surely two and two would make five. That is, the result to the mind of contemplating two twos would be to count five." The answer to this reasoning has been already given by Archdeacon Lee in his Essay on Miracles. The "five" in this case is not the sum of two and two, but of two and two plus the new creature, i.e., of two and two plus one.

The sense in which Sir W. Hamilton himself uses the word conception is explained in a note to Reid's Works, p. 377--namely, the combination of two or more attributes in a unity of representation. The second sense which Mr. Mill imagines is simply a mistake of his own. When Hamilton speaks of being "unable to conceive as possible," he does not mean, as Mr. Mill supposes, physically possible under the law of gravitation or some other law of matter, but mentally possible as a representation or image; and thus the supposed second sense is identical with the first. The third sense may also be reduced to the first; for to conceive two attributes as combined in one representation is to form a notion subordinate to those of each attribute separately. We do not say that Sir W. Hamilton has been uniformly accurate in his application of the test of conceivability; but we say that his inaccuracies, such as they are, do not affect the theory of the conditioned, and that in all the long extracts which Mr. Mill quotes, with footnotes, indicating "first sense," "second sense," "third sense," the author's meaning may be more accurately explained in the first sense only.

It is marvellous that it should not have occurred to Mr. Mill, while he was writing this passage, "How comes this large number to be a 'whole' at all; and how comes it that 'this whole,' with all its units, can be written down by means of six digits?" Simply because of a conventional arrangement, by which a single digit, according to its position, can express, by one mark, tens, hundreds, thousands, &c., of units; and thus can exhaust the sum by dealing with its items in large masses. But how can such a process exhaust the infinite? We should like to know how long Mr. Mill thinks it would take to work out the following problem:--"If two figures can represent ten, three a hundred, four a thousand, five ten thousand, &c., find the number of figures required to represent infinity."

Precisely the same misconception of Hamilton's position occurs in Professor De Morgan's paper in the Cambridge Transactions, to which we have previously referred. He speaks (p. 13) of the "notion, which runs through many writers, from Descartes to Hamilton, that the mind must be big enough to hold all it can conceive." This notion is certainly not maintained by Hamilton, nor yet by Descartes in the paragraph quoted by Mr. De Morgan; nor, as far as we are aware, in any other part of his works.

Infinite divisibility stands or falls with infinite extension. In both cases Mr. Mill confounds infinity with indefiniteness. But with regard to an absolute minimum of space, Mr. Mill's argument requires a separate notice.

"It is not denied," he says, "that there is a portion of extension which to the naked eye appears an indivisible point; it has been called by philosophers the minimum visibile. This minimum we can indefinitely magnify by means of optical instruments, making visible the still smaller parts which compose it. In each successive experiment there is still a minimum visibile, anything less than which cannot be discovered with that instrument, but can with one of a higher power. Suppose, now, that as we increase the magnifying power of our instruments, and before we have reached the limit of possible increase, we arrive at a stage at which that which seemed the smallest visible space under a given microscope, does not appear larger under one which, by its mechanical construction, is adapted to magnify more, but still remains apparently indivisible. I say, that if this happened, we should believe in a minimum of extension; or if some à priori metaphysical prejudice prevented us from believing it, we should at least be enabled to conceive it."--(P. 84.)

The natural conclusion of most men under such circumstances would be, that there was some fault in the microscope. But even if this conclusion were rejected, we presume Mr. Mill would allow that, under the supposed circumstances, the exact magnitude of the minimum of extension would be calculable. We have only to measure the minimum visibile, and know what is the magnifying power of our microscope, to determine the exact dimensions. Suppose, then, that we assign to it some definite magnitude--say the ten billionth part of an inch,--should we then conclude that it is impossible to conceive the twenty billionth part of an inch?--in other words, that we have arrived at a definite magnitude which has no conceivable half? Surely this is a somewhat rash concession to be made by a writer who has just told us that numbers may be conceived up to infinity; and therefore, of course, down to infinitesimality.

Mr. Mill concludes this chapter with an assertion which, even by itself, is sufficient to show how very little he has attended to or understood the philosophy which he is attempting to criticise. "The law of Excluded Middle," he says, "as well as that of Contradiction, is common to all phenomena. But it is a doctrine of our author that these laws are true, and cannot but be known to be true, of Noumena likewise. It is not merely Space as cognisable by our senses, but Space as it is in itself, which he affirms must be either of unlimited or of limited extent" (p. 86). At this sentence we fairly stand aghast. "Space as it is in itself!" the Noumenon Space! Has Mr. Mill been all this while "examining" Sir William Hamilton's philosophy, in utter ignorance that the object of that philosophy is the "Conditioned in Time and Space;" that he accepts Kant's analysis of time and space as formal necessities of thought, but pronounces no opinion whatever as to whether time and space can exist as Noumena or not? It is the phenomenal space, "space as cognisable by our senses," which Sir W. Hamilton says must be either limited or unlimited: concerning the Noumenon Space, he does not hazard an opinion whether such a thing exists or not. He says, indeed (and this is probably what has misled Mr. Mill), that the laws of Identity, Contradiction, and Excluded Middle, are laws of things as well as laws of thought; but he says nothing about these laws as predicating infinite or finite extension. On the contrary, he expressly classifies Space under the law of Relativity, the violation of which indicates what may exist, but what we are unable to conceive as existing. Briefly, the law of Excluded Middle (to take this instance alone) is a law of things only in its abstract form, "Everything must be A or not A" (extended, if you please, or not extended); but in its subordinate form, "Everything extended must be extended infinitely or finitely," it is only applicable, and only intended by Hamilton to be applied, to those phenomena which are already given as extended in some degree.

Discussions, p. 603.

We have now examined the first six chapters of Mr. Mill's book, containing his remarks on that portion of Sir W. Hamilton's philosophy which he justly regards as comprising the most important of the doctrines which specially belong to Hamilton himself. The next chapter is an episode, in which Mr. Mill turns aside from Sir W. Hamilton to criticise Mr. Mansel's Bampton Lectures. As our limits do not permit us to carry on the argument at present through the remainder of Mr. Mill's remarks on Hamilton himself, we shall conclude our notice with a few words on this chapter, as closing the properly metaphysical portion of Mr. Mill's book, and as affording ample proof that, in this department of philosophy at least, Mr. Mill's powers of misapprehension do not cease when Sir W. Hamilton is no longer their object.