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.”[BB]
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;[BC] 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.