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orders, and so the different ways in which the time series is capable of being formed can be actually realized, but the whole series remaining the same, it might have been seen, a priori, that this must be so, as it is not necessary actually to realise the different orders; it is sufficient to see that the arrangement of the perfectly homogeneous units may be contemplated from two different points of view; yet the assertion of the fact that they can be so contemplated is not an analytical judgment. It appears, then, that if it be assumed that the science of number depends altogether on time and its laws, and not at all on the independent laws of space, yet a set of objects in space must exemplify its truths, and can be so arranged as to exhibit the truth of the commutative law in a peculiarly striking manner.
The second question which was proposed seems more difficult of solution. The figure which shows that the square of a line is equal to the sum of the squares of its two parts, and twice the rectangle under them shows equally that the square of the sum of two numbers is equal to the sum of their squares, together with twice their product, and yet the units of the numbers are not exhibited as separate objects, nor does it seem easy to see how the mode in which the figure in space is constituted can be the result of the laws of time. That such, however, is the case can, I think, be made apparent to the student of Kant's philosophy.
Time is the primitive quantum, and all other quanta are such only in so far as they can be generated by a synthesis in time.
In space all objects are quanta, and are cogitable only through the synthesis that is the successive contemplation and conjuction of their parts. Number is the pure schema of quantity, and may be regarded either subjectively or objectively. Regarded subjectively, it is the act of successively intuiting a series of homogeneous units, and of combining them into a whole ; regarded objectively, it is the result of such an act. In either case it includes a relation to time, and cannot therefore be reduced to an image which is purely spatial (Kant Critique of the Pure Reason, translated by J. M. D. Meicklejohn, p. 110). Yet an image (such as an aggregate of dots in space) considered as representing a collection of units can be cogitated only by successively intuiting the units in time, and, being always capable of being so cogitated, it must exhibit the laws of time. As exhibiting the laws of time, and as suggesting and requiring the successive intuition of its units, a collection of dots in space may be termed an image of the number (Critique, p. 109). The synthesis by which such a collection of units is cogitated differs, as Kant (p. 128) observes, from that which generates a continuous quantum in that it is interrupted. Nevertheless, by selecting a definite portion of the continuous quantum as unit, the continuous synthesis may be regarded as broken up into successive repetitions of this unit, and it is only in this way that the magnitude of the quantum can be estimated (Critique, p. 181). Number, considered subjectively, is thus the “representation of the general procedure of the imagination to present its image to the concept" (Critique, p. 109) of quantity, and is, therefore, as has been before stated, the schema of quantity. To the laws of number, then, every quantum must be subject; and if its quantity is discoverable in apprehension (Critique, p. 127), in other words, if it is extensive, numerical truths will be discoverable in it intuitively.
Having already, as I believe, shown that the whole science of number might be deduced from intuitions in time alone, I have now further shown that those facts which might lead us to conclude that it could also be based on the independent laws of space do not warrant such a conclusion.
It is easy to advance another step, and to assert that in every case numerical truths are the result of laws of time. How, indeed, could they be otherwise? A number, as such, cannot be cognized except by counting, and counting is an act which must take place in time, and be subject to its laws.
If it be asserted that on the theory of Sir W. Hamilton we can be conscious of as many as six objects simultaneously, it may be replied that this is undoubtedly true, if the six objects be regarded as producing a single sensation which indicates that there are six (just as a perception of sight may indicate that the visible object is rough to the touch), but that it is not true that the representation of six, as six, can be originally gained by a single glance without counting the units separately. It
may still be said that we must be conscious at least of two as two simultaneously, for that otherwise all comparison would be impossible. Mr. Mill has, I think, disposed of this argument. Comparison is probably nothing more than a rapid change from one state of consciousness to another. When the nature of the change is stated we are said to compare the two representations. For example, when we compare two sounds, and say one is louder than the other, we mean that in passing from one sensation to the other a certain change would take place, and similarly in other cases. Without however entering into the merits of the controversy between Hamilton and Stewart, this at least may be regarded as certain, that in any case where more than one object is present, we can attend to each separately and successively, and that it is because we can do this that we judge that a number of objects is present.
There are laws of space which may be said to be independent of time, such as, space has three dimensions ; two points determine a right line, &c.; though even these contain some reference to time, but it is not on such laws that the numerical properties of space depend.
It remains only to inquire what was the opinion of Kant himself. Mr. Mahaffy, who probably knows more of Kant than any other English author, maintains that Kant bases arithmetic not on time but on space (Fischer on Kant's Critick, translated by J. P. Mahaffy, with notes, &c., ps. xxix, 15, 95).
In his recent work (The Critical Philosophy for English Readers) Mr. Mahaffy adheres to the same opinion, and seems to think that his remarks in the notes to Fischer have put an end to all controversy on the subject. In this opinion I am unable to coincide, and in refuting Mr. Mahaffy, which I hope to be able to do, I shall gladly avail myself of the assistance afforded by his notes to Fischer on Kant.
The quotations which I shall make from Kant himself are taken from Mr. Meiklejohn's translation of the Critique of the Pure Reason.
Mr. Mahaffy's first reference to Kant's theory of the nature of Arithmetical truths is in p. xxix of the Introduction to Fischer's Commentary, from which I quote the following :
“In basing arithmetic on synthetical axioms, Kant seems not to have considered these axioms to extend to any numbers beyond the range of ordinary intuition. If, as Sir William Hamilton thinks, we can intuite six objects simultaneously, then the original axioms will be limited to the addition and subtraction of units within this number. But within the sum, whatever it may be, which can be intuited at once, the adding and subtracting of numbers is a process directly intuitive, and we should be careful how we speak of the act of adding or the result produced, as if there were any mediate inference or manipulation of the units during which they did not each and all remain, actually before us.
When we come to higher numbers, the association school seems to think our principle is at fault, for that we add and subtract large numbers with equal certainty is obvious, and surely we can never have any evidence on the subject from direct intuition. Mr. Mansel, who bases arithmetic on time, says that we must have been conscious of even these large numbers at some time or other in some succession of thoughts, and that this is sufficient. Sufficient it certainly would be, but its truth is very doubtful. Kant appears more correct in deducing arithmetic from space, and on this view we may hold that our knowledge of all the higher numbers and the processes we perform with them, are mere cogitationes caecae sive symbolicae."
The theory here put forward by Mr. Mahaffy as that of Kant seems to be that, by looking at a collection of objects, we gain instantaneously a representation of the corresponding number, and that when in this way a knowledge of two numbers has been obtained, by putting the two collections together, and looking at the result, we instantaneously arrive at a knowledge of the number which is the sum of the two former, but that this mode of procedure is applied only to numbers whose sum is not greater than six, and that beyond this point our knowledge with respect to numbers is merely a cogitatio caeca sive symbolica. I am not at present concerned with the absolute truth or falsehood of this theory, but merely with the question as to whether it was held by Kant.
It is an unfortunate circumstance for Mr. Mahaffy, that the example selected by Kant in the first passage, where he speaks of the mode of arriving at arithmetical truths, introduces so large a number as twelve, which, on Mr. Mahaffy's principles, is beyond the reach of anything but a cogitatio caeca sive symbolica. However, as the limit of direct intuition is taken from Hamilton, not Kant, we can not lay very great stress on this point.
Let us now examine the mode in which Kant describes the origin of the judgment 7 + 5 = 12 (Critique, p. 10). “I first take the number 7,” says Kant, “and, for the conception of 5, calling in the aid of the fingers of my hand as objects of intuition, I add the units, which I before took together to make up the number 5, gradually, now, by means of the material image, my hand, to the number 7, and by this process I at length see the number 12 arise. Arithmetical propositions are, therefore, always synthetical, of which we may become more clearly convinced by trying large numbers."