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Schütz says, “Necessario scribendum vel quam olim domo sua, vel quam consul domo sua.” There is plainly an allusion to the consulship of Bibulus and Cæsar, when the former did not leave his house for eight months. But instead of olim or consul I suggest DOMI, a word which would have been very likely to fall out before domo. Domi means at Rome, as opposed to the present militia of Bibulus. So Ter. Ad. 3. 4. 49, una semper militiae et domi fuimus ; so also Liv. XXVIII. 12. in hostium terra per annos tredecim tam procul ab domo, and, in the same chapter, nec ab domo quidquam mittebatur—that is, 'no supplies were sent to them from home' (from Carthage).
ON THE FOUNDATION OF THE SCIENCE OF NUMBER,
ACCORDING TO THE PHILOSOPHY OF KANT. By FRANCIS A. TARLETON, LL. D., Fellow of Trinity College, Dublin.
THERE is no part of the Philosophy of Kant more deserving of unqualified admiration than the account which that philosopher gives of the nature and foundation of the science of Mathematics. The existence of that science had, up to the time of Kant, been an insoluble puzzle. Philosophers had to account, on the one hand, for the number and variety of the truths which it embraced, its great value, and extensive application to objects of experience, and on the other, for its necessity, its dependence on principles immediately evident to the mind of every man, and the theoretical possibility of its being completely evolved from the cogitations of a solitary individual without any external aid whatever. The majority of philosophers appear to have been so much influenced by the characteristics of the latter class as to ignore those of the former. Archbishop Whately, in his Treatise on Logic, went so far as to maintain that Geometrical truths are deduced from arbitrary hypotheses by processes of pure reasoning. That such a theory could have been propounded by a writer of such high character, half a century after the publication of the “Critique of the Pure Reason,” may well excite our astonishment.
At present, no one having any pretensions to a knowledge of mental science would assent to Whately's theory. The followers of Kant and his opponents—those who admit, and those who deny a priori elements in human knowledge, now agree in admitting that the axioms on which Geometry is based are essentially synthetical, and must be arrived at by intuitions in space.
It has not, however, been pointed out with sufficient clearness by any writer with whom I am acquainted, except Kant, that after the definitions have been laid down, and the axioms assented to, no progress could be made by means of a train of reasoning alone. Pure reasoning is not, indeed, more intimately connected with Geometry than with any other science. The essential element in a Geometrical demonstration, as Kant has pointed out with great clearness, is the construction. By means of the construction, a number of properties of space are perceived by immediate intuition, and these are connected together by a train of reasoning so as to lead to the property required. There are, I believe, only two propositions in the first book of Euclid which are deduced from the preceding by reasoning alone. In general, the process by which Geometrical truths are arrived at is no more a train of
pure reasoning, than that by which a grocer arrives at the conclusion that the parcel of tea he is selling weighs a pound.
To consider why, and in what sense, Geometry is peculiarly a demonstrative science would lead me too far from my present subject. The question is discussed in the most complete manner by Kant.
Geometry, then, it is allowed by all, rests on intuitions in space; and the only question now at issue between different schools of philosophy is, whether these intuitions are a priori or not.
With respect to the other department of Mathematics, no such uniformity of opinion exists. Even those philosophers who profess to follow Kant are at variance, some holding, with Mansel and Kuno Fischer, that arithmetical truths are dependent on the laws of Time; others maintaining, with Mr. Mahaffy and Mr. Monck, that they must be based on intuitions in Space. It becomes, then, an interesting question to examine
which of these theories is correct, and to inquire what was the opinion of Kant himself on this matter.
The science of number contains two kinds of truths-those relating to the composition of particular numbers, such as 5 + 2
7 ; 3 X 4 12, and those which are true of any numbers whatever, such as ab= ba; (a + b)2 = a + 2ab + b2.
Algebra is principally concerned with truths of the latter class; while Arithmetic, as distinguished from Algebra, is chiefly occupied with those of the former. It must be remembered, however, that without making use of some general principles, our knowledge with respect to particular numbers would be extremely limited. general principles were made use of, the sum of two large numbers would be practically unattainable, as it would be necessary to add each unit separately. Thus, to find the sum of 4678 and 3784 would require at least two hours, and the calculator would be almost certain to make mistakes.
Mansel appears to have overlooked the important truth stated above, and, consequently, fails altogether in accounting for the formation of judgments with respect to large numbers. He also, as it seems to me, is in error in stating that pure Arithmetic contains no demonstration.' His general theory that Arithmetic is dependent on intuitions in time, I believe to be correct, and I shall now endeavour to show that intuitions in space are not required to enable us to arrive at the truths belonging to either of the classes which I have mentioned, but that the whole science of number may be based on intuitions in time alone.
A number is a collection of homogeneous units.
Any series of distinct mental states capable of being separate objects of attention (for example, the sensations caused by a series of similar sounds) will supply the units; and when, by an act of the Understanding, an unity is attributed to the series, a notion of the corresponding
Mansel's " Prolegomena Logica,” p. 115.
number is obtained. In this manner, by means of a series of similar sounds, the notions of 1, 2, 3, 4, 5, &c., would be reached.
By the application of language, an immense assistance might be given to the memory, and the readiest way of producing the series would be by the use of the calculator's voice. Thus, by saying one, two, three, four, five, six, seven, eight, nine, ten, a notion of the number ten would be obtained just as well as by placing ten visible objects successively next each other. As a matter of fact, those who are not very practised arithmeticians obtain the sum of two numbers by successive acts of addition, each of which indicates a new homogeneous unit; thus, an inexperienced arithmetician, when adding 7 and 5, would say seven and one are eight, seven and two are nine, &c.; arriving finally at seven and five are twelve. No appeal to visible objects would be necessary, and the certainty produced would be just as great as if seven balls were placed beside five balls, and the whole collection then counted.
Having obtained the notions of several numbers, the next step would be to perceive that the addition of two smaller numbers would produce a larger number which could be reached otherwise by a series of homogeneous units. Thus, it could be ascertained that 2 + 1 = 3 ; 2 + 2 = 4, &c. Conversely a larger number could be analysed into a collection of smaller ones, and the general principle obtained that every number might be considered as the sum of a series of smaller numbers.
I have here made use of the word analysed, but I do not mean that such a judgment as 10 = 7 + 3 is analytical in the Kantian sense. It results from the analysis of an intuition, but could never be reached by the analysis of a mere notion. When 7 is removed from the series of 10 homogeneous units, it is necessary to count to ascertain how many remain; and this would be so if 10, 7, and 3 were all given beforehand. The remark just made supplies, as it appears to me, a defect in Mansel's argu