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such, and the propositions belonging to the fundamental operations of arithmetic-addition and subtraction. (Distinguish between postulates, and some axioms which are logical, not geometrical principles, and depend solely on the laws of thought.) (Though in some things, as in numbers, besides adding and subtracting, men name other operations, as multiplying and dividing, yet are they the same ; for multiplication is but adding together of things equal; and division but subtracting of one thing as often as we can. Hobbes, Leviathan, p. i., chap. 5.) The necessity of these judgments results from the existence in the mind of the a priori forms of intuition—Space and Time. The axioms of geometry are self-evident statements concerning magnitudes in space ; such as that two straight lines cannot enclose a space.
Their self-evidence or necessity is to be explained by the circumstance that the presented intuition, as well as the representative thought, is derived from within, not from without. For geometrical propositions are primarily necessary, not as truths relating to objects without the mind, but as thoughts relating to objects within : their necessity, as regards real objects, is only secondary and hypothetical. If there exist anywhere in the world two perfect straight lines, those lines cannot enclose a space ; but if such lines exist nowhere but in my imagination, it is equally true that I cannot think of them as invested with the contrary attributes. This necessity of thonght is dependent on a corresponding necessity of intuition. The object of which pure geometry treats is not dependent on sensation, but sensation on it : it is a condition under which alone sensible experience is possible ; and therefore its characteristics must accompany all our thoughts concerning any possible object of such experience ; for, however much we may abstract from the attributes of this or that particular phenomenon of experience, we are clearly incompetent to deprive it of those conditions under which alone, from the constitution of our minds, experience itself is possible. We can perceive only as we are permitted by the laws of our perceptive faculties, as we can think only in accordance with the laws of the understanding. If, then, by a law of my perceptive faculty, I am compelled to regard all objects as existing in space, the attributes which are once presented to me as the properties of a given portion of space, such as the pair of straight lines now present to my sight or imagination, must necessarily be thought as existing in all space and at all times. For to imagine a a portion of space in which such properties are not found, would not be to imagine merely a different combination of sensible phenomena, such as continually takes place without any change in the laws of sensibility : it would be to imagine myself as perceiving under other conditions than those to which, by a law of my being, I am subjected. But a condition, though potentially existing in the original constitution of the mind, is actually manifested only in conjunction with that of which it is the condition. Space, there
fore, and its laws, are first made known to consciousness on the occasion of an actual phenom. enon of sense. Hence the twofold character of geometrical principles : empirical, as suggested in and through an act of experience ; necessary, as relating to the conditions under which alone such experience is possible to human faculties.
“ Arithmetic is related to Time as Geometry to Space; and the necessity of its propositions may be explained upon similar principles. The two sciences, however, present some important features of distinction. Most of the proposi. tions of geometry are deductive : it contains very few axioms, properly so called, and its processes consist in the demonstration of a multitude of dependent propositions from the combination of these axioms with certain logical principles of thought in general. On the other hand, the fundamental operations of arithmetic -addition and subtraction-present to us a vast number of independent judgments, every one of which is derived immediately froin intuition, and cannot, by any reasoning process, be deduced from any of the preceding ones. Although
( it is simpler to regard addition and subtraction as independent processes, yet no result of either can be derived from a preceding result of the same operation.) Pure geometry cannot advance a step without demonstration, and its processes are therefore all reducible to the syllogistic form. Pure arithmetic contains no demonstration ; and it is only when its calculus is applied to the solution of particular problems that reasoning takes place, and the laws of the syllogism become applicable. It is not reasoning which tells
. us that two and two make four ; nor, when we have gained this proposition, can we in any way deduce from it that two and four make six. We must have recourse, in each separate case, to the senses or the imagination (memory), and by counting up the individual succession corresponding to each term, intuitively perceive the resulting sum. The intuition thus serves nearly the same purpose as the figure in a geometrical demonstration ; with the exception that, in the lat
; ter case, the construction is adopted to furnish premises to a proposed conclusion ; while in the former it gives us a judgment which we have no immediate intention of applying to any
further use. “ The intuition in the case of arithmetic is furnished by the consciousness of successive states of our own minds. Setting aside all other characteristics of those states, save that of their succession in time, we have the immediate consciousness of one, two, three, four, etc. A purely natural arithmetic would consist in carrying on this series, with no other relation between its members but that of succession, until the memory became unable to continue the process. The artificial methods by which calculation is facilitated and extended, such as that of a scale of notation, in which the series recommences after a certain number of members, vastly increase the utility of the calculus, but do not affect its psychological basis. To construct the science of arithmetic in all its essential features, it is only necessary that we should be conscious of a succession in time, and should be able to give names to the several members of the series ; and since in every act of consciousness we are subject to the condition of succession, it is impossible in any form of consciousness to represent to ourselves the facts of arithmetic as other than they
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“The necessity of propositions in geometry and arithmetic is thus derived from their relation to the universal forms of intuition—Space and Time. We can suppose the possibility of beings existing whose consciousness has no relation to space or time at all. This is no more than to admit the possible existence of intelligent beings otherwise constituted than ourselves, and consequently incomprehensible by us. pose the existence of geometrical figures or arithmetical numbers such as those with which we are now acquainted, is to suppose the existence of
space and time as we are now conscious of them, and therefore relatively to beings whose mental constitution is so far similar to our own. Such a supposition necessarily carries with it all the mathematical relations in which space and time, as given to us, are necessarily thought. For mathematical judgments strictly relate only to objects of thought as existing in my mind, not to distinct realities existing in relation to my mind. They therefore imply no other existence than that of a thinking subject, modified in a certain manner. Destroy this subject, or change its modification, and we cannot say, as