Читать книгу Fundamental Philosophy, Vol. 2 (of 2) - Balmes Jaime Luciano - Страница 5

29. The idea is a very different thing from the sensible representation, but it has certain necessary relations with it which it will be well to examine. When we say necessary, we speak only of the manner in which our mind, in its actual state, understands, abstracting the intelligence of other spirits, and even that of the human mind when subject to other conditions than those imposed by its present union with the body. So soon as we quit the sphere in which our experience operates, we must be very cautious how we lay down general propositions, and take care not to extend to all intelligences qualities which are possibly peculiar to our own, and which, even with respect to it, will perhaps be entirely changed in another life. Having made these previous observations, which will be found of great utility to mark the limits of things there is danger of confounding, we now proceed to examine the relations of our ideas with sensible representations.

30. A classification of our ideas into geometrical and non-geometrical naturally occurs when we fix our attention upon the difference of objects to which our ideas may refer. The former embrace the whole sensible world so far as it can be perceived in the representation of space; the latter include every kind of being, whether sensible or not, and suppose a primitive element which is the representation of extension. In their divisions and subdivisions the latter present simply the idea of extension, limited and combined in different ways; but they offer nothing in relation to the representation of space, and even when they refer to it, they only consider it inasmuch as numbered by the various parts into which it may be divided. Hence the line which in mathematics separates geometry from universal arithmetic; the former is founded upon the idea of extension, whereas the latter considers only numbers, whether determinate, as in arithmetic properly so called, or indeterminate, as in algebra.

31. Here we have to note the superiority of non-geometrical to geometrical ideas, – a superiority plainly visible in the two branches of mathematics, universal arithmetic and geometry. Arithmetic never requires the aid of geometry, but geometry at every step needs that of arithmetic. Arithmetic and algebra may both be studied from their simplest elementary notions to their highest complications without ever once involving the idea of extension, and consequently without making use of one single geometrical idea. Even infinitesimal calculus, in a manner originating in geometrical considerations, has been emancipated from them and formed into a science perfectly independent of the idea of extension. On the contrary, geometry cannot take a single step without the aid of arithmetic. The comparison of angles is a fundamental point in the science of geometry, but it cannot be made except by measuring them; and their measure is an arc of the circumference divided into a certain number of degrees, which must be counted; and thus we come to the idea of number, the operation of counting, that is, into the field of arithmetic.

The very proof by superposition, notwithstanding its eminently geometrical character, stands in need of numeration, inasmuch as the superposition is repeated. We do not require the idea of number to demonstrate by means of superposition the equality of two arcs perfectly equal; but in order to appreciate the relation of their quantity we compare two unequal arcs and follow the method of placing the less upon the greater several times, we count, we make use of the idea of number, and find we have entered upon the ground of arithmetic. We discover the equality of two radii of a circle, when we compare them by superposition, abstracting the idea of number; but if we would know the relation of the diameter to the radii, we employ the idea of two; we say the diameter is twice the radius, and again enter the domains of arithmetic. As we proceed in the combination of geometrical ideas, we make use of more and more arithmetical ideas. Thus the idea of the number three necessarily enters into the triangle; and the sum of three and the sum of two both enter into one of its most essential properties; the sum of the three angles of a triangle is equal to two right angles.

32. The idea of number cannot be replaced by the sensible intuition of the figure whose properties and relations are under discussion. In many cases this intuition is impossible, as, for example, in many-sided figures. We have little difficulty in representing to our imagination a triangle, or even a quadrilateral figure, but the difficulty is greater in the case of the pentagon, and greater still in the hexagon and heptagon; and when the figure attains a great number of sides, one after another escapes the sensible intuition, until it becomes utterly impossible to appreciate it by mere intuition. Who can distinctly imagine a thousand-sided figure?

33. This superiority of non-geometrical over geometrical ideas is very remarkable, since it shows that the sphere of intellectual activity expands in proportion as it rises above sensible intuition. Extension, as we have before seen,3 serves as the basis not only of geometry, but also of the natural sciences, inasmuch as it represents in a sensible manner the intensity of certain phenomena; but it can by no means enable us to penetrate their inmost nature, and guide us from that which appears to that which is. This and other subordinate ideas are, so to speak, inert, and from them springs no vital principle to fecundate our understanding, and still less the reality; they are an unfathomable depth in which our intellectual activity may toil, perfectly certain of never finding any thing in it which we ourselves have not placed there; they are a lifeless object which lends itself to all imaginable combinations without ever being capable of producing any thing, or of containing any thing not given to it. The naturalists in considering inertness as a property of matter, have perhaps regarded more than they are aware the idea of extension, which presents the inertness most completely.

34. The ideas of number, cause, and substance abound in results, and are applicable to all branches of science. We can scarcely speak without expressing them; it might almost be said that they are constituent elements of intelligence, since without them it vanishes like a passing illusion. They extend to every thing, apply to every thing, and are necessary, whenever objects are offered to the intellectual activity, in order that the intellect can perceive and combine them. It makes no difference whether the objects be sensible or insensible, whether there be question of our intelligence or of others subject to different laws; whenever we conceive the act of understanding we conceive also these primitive ideas as elements indispensable to the realization of the intellectual act. They exist and are combined independently of the existence, and even of the possibility, of the sensible world; and they would also exist in a world of pure intelligences, even if the sensible universe were nothing but an illusion or an absurd chimera.

On the other hand, take geometrical ideas and remove them from the sensible sphere; and all that you base upon them will be only unmeaning words. The ideas of substance, cause, and relation do not flow from geometrical ideas; if we regard them alone, we see an immense field extending into regions of unbounded space; but the coldness and silence of death reign there. If we would introduce beings, life, and motion into this field we must seek them elsewhere; we must use other ideas, and combine them, so that life, activity, and motion may result from their combination, in order that geometrical ideas may contain something besides this inert, immovable, and vacant mass, such as we imagine the regions of space to be beyond the confines of the world.

35. Geometrical ideas, properly so called, as distinguished from sensible representations, are not simple ideas, since they necessarily involve the ideas of relation and number. Geometry cannot advance one step without comparing them; and this comparison almost always takes place by the intervention of the idea of number. Hence it is that geometrical ideas, apparently so unlike purely arithmetical ideas, are really identical with them so far as their form or purely ideal character is concerned; and are only distinguishable from them when they refer to a determinate matter, such as extension as presented in its sensible representation. The inferiority therefore of geometrical ideas already mentioned, only refers to their matter, or to their sensible representations, which are presupposed to be an indispensable element.

36. Another consequence of this doctrine, is the unity of the pure understanding, and its distinction from the sensitive faculties. For, the very fact that the same ideas apply alike to sensible and to insensible objects, with no other difference than that arising from the diversity of the matter perceived, proves that above the sensitive faculties there is another faculty with an activity of its own, and elements distinct from sensible representations. This is the centre where all intellectual perceptions unite, and where that intrinsic force resides, which, although excited by sensible representations, develops itself by its own power, makes itself master of these impressions, and converts them, so to speak, by a mysterious assimilation, into its own substance.

37. Here we repeat what we have already remarked, concerning the profound ideological meaning involved in the acting intellect of the Aristotelians, so ridiculed because not understood. But we leave this point and proceed to the careful analysis of geometrical ideas, to discover, if possible, a glimpse of some ray of light amid the profound darkness which envelops the nature and origin of our ideas.