The Philosophy of Auguste Comte

by Lucien Lévy-Bruhl

In the philosophical study of abstract mathematics, Comte proceeds successively from arithmetical to algebraical calculation, and from the latter to the transcendent analysis or differential and integral calculus. After having stated the manner in which this calculus is presented according to Leibnitz and to Newton, he adopts that of Lagrange, which appears to him the most satisfactory. It is true that at the end of his life his admiration for the author of the Mécanique analytique had greatly diminished. Without here entering into the detail of questions, we will limit ourselves to the indication of a consideration upon the bearings of abstract mathematics, which appears to be of capital importance to Comte. Whether it be a question of ordinary analysis, or especially of transcendental analysis, Comte brings out at once the extreme imperfection of our knowledge, and the extraordinary fecundity of their applications. He can only solve a very small part of the questions which come before us in these sciences. However, “in the same way as in ordinary analysis we have succeeded in utilising to an immense degree a very small amount of fundamental knowledge upon the solution of equations, so, however little advanced geometers may be up to the present time in the science of integrations, they have none the less drawn, from these very few abstract notions the solution of a multitude of questions of the first importance, in geometry, in mechanics, in thermology, etc., etc.”[105] The reason of this is that the least abstract knowledge naturally corresponds to a quantity of concrete researches. The most powerful extension of intellectual means which man has at his disposal for the knowledge of nature consists in his rising to the conception of more and more abstract ideas, which are nevertheless positive. When our knowledge is abstract without being positive, it is “fictitious” or “metaphysical.” When it is positive without being abstract, it lacks generality, and does not become rational. But when, without ceasing to be positive, it can reach to a high degree of abstraction, at the same time it attains the generality, and, along the lines of its furthest extension, the unity which are the end of science.

Hence the importance of Descartes’ fine mathematical discovery, and also of the invention of differential and integral calculus, which may be considered as the complement to Descartes’ fundamental idea concerning the general analytical representation of natural phenomena. It is only, says Comte, since the invention of the calculus, that Descartes’ discovery has been understood and applied to the whole of its extent. Not only does this calculus procure an “admirable facility” for the search after the natural laws of all the phenomena; but, thanks to their extreme generality, the differential formulæ can express each determined phenomenon in a single equation, however varied the subjects may be in which it is considered. Thus, a single differential equation gives the tangents of all curves, another expresses the mathematical law of every variety in motion, etc.

Infinitesimal analysis, especially in the conception of Leibnitz, has therefore not only furnished a general process for the indirect formation of equations which it would have been impossible to discover directly, but in the eyes of the philosopher it has another and a no less precious advantage. It has allowed us to consider, in the mathematical study of natural phenomena, a new order of more general laws. These laws are constantly the same for each phenomenon, in whatever objects we study it, and only change when passing from one phenomenon to another “where we have been able moreover, in comparing these variations, to rise sometimes, by a still more general view, to a positive comparison between several classes of various phenomena, according to the analogies presented by the differential expressions of their mathematical laws.”[106] Comte cannot contemplate this immense range of transcendent analysis without enthusiasm. He calls it “the highest thought to which the human mind has attained up to the present time.” The highest, because being the most profoundly abstract among all the positive notions, this thought reduces the most comprehensive range of concrete phenomena to rational unity.

As the consideration of analytical geometry suggested to Descartes the idea of “universal mathematics,” which lies at the basis of his method, so we can think that philosophical reflection upon transcendental analysis led Comte to the idea of those “encyclopædic laws,” which hold such an important place in his general theory of nature. For these encyclopædic laws, analogous as they are to the differential formulæ spoken of by Comte, are equally verifiable in orders of otherwise irreducible phenomena, and allow us to conceive them as convergent.

III.

Geometry is the first portion of concrete mathematics. Undoubtedly the facts with which it deals are more connected among themselves than the facts studied by the other sciences, and this allows us easily to deduce some of these facts once the others are given. But there is a certain number of primary phenomena which, not being established by any reasoning, can only be founded upon observation, and which stand as the basis of all geometrical deductions.[107] Although very small, this part of observation is indispensable because it is the initial one, and never can quite vanish.

In this way, metaphysical discussions upon the origin of geometrical definitions and space are set aside. Comte here adopts d’Alembert’s opinion. The latter had said: “The true principles of the sciences are simple recognised facts, which do not suppose any others, and which consequently can neither be explained nor questioned: in geometry they are the properties of extension as apprehended by sense. Upon the nature of extension there are notions common to all men, a common point at which all sects are united as it were in spite of themselves, common and simple principles from which unawares they all start. The philosopher will seize upon these common primitive notions to make them the basis of the geometrical truths.”[108]

Extension is a property of bodies. But, instead of considering this extension in the bodies themselves, we consider it in an indefinite milieu which appears to us to contain all the bodies, of the universe and which we call space. Let us think, for instance, of the impression left by a body in a fluid in which it might be immersed. From the geometrical point of view this impression can quite conveniently be substituted to the body itself. Thus, by a very simple abstraction, we divest matter of all its sensible properties, only to contemplate in a certain manner its phantom, according to d’Alembert’s expression. From that moment we can study not only the geometrical forms realised in nature, but also all those which can be imagined. Geometry assumes a “rational” character.

Similarly, it is by a simple abstraction of the mind that geometry regards lines as having no thickness, and surfaces as being without depth. It suffices to conceive the dimension to be diminished as becoming gradually smaller and smaller until it reaches such a degree of thinness that it can no longer fix the attention. It is thus that we naturally acquire the “real idea” of surface, then of the line, and then of the point. There is therefore no necessity to appeal to the a priori.

Thus constituted, the object of geometry is the measurement of extension. But since this measurement can hardly ever be directly taken by superposition, the aim of geometry is to reduce the comparison of all kinds of extensions, volumes, surfaces or lines to simple comparisons of straight lines, the only ones regarded as capable of being immediately established.”[109] The object of geometry is of unlimited extent, for the number of different forms subject to exact definitions is unlimited. In regarding curved lines as generated by the movement of a point subject to a certain law, we can conceive as many curves as laws.