The Philosophy of Auguste Comte

by Lucien Lévy-Bruhl

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.

The human mind, in order to cover this immense field, the extension of which it was very late in apprehending, may pursue two different methods. Perfect geometry would, indeed, be the one which would demonstrate all the properties of all imaginable forms, and this can be obtained in two ways. Either we can successively conceive each of the forms, the triangles, the circle, the sphere, the ellipse, etc., and seek for the properties of each one of them. Or else we can group together the corresponding properties of various geometrical forms, in such a way as to study them together, and, so to speak, to know beforehand their application to such and such a form which we have not yet examined. “In a word,” says Comte, “the whole of geometry can be ordered, either in relation to bodies which are being studied, or in relation to phenomena which are to be considered.” The first plan is that of the geometry of the ancients, or special geometry; the second is that of the geometry since Descartes, or general geometry.[110]

At its origin geometry could only be special. The ancients, for instance, studied the circle, the ellipse, the parabola, etc., endeavouring, in the case of each geometrical form, to add to the number of known properties. But, if this line of advance had been the only one which could be followed, the progress of geometry would never have been a very rapid one. The method invented by Descartes has transformed this science, by enabling it to become general, and to abandon the individual study of geometrical forms for the common study of their properties. This revolution has not always been well understood. Often in teaching mathematics, its bearings are not sufficiently shown. From the manner in which it is usually presented, this “admirable method” would at first seem to have no other end than the simplification of the study of conic sections or of some other curves, always considered one by one according to the spirit of ancient geometry. This would not be of great importance. The distinctive character of our modern geometry consists in studying in a general way the various questions relating to any lines or surfaces whatever by transforming geometrical considerations and researches into analytical considerations and researches.[111]

All geometrical ideas necessarily relate to the three universal categories; magnitude, form, position. Magnitude already belongs to the domain of quantity. Form can be reduced to position, since every form can be considered as the result of the advance of a point, that is to say of its successive positions. The problem is therefore to bring all ideas of situation whatever back to ideas of magnitude. How did Descartes solve it? By generalising a process which we may say is natural to the human mind, since it comes spontaneously into being under the stress of necessity. Indeed, if we must indicate the situation of an object without showing it immediately, do we not refer it to others which are known, by stating the magnitude of geometrical elements by which we conceive the object to be connected with them? Geographers act in the same way in their science to determine the longitude and latitude of a place, and astronomers to determine the right ascension and the declination of a star. These geographical and astronomical co-ordinates fulfil the same office as the Cartesian co-ordinates. The only difference, but it is a capital one, consists in the fact that Descartes carried this method to the highest degree of abstract generality thus giving it its maximum of fertility and power.

Although general geometry is infinitely superior to special geometry it cannot, nevertheless, altogether dispense with the latter. As the ancients did, so it will always be necessary to begin with special geometry. For general geometry rests upon the use of calculation. But if, as Comte has said, geometry is truly a science of facts calculation will evidently never be able to supply us with the first knowledge of these facts. In order to lay the foundations of a natural science simple mathematical analysis would never suffice, nor could it give a fresh demonstration of it, when these foundations have already been laid. Before all things a direct study of the subject is necessary, until the precise relations are discovered. “The application of mathematical analysis can never begin any science whatever, since it could never take place except when the science has been sufficiently elaborated to establish, in relation to the phenomena under consideration, some equations which might serve as a starting-point for analytical work.”[112] In a word, the creation of analytical geometry does not prevent geometry from remaining a natural science. Even when it has become as purely rational as possible, it none the less remains rooted in experience.