CHAPTER I MATHEMATICS

 In the eyes of philosophers, mathematics has always occupied a privileged place among the sciences. Plato located their object in an intermediate region between the world of sensible phenomena and that of intelligible realities. On the one hand mathematical objects, and in particular the geometrical figures, appeal to the imagination as sensible things; on the other hand, mathematical truths like ideas and the relations between ideas, are characterised by immutable and eternal fixity. This is why the study of mathematics is an excellent preparation for philosophy, which is the science of ideas. While still leaving to the mind the help of direct sensible perception, it accustoms it to permanent truth. During the whole of antiquity the science of mathematics, as the name indicates, was pre-eminently the science. The science of physics, less sure of its object and of its method, was hardly distinguished from philosophical speculation, and lent itself with difficulty to the purely scientific form.   For Plato then, and for those who followed him, mathematics has characteristics which distinguish it from the study of phenomena. In a certain measure, it partakes of the nature of science, conceived as bearing upon what is, upon the absolute reality which is neither subject to change nor to motion. It is true that they start from definitions and hypotheses. But, once the principles are126 established, they are developed a priori by a succession of necessary demonstrations like the dialectics of ideas.   This conception offers a mixture of metaphysical and positive elements. It implies that the object of science is reality such as it is in itself; but, at the same time, it sees in the demonstration the essential character of science. A long evolution, which culminates in Comte’s doctrine, has driven the metaphysical elements out of science while the other elements subsist in it still. Far from saying with Plato or with his successors that there is no science of the phenomenon or of that which passes away, Comte thinks on the contrary, that the only object of science is phenomenal reality so far as it is subject to laws. Science has not to search for causes or substances; it suffices for it to determine invariable relations.   If the mathematical sciences have long been the only sciences properly so called, and if to-day they are still more advanced than any others, it is because the geometrical and mechanical phenomena are indeed the simplest of all, and those which are most naturally connected among themselves. The period during which they could be studied by observation could therefore be very short, so short that it is even not absurd to maintain that it never existed, and that, in this case, rational knowledge was not preceded by the empirical establishment of facts. But the difference between mathematics and the other sciences none the less remains one of degree and not of kind. The Science of Mathematics is in advance of the other sciences; but all work on common ground. In a word, like all other sciences it is a natural science.   This endeavour to present the whole of the sciences as homogeneous, that is to say, to avoid two distinct classes being formed of mathematics on the one hand, and of the sciences of nature on the other, had already been attempted before Comte. This endeavour imposed itself, so to speak,127 upon modern philosophers, from the time when Descartes sought for a universal method for science conceived as a whole. Comte, who saw very well the defect in the Cartesian conception, in which the ascendency of mathematics was still too much felt, did not, however, deny that his own conception proceeded from that of Descartes. In another form, the idea of the homogeneity of the sciences is also found in Leibnitz and even in Kant. Does not the Critique de la raison pure show that mathematics on the one hand, and physics on the other, equally rest upon principles which are synthetic a priori? In the Prolégomenes à toute métaphysique future just as the chapter corresponding to l’esthétique transcendentale is entitled “How are pure mathematics possible a priori?” so the chapter corresponding to the Logique transcendentale bears as its title “How are pure physics possible a priori?” On another plan Comte’s theory is parallel to Kant’s. Here as there mathematics as well as physics rests upon synthetic principles—“superior to experience,” says Kant—proceeding from experience, says Comte. The latter, it is true, did not know Kant’s theory, and, had he known it he would not have accepted it. But the analogy of tendency subsists none the less beneath the diversity of doctrines.   The immediate antecedent of Comte’s theory is found in d’Alembert. The author of the Discours préliminaire had said, “We will divide the science of nature into physics and mathematics.” II.   Every science has its origin in the art corresponding to it. Mathematics arose out of the art of measuring magnitudes. Indeed this art would be very rudimentary if we only practised direct measurement. Among the magnitudes which interest us there are very few which we can measure thus.128 Consequently the human mind had to seek some indirect way of determining magnitudes.   In order to know the magnitudes which do not allow of direct measurement, we must evidently connect them with others which are capable of being immediately determined, and according to which we succeed in discovering the former, by means of the relations which exist between them and the latter. “Such is the precise object of mathematical science in its entirety.”102 We see immediately how extremely vast it is. If we must insert a large number of intermediaries between the quantities which we desire to know, and those which we can measure immediately, the operations may become very complicated.   Fundamentally, according to Comte, there is no question, whatever it may be, which cannot be finally conceived as consisting in determining one quantity by another, and consequently which does not depend ultimately upon mathematics. It will be said that we must take into account not only the quantity, but also the quality of the phenomena. This objection, decisive in the eyes of Aristotle, who could not conceive that we could legitimately [Greek: metaballein] [Greek: eis allo genos], no longer holds good for modern thinkers. Since Descartes’ time, they have seen analysis applied to geometrical, mechanical and physical phenomena. There is no absurdity in conceiving that what has been done for these phenomena is possible for the others. We must be able to represent every relation between any phenomena whatever by an equation, allowing for the difficulty of finding this equation and of solving it.103 As a matter of fact, we are quickly stopped by the complexity of the data. In the present state of the human mind there are only two great categories of phenomena of which we regularly know the equations: these are geometry and mechanics.   129   This being established, the whole of mathematical science is divided into two parts: abstract and concrete mathematics. The one studies the laws of geometrical and mechanical phenomena. The other is constituted by the calculus, which, if we take this word in its largest sense, applies to the most sublime combinations of transcendent analysis, as well as to the simplest numerical operations. It is purely “instrumental.” Fundamentally, it is nothing else than an “immense admirable extension of natural logic to a certain order of deductions.”   This part of mathematical science is independent of the nature of the objects which it examines, and only bears upon the numerical relations which they present. Consequently, it may happen that the same relations may exist among a great number of different phenomena. Notwithstanding their extreme diversity these phenomena will be considered by the mathematician as presenting a single analytical question, which can be solved once for all. “Thus, for instance, the same law which reigns between space and time when we examine the vertical fall of a body in vacuo, is found again for other phenomena which present no analogy with the former nor among themselves; for it also expresses the relation between the area of a sphere and the length of its diameters; it equally determines the decrease in intensity of light or of heat by reason of the distance of the objects lighted and heated, etc.”104 We have no general method which serves indifferently for establishing the equations of any natural phenomena whatever: we need special methods for the several classes of geometrical, optical, mechanical phenomena, etc. But, whatever may be these phenomena, once the equation is established, the method for solving it is uniform. In this sense, abstract mathematics is really an “organon.”   Geometry and mechanics, on the contrary, should be regarded as real natural sciences, resting as the others do130 upon observation. But, adds Comte, these two sciences present this peculiarity, that in the present state of the human mind, they are already used, and will continue to be used as methods far more than as direct doctrine. In this way mathematics is in fact “instrumental,” not only in abstract parts, but also in its relatively concrete parts. It is entirely used as a “tool” by the more complicated sciences, such as astronomy and physics. It is truly the real logic of our age.   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.,131 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 formul? 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 natural132 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 “encyclop?dic laws,” which hold such an important place in his general theory of nature. For these encyclop?dic laws, analogous as they are to the differential formul? 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 primary133 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 that134 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 is135 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 we136 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 remain137ing a natural science. Even when it has become as purely rational as possible, it none the less remains rooted in experience. IV.   The second part of concrete mathematics (mechanics) is also one of the natural sciences which owes its marvellous progress to analysis. Here again we must distinguish the data which are at the basis of science, and which are facts, from the abstract development undergone by this science because of the simplicity of these facts and the precision of the relations which exist between them. The distinction between what is “really physical” and what is “purely logical”113 is not always an easy one. We must, however, separate facts furnished by experience, from artificial conceptions whose object is to facilitate the establishment of general laws of equilibrium and of motion.   Only to consider inertia in bodies is a fiction of this kind. Physically the force of inertia does not exist. Nature nowhere shows us bodies which are devoid of internal activity. We term those which are not alive inorganic, but not inert. Were gravitation alone common to all molecules, it would suffice to prevent the conception of matter as devoid of force. Nevertheless, mechanics only considers the inertia of bodies. Why? Because this abstraction presents many advantages for the study, “without, moreover, offering disadvantages in the application.” Indeed, if mechanics had to take into account the internal forces of bodies and the variations of these forces, the complications would immediately become such that the facts could never be submitted to calculation. Mechanics would run the risk of losing its character as a mathematical science. And, on the other hand, as it only138 considers the movements in themselves, regardless of their mode of production, it is always lawful for mechanics to replace, if necessary, the internal forces by an equivalent external force” applied to the body. The inertia of matter is therefore an abstraction, the end of which is to secure the perfect homogeneity of mechanical science, by allowing us to consider all moving bodies as identical in kind, and all forces as of the same nature.   The “physical” character of this science is again evident from the consideration of the three fundamental laws upon which it rests.114   The first, called Kepler’s law, is thus defined: “All movement is naturally rectilinear and uniform; that is to say, any body subject to the action of a single force which acts upon it instantaneously, moves constantly in a straight line with invariable speed.” It has been said that this law is derived from the principle of sufficient reason. The body must continue in a straight line because there is no reason why it should deviate from it more on one side than on the other. But, answers Comte, how do we know that there is no reason for the body to deviate, except precisely because we see that it does not deviate? The reasoning “reduces itself to the repetition in abstract terms of the fact itself, and to saying that bodies have a natural tendency to move in a straight line, which is precisely the proposition which we have to establish.” It is by similar arguments that the philosophers of antiquity, and especially Aristotle, had, on the contrary been led to regard circular motion as natural to the stars, in that it is the most perfect of all, a conception which is only the abstract enunciation of a imperfectly analysed phenomenon. The tendency of bodies to move in a straight line with constant speed is known to us by experience.   The second fundamental law of mechanics, called Newto139n’s law, expresses the constant equality of action and reaction. It is pretty generally agreed to-day to consider this law as resulting from the observation of facts. Newton himself understood it so.   Finally the third law establishes that “every movement exactly possessed in common by all the bodies of any system does not alter the particular movements of those different bodies in respect to each other; but those movements continue to take place as if the whole of the system was motionless.” This law “of the independence or of the coexistence of movements” was formulated by Galileo. It is no more a priori than the two preceding ones. How could we be sure, if experience did not show it to us, that a common motion communicated to a system of bodies moving in relation to one another, would change nothing in their particular motions? When his law was made known by Galileo, on all hands there arose a cloud of objections, tending to prove a priori that this proposition was false and absurd. It was only admitted later when, in order to examine it, the logical point of view was set aside for the physical point of view. It was then seen that experience always confirmed this law, and that, if it ceased to operate, the whole economy of the universe would be thrown into utter confusion. For instance, the movement of the translation of the earth in no way affects the mechanical phenomena which take place upon the surface or within the globe. As the law of the independence of motions was unknown when the theory of Copernicus appeared, an objection was put to him which was thought to be drawn from experience. He was told that if the earth moved round the sun all the movements which take place upon it or within it would be modified by the action. Later on when Galileo’s law became known, the fact was explained and the objection disappeared.   Once these three laws are established, mechanics has140 sufficient foundation. Henceforth the scientific edifice can be constructed by simple logical operations, and without any further reference to the external world. But this purely rational development no more transforms mechanics into an a priori science than the application of analysis deprives geometry of its character as a natural science. What proves this, in one case as in the other, is the possibility of passing from the abstract to the concrete and of applying the results obtained to real cases, merely restoring the elements which science had been compelled to set aside. If it were possible entirely to constitute the science of mechanics according to simple analytical conceptions, we could not imagine how such a science could ever become applicable to the effective study of nature. What guarantees the reality of rational mechanics is precisely its being founded upon some general facts, in a word, upon the data of experience.   Comte could assuredly not foresee the controversies which to-day bear upon the principles of mechanics and which have been summed up by Mr. Poincaré in an article upon Hertz’s mechanical theories.115 Mr. Poincaré says that the principles of Dynamics have been stated in many ways, but nobody sufficiently distinguished between what is definition, what is experimental truth, and what is mathematical theorem. Mr. Poincaré is satisfied neither with the “classical” conception of mechanics, whose insufficiency has been shown by Hertz, nor with the conception with which Hertz wishes to replace it. In any case it is a high philosophical lesson to see the classical system of analytical mechanics—a system constructed with such admirable accuracy, and made by Laplace to arise altogether, as Comte says, out of a single fundamental law,—to see it after a century labouring under grave difficulties, not unconnected with the progress of physics.   141   Might not this be an argument in support of the theory of d’Alembert and of Comte on the nature of concrete mathematics? Geometry and mechanics would only differ from the other natural sciences by the precision of the relations between the phenomena of which they treat, by the facility which they have for dealing with these relations by means of calculus and analysis, and, consequently, by assuming an entirely rational and deductive form. For the extraordinary power of the instrument should not hide from us the nature of the sciences which make use of it. These, like the others, bear upon natural phenomena. Only, as these phenomena are the most simple, the most general and the most closely allied of all, these sciences are also those which respond in the best way to the positive definition of science. They have “very easily and very quickly replaced empirical statement by rational prevision.” They are composed of laws and not of facts. But, conforming in this again to the positive definition of science, they are empirical in their origin, and they remain relative in the course of their development.   Thus positive philosophy, having reached the full consciousness of itself, reacts upon the conception of the sciences which have most contributed to its formation. When the philosophy is universally accepted the idea that a science can be a priori, that is both absolute and immutable, will have disappeared. Precisely because it is the most perfect type of a positive science, mathematics will no longer claim these characteristics, and its ancient connection with metaphysics will be finally severed.