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 only 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 Newton’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 has 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.
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.
[CHAPTER II]
ASTRONOMY
The object of astronomy is the discovery of the laws of the geometrical and mechanical phenomena presented by the celestial bodies; and, by the knowledge of these laws to obtain the precise and rational prevision of the state of our system at any given period whatever. It is in a word, “the application of mathematics to celestial phenomena.”[116]
Mr. H. Spencer has taken occasion of this definition to criticise the place assigned by Comte to astronomy in his classification of the sciences. He makes him contradict himself. He says: you term fundamental sciences the abstract sciences which do not study beings in nature, but the laws which govern phenomena in those beings; by what right is astronomy placed among these sciences, between mathematics and physics? Is not the object of astronomy the study of certain beings in nature? In what does the application of mathematics to celestial phenomena differ from their application to other cases? It appears evident that here Comte introduces into the series of abstract sciences a science which is really concrete, or at least, according to Mr. Spencer’s expression, abstract-concrete.
Comte had foreseen the objection. The answer which he makes throws a strong light upon the sense in which he understands the words “abstract” and “general” as applied to the sciences. He partly accepts the objection. The true astronomical notions, he says, only differ from purely mathematical notions by their special restriction to the celestial case; and this, at first sight, must appear contrary to the essentially abstract nature of the speculations which belong to the first philosophy. But on the other hand, these speculations bear upon the phenomena given in experience, and the order of the abstract sciences should reproduce the real order of dependence of the phenomena. Thus the first of these sciences, mathematics, determines the essential laws of the most general phenomena, which are common to all material beings (form, position, movement). Now, are not the most general phenomena after these, those “of which the the continuous ascendency inevitably dominates the course of all the other phenomena?”[117] In other words, before passing to the study of physical, chemical, biological phenomena, etc., it is indispensable to know the general laws of the milieu in which these phenomena are manifested. Outside of this milieu, they would be impossible, or at any rate, it so conditions them that, were it otherwise, these phenomena would also be different from what they are.
The character of generality which, with that of abstraction, is made use of to institute the hierarchy of phenomena is thus reduced to the idea of dependence. It is the consideration of this dependence which assigns to astronomy its place between mathematics and physics in the encyclopædic ladder of the sciences. Considered singly in themselves, the phenomena studied by astronomy are purely geometrical and mechanical. They would not, therefore, constitute the object of a science distinct from mathematics. But positive philosophy considers everything from the point of view of humanity. Now, for humanity, this “special case” is of unequalled importance. All the other phenomena given to us by experience (except the mathematical phenomena) depend, in a more or less direct manner, upon astronomical phenomena. The knowledge of astronomical laws is therefore the necessary condition for the knowledge of all the others. Thus, the infringement of the principle of the hierarchy of fundamental sciences is only apparent. An analogous case is found in chemistry. The analysis of air and water is incorporated in abstract chemistry, because air and water constitutes the general milieu, “in which all ulterior phenomena occur.”[118]
The place given to astronomy is therefore justified. This science, moreover, remains abstract. For it to be a concrete science, all aspects of the existence of celestial bodies would have to be studied and considered in their relations, to each other in it. But, on the contrary, astronomy only studies the geometrical and mechanical phenomena in the celestial bodies, all physical and chemical considerations, etc., being eliminated. Comte concludes that in passing on to the celestial case mathematics does not lose its abstract nature. It only becomes more developed in the case of a special example, whose extreme importance demands such a specialisation.
The abstract character of astronomy belongs to it almost a priori. The facts upon which it rests are only revealed to us by one of our senses, the most intellectual of them indeed, but by which we are only informed of the mathematical properties of bodies. Our eyes alone touch the stars. There is no astronomy for a blind race. Dark stars, if such there be, are for ever hidden from us. All that is given to us, therefore, is the shape, the position and the motion of visible celestial bodies. We can never by any means know how to study their chemical composition, nor their mineral structure, nor a fortiori the nature of the organic bodies which may live upon them. Comte might have formulated in less categorical terms affirmations which were soon to be contradicted by spectral analysis and by photography. But he was confirmed in the entirely abstract and mathematical conception which he had of astronomy by his persuasion that no discoveries of so far-reaching a nature were possible.
Thus, astronomy appeared to be an excellent type of a positive science, because it is at once natural and abstract, and in it these two characteristics are equally apparent, which was not the case in mathematics. In this science the share of observation is so limited, so transient, that it becomes inappreciable. In astronomy, on the contrary, determination of certain facts evidently plays a part in the science. But, at the same time, nowhere do we see more clearly that science does not consist in the mere apprehension of facts. Here they are so simple, and moreover so uninteresting, that their connexion and the knowledge of their laws alone deserves the name of science. In general, what is an astronomical fact? None other than this: such a star has been seen at such a precise instant, and under such an angle duly measured. The more or less profound elaboration of these observations is indispensable to science, even in its most imperfect state. Astronomy, says Comte, did not really come into being when the priests of Egypt or Chaldea made a series of more or less exact empirical observations in the heavens; but only when the first Greek philosophers began to reduce the general phenomenon of diurnal motion to a few geometrical laws.[119]
Of all the natural sciences, after mathematics, astronomy is also the most perfectly free from all theological and metaphysical considerations. From every point of view it is positive. Astronomers no longer have recourse to a Providence, which as the intelligent cause of the order of the celestial world, would in its turn, witness to the existence of this cause. They do not inquire any more into the intimate nature of forces (gravitation, attraction, etc.). Astronomy is content to determine the invariable relations of phenomena with the greatest possible precision. It is here that philosophical minds can study the essential characteristics of a positive science. In it they will also see how disinterested it must be in order to become useful. “Without the highest speculations of geometers upon celestial mechanics, which have so greatly increased the precision of astronomical tables, it would be impossible to determine the longitude of a ship with the degree of accuracy which is now attainable.”[120]
Finally no science has exercised a greater influence upon the evolution of the human mind than this one. The great epochs in astronomy are also those in cosmological philosophy. The desperate resistance which was offered by theological dogmatism to Galileo’s discovery responded to a just apprehension of the consequences involved in this discovery. To admit that the earth was not the centre of the world was to take a first and a decisive step in the way which leads away from the anthropocentric prejudice. It was like pledging oneself to substitute sooner or later the relative point of view to the absolute one in philosophy. It was introducing the positive spirit, to-day in speculative physics, to-morrow in speculative ethics.