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, 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. 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.
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 do 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.