Considerations On The Mathematical Sciences
( Originally Published 1913 )
THE object of Mathematical science is the measurement of magnitudes. Direct measurement, by simple immediate comparison of one magnitude with another known one, is seldom possible ; and hence the necessity for the formation of a science of measurement. We discover the relation of a magnitude, not susceptible of immediate measurement, to another which is susceptible of measurement what function the one is of the other. Then, in any given case, we can, from the immediate measurement of the one quantity, indirectly arrive at the measurement of the other. Thus (to take a familiar example) the height from whence a body has fallen, and the time of its fall, have always a fixed relation. The two magnitudes are functions of each other. Hence, in the case where we can measure the time of the fall of a body from a precipice, the time gives us the height ; and in an inverse case, we can tell the time a body would take to fall vertically from the moon to the earth, from our knowledge of the distance between the moon and the earth. So, also, from knowing the fixed relations between the sides and angles of a triangle, we can in any given case, from a direct measurement of some of these parts, ascertain the measurement of the remaining. The unknown magnitude, however, may not be ascertained by a knowledge of its relation merely to one other ; it may be, that we require to know what function the unknown magnitude is of a second, and the second of a third, and the third of a fourth, and so on through a long chain; none of the series except the last term being capable of immediate measurement. But the principle in all the cases, simple or complex, is identical.
The exact definition of mathematical science may therefore be arrived at by assigning as its object the indirect measurement of magnitudes, and by saying that our aim in it always is to determine one magnitude from another, by means of the exact relations which exist between them. This way of defining mathematics, instead of giving the idea of an art only, as all definitions have hitherto done, characterizes directly a true science, and shows it, at once, to be composed of a vast series of intellectual operations, which may evidently become very complicated, by reason of the chain of intermediate terms which it may be requisite to establish between the unknown quantities and those allowing of direct measurement, of the number of co-existing variables in a given question and of the nature of the relations among all these various magnitudes presented by the phenomena under consideration. According to this definition, it is in the very spirit of mathematics always to regard all quantities which any phenomenon whatever ' can present to us, as mutual relations, so that they may be deduced from each other. Now, there is evidently no phenomenon that cannot give room for considerations of this kind ; whence naturally result the indefinite extent' and even the strict logical universality of mathematics.
The foregoing explanations clearly justify the employment of the name used to designate the science in question. This appellation, which has now received so fixed an acceptation, signifies by itself simply science in general. This designation, which in Greek usage was quite exact, seeing that they had no other real science, has been only retained by the moderns to indicate that mathematics is the science par excellence. And indeed definition given above (leaving out of account the different degrees of precision) is nothing but the definition of every real science whatever : for has not each of them for its necessary aim to determine one phenomenon from another, by means of the relations which exist among them ? All science consists in the coordination of facts. If our different observations were entirely isolated, there would be no science. We may even say that, in so far as the different phenomena will permit, science is essentially destined to dispense with all direct observation, by allowing us to deduce the greatest possible number of results from the smallest possible number of immediate data. It is in this that lies the real use, in speculation, as well as in action, of the laws which we are discovering among natural phenomena. In this view, Mathematics only pursue, with regard to subjects truly within their province, the same kind of inquiries as are followed out in greater or less degree by each of the exact sciences, in their respective spheres,with this difference, that mathematics carries them to the highest possible point, both with respect to quantity and quality.
It is, then, by the study of mathematics, and it alone, that we can obtain a just and comprehensive idea of what a science really is. It is in that study we ought to learn precisely the general method always followed by the human mind in its positive researches ; for nowhere else are questions resolved so completely, and deductions prolonged so far with extreme rigour. It is there, too, that our intelligence has given the greatest proofs of its power, since the ideas dealt with are the most entirely abstract possible in positive science. All scientific education which does not commence with this study, is therefore and of necessity defective at its foundation.
Hitherto Comte has been speaking of mathematics in their totality. Now let us glance at their principal divisions.
In the complete analysis of a mathematical question, the science is seen spontaneously separating itself into two great divisions. In the first place we have to ascertain the precise relations actually existing between the quantities under consideration. Thus, in order to deter mine the height from which a body has fallen, from the time of the fall, we have to discover the equation between height and time. This constitutes the concrete part of mathematics.
In the second place, we have a pure question of numbers before us. Having the equation, we have simply to determine the unknown numbers from the known. The height being a known multiple of the second power of the time (such being the equation in the particular case referred to), we have to perform the numerical operation of finding the one from the other. This is the abstract part of mathematics.
Sometimes the concrete part is the more difficult ; sometimes the abstract ; and these two great branches of mathematics may be considered as equal in extent and in difficulty. They are as distinct in their object as in the nature of the inquiries embraced by them. Concrete mathematics depend upon the kind of phenomena under consideration, and are essentially experimental, i. e. physical. Abstract mathematics are independent of the objects examined, except as to their numerical relations ; they are purely logical, i. e. rational.
Concrete Mathematics, having for their object the discovery of the equations of phenomena, ought ą priori to be composed of as many distinct sciences as there are categories of phenomena. Practically, however, the only two great categories of phenomena, of which we can always know the equations, are the Geometrical and Mechanical. Hence Concrete Mathematics subdivide themselves into the sciences of Geometry and of Mechanics. These two are natural fundamental sciences, inasmuch as all natural effects can be conceived as simple and necessary results either of extension or of movement.
Abstract Mathematics, on the other hand, are composed of the Calculus in its widest sense, embracing all numerical operations from the simplest to the highest combinations of transcendental analysis. They have for their object the resolution of all questions of numbers, starting from the equations yielded by concrete mathematics.
It is of importance to notice that the fundamental division of mathematics is only an application of the general principle of Classification established in a preceding section, viz. the hierarchy of the different positive sciences. If, in fact, we compare the Calculus on the one side, with Geometry and Mechanics on the other, we truly find, as respects the ideas considered in each of these two primary divisions of mathematics, the essential characteristics of our Encyclopędical Method. The analytical ideas of the Calculus are evidently more abstract, and also more general and more simple, than geometrical or mechanical ideas. Although the principal conceptions of Mathematical Analysis regarded under the historical point of view, were formed under the influence of geometrical or mechanical considerations (with whose progress that of the Calculus has been closely connected), Analysis, nevertheless, is, under the logical point of view, essentially independent of Geometry and Mechanics, while the latter are, on the contrary, necessarily founded on the former.
Mathematical Analysis is therefore, according to the principles laid down, the true and rational basis of the complete system of our positive knowledge. It is the first and the most perfect of all the fundamental sciences. The ideas which form its subject matter are the most universal, the most abstract, and the most simple that we can actually conceive.
This peculiar characteristic of Mathematical Analysis allows us easily to explain why it affords so powerful an nstrument when properly used, not only of giving additional precision to our real knowledge (which is self-evident), but also of establishing an infinitely more perfect coordination in the study of phenomena which permit its application. For, the conceptions having been generalized and simplified to the highest possible degree, so that a single analytical question resolved abstractly contains the implicit solution of a number of different physical questions, the result must necessarily be that the human mind will have a greater facility in perceiving the relations between phenomena which at first sight appear altogether isolated from each other, and from which we thus come, by considering it apart, to make out all they have in common. It is thus that in examining the progress of the intellect in the solution of important questions of Geometry and Mechanics, we see that, by the intervention of Analysis, there have naturally come to light the most frequent and the most unexpected similarities among problems which did not at first appear to present any connection, and which in the end we often regard as identical. How could we, for example, have perceived without the aid of analysis the least analogy between the determination of the direction of a curve to each of its points, and that of the velocity acquired by a body at each instant of its varied movement? Questions, which, however different they may be, are only one in the eyes of a geometrician.
And in like manner it is easy to understand the high state of perfection of Mathematical Analysis, compared with the other sciences, which is owing not to its signs, but to the extreme simplicity of its ideas.
Comte concludes by showing the real extent of the domain of the mathematical science. In the purely logical point of view, it is necessarily universal. Every question can be conceived as being ultimately resolved into one of number. But its domain is practically circumscribed to the less complex questions of Inorganic Physics, on two accounts;
First: Because the different quantities presented in the more complex questions of Inorganic Physics, and in all Organic questions, do not permit fixed numbers, so as to give us the requisite equation; the numerical variability of their phenomena is extreme, and bids defiance to our powers of observing and fixing their value;
Secondly : Because even if we knew the mathematical law of each agent, we could not solve the corresponding mathematical problem, by reason of the great complexity of the conditions.
Want of space prevents me from giving this part of Comte's exposition at length ; but the reader will be at no loss to find illustrations of both cases.
Passing over the six profound and instructive chapters which follow on Abstract Mathematics or Mathematical Analysis we come to the preliminary chapter on Geometry.
Geometry is not, as many have supposed, a purely rational science, independent of observation; certain primitive phenomena, not established by reasoning, but founded on observation, must constitute the basis of its deductions. It has a scientific superiority to Mechanics, and precedes it because it is more universal, more simple, and more independent than Mechanics. Every body in nature may give rise to geometrical as well as mechanical questions ; and we never have the latter without the former but even if the universe were to become immoveable, we should still have geometrical questions to solve.
This is the definition of geometry : it has to measure extension. But direct measurement of a solid or a surface by superposition of another solid or surface is, as a general rule, impracticable. There are always, however, in the case of a solid or of a surface, certain lines whose measurement will give the measurement of the solid or surface. In like manner, a curved line may be measured by certain right lines related to it ; and right lines themselves may in their turn be measured by their relations to other right lines susceptible of immediate comparison.
We can thus form a very precise idea of the science of geometry, by assigning it the general object of ultimately reducing comparisons of all species of extensions, solids, surfaces, or lines, to single comparisons of right lines, which are the only comparisons considered susceptible of being immediately made.
The extent of the science is necessarily indefinite ; for the variety of lines, surfaces, and volumes is indefinite. To measure these various natural forms, as they offer themselves, we require to be prepared by a general study, and by a special examination of certain hypothetical and more simple forms. Hence it is not enough to confine ourselves, as the ancient geometricians did, to the study of certain simple forms directly furnished by nature, or of others deduced from them ; we prepare ourselves for all imaginable forms by the abstract or modern Geometry, which we owe to Descartes, and which reduced the invention of forms to that of equations of right lines. Each equation, and consequently each form, could thus be specially studied. These equations being infinite in number, prepare us for all forms. But there are certain geometrical questions which at first sight do not appear to fall within Comte's definition. These refer to the properties of particular lines or surfaces. A single form may have many properties, each of which may more conveniently than the others lead to a solution in a particular case. By copious illustration, he shows how these questions really and essentially serve the purpose of facilitating measurement.
He then proceeds to point out the two different Methods which may be pursued in forming the Science of Geometry. He discards the phrases, Synthetical and Analytical Geometry, usually employed to designate them.. The one would he better characterised as the Geometry of the Ancients; the other as the Geometry of the Moderns. But instead of these historical appellations; he employs the term Special Geometry, for the former ; General Geometry, for the latter. The radical difference between them, hitherto but imperfectly comprehended, seems really to consist in the very nature of the questions considered. In fact, Geometry supposed to be arrived at complete perfection, ought, as we have seen, on the one hand to embrace all imaginable forms, and on the other, to discover all the properties of each form. According to this double consideration, it is susceptible of being treated in two essentially distinct ways : either in grouping together all the questions, however different they may be, which concern the same form, and treating separately those relating to different bodies, whatever analogy may exist between them; or, on the contrary, in uniting under one and the same point of view all similar questions to whatever different forms they may belong, and separating the questions relative to the properties of the same body that are really different. In a word, the ensemble of Geometry can be fundamentally arranged either with reference to the bodies studied, or with reference to the phenomena to be considered. The first plan, which is the most natural, was that of the Ancients ; the second, infinitely more rational, is that of the Moderns since the time of Descartes. Such is, in fact, the chief characteristic of ancient geometry, where we study, one by one, different lines and different surfaces, never passing to the examination of a new form until we believe we have exhausted every thing of interest which the known forms can give us. In this mode of procedure, when we undertake the study of a new curve, our labours upon the preceding ones do not directly afford any essential help, except in the geometrical exercise which the mind has obtained. In a word, the Geometry of the Ancients was, to use the expression above proposed, essentially special. In the system of the Moderns, Geometry is, on the contrary, essentially general; that is, relative to any forms whatever. It is easy to understand that all geometrical questions of any interest can be proposed in reference to every imaginable form. The very few questions which are truly peculiar to this or that form are of the very least importance. This being granted, Modern Geometry essentially consists in making abstraction of every question relative to the same geometrical phenomenon in whatever body it may be considered, in order to treat it apart in a completely general way. The application of universal theories thus constructed for the special determination of the phenomenon under consideration in each particular body, is no longer regarded but as a secondary work, to be executed according to invariable rules, and whose success is certain beforehand. But we attach no real importance except to the conception and the complete solution of a new question belonging to any form whatever. Operations of this kind are alone regarded as advancing science. The attention of Geometricians being thus freed from the examination of the particular properties of different forms, and wholly directed towards general questions, they have been enabled to rise to the consideration of new geometrical notions, which, when applied to the curves studied by the ancients, have led to the discovery of important properties that they had not so much as suspected. Such is Geometry, since the radical revolution effected by Descartes in the general system of the science.
After pointing out the practical and incomparable ,superiority of the Modern over the Ancient method, Comte concludes by observing that we cannot dispense with the study of the latter. Historically speaking, it was required to enable Descartes to found the Modern method ; and dogmatically it serves as the preliminary basis to General Geometry, in so far as it furnishes to the latter those concrete equations which are the ground work of its analytical processes.
The 11th, 12th, 13th, and 14th lectures are devoted to the subjects of special and general Geometry. Let us pass to the 15th, which is entitled " Philosophical considerations on the fundamental principles of rational Mechanics."
Mechanical phenomena follow Geometrical phenomena, in the order of simplicity, generality, and independence. The philosophical character of the Science of Mechanics (or, more properly speaking, rational Mechanics) is influenced to a greater degree than Geometry by a remnant of the Metaphysical habits of thought. A complete confusion exists in many minds between the abstract and concrete points of view in this science. The distinction is not properly made between the parts of it that are purely physical and those that are purely rational. The progress of this science for a century past has been due so much to Mathematical Analysis that the notion of mechanics being mere cases of Analysis obtained an easy acceptance. Its fundamental principles 'were supposed capable of establishment a priori it being forgotten that Analysis is only a means of deduction, and that if Mechanics were founded on it solely, it would not be really applicable to . the study of nature, as we find it to be. Comte's object in the 15th lecture is to free the subject from these Meta-physical notions, and make the separation between the experimental and rational parts of Mechanics apparent and distinct.
Let us commence by pointing out precisely the general object of the science. We are in the habit of remarking, and justly so, that Mechanics eschew the consideration not only of the first causes of movements, which are beyond the pale of Positive Philosophy, but also the circumstances of their production, which, although really forming an interesting subject of positive study in different branches of Physics, are quite without the province of Mechanics. That science is confined to a consideration of movement in itself, without enquiring into the manner in which it was produced. Hence, forces are nothing in Mechanics but movements produced or tending to produce themselves ; and two forces which impress on a body the same velocity in the same direction, are regarded as identical, however different their origin.
But although this manner of viewing the subject is fortunately now quite familiar to us, it is still left to Geometricians to effect an essential reform, if not in the conception itself; at least in our habitual language, in order to get rid entirely of the ancient metaphysical notion of forces, and to make out more exactly than has yet been done, the true point of view of Mechanics. We can now in a very precise manner characterise the general problem of rational Mechanics. It consists in determining the effect which different forces, acting simultaneously, will produce upon a given body, when we know the simple movement which would result from the separate action of each of them ; or, taking the question inversely, in determining the simple movements whose combination would produce a known compound movement. This enumeration shows exactly what are, of necessity, the known and the unknown terms of any mechanical question. We see that the study of the action of a single force is, properly speaking, never within the domain of rational Mechanics, where it is always supposed to be known, because the second general problem is never susceptible of resolution, except as being the converse of the first. The whole of Mechanics, therefore, bears essentially on the combination of forces, whether there results from that concourse a movement whose different circumstances it is necessary to study, or whether the body, owing to their mutual neutralization, is in a state of equilibrium, whose characteristic conditions are required to be determined. These two general problems, the one direct, the other inverse, the solution of which constitutes the science of Mechanics, have an equal importance as respects their application ;for sometimes the simple movement can be studied by observation, while the compound resultant can only be got at by theory, and vice versa. Comte gives several familiar illustrations of this.
Having thus expounded the general aim of Mechanics, Comte next considers the fundamental principles on which the science rests. As a preliminary step, he examines at length an important and necessary philosophical artifice used in Mechanics, without which no proposition on the abstract laws of equilibrium or movement could be established. This is the assumption that all bodies are inert ; that is, not that they are subject to what is called the law of inertia (a different point altogether), but that they are of themselves incapable of spontaneously modifying the action of forces applied to them. It is in reality a pure assumption, for every body, animate and inanimate, is, to a greater or less extent, in a state of spontaneous activity or movement. The contrary belief is a remnant of the old metaphysical notion that matter is by its nature essentially inert, and that all states of activity and movement are produced from without a notion in keeping with that stage of the mental development wherein movement is explained by supernatural entities or causes, but absolutely inconsistent with the positive point of view. Comte shows how the supposition of a body's inertness is made in Mechanics without impropriety. Movements in abstract mechanics being considered, as already observed, without reference to their mode of production, it matters not whether they come from within or from without. We can take the equivalent of the former in the latter.
It would be superfluous to say much, to make manifest the indispensable necessity of supposing bodies in this state of complete passiveness, where we have to consider only the external forces which are applied to them (as, for example, the movement of a falling body by the assumed entity attraction), for establishing the abstract laws of equilibrium and movement. We may conceive that if it were necessary to take into account any modification whatever that a body in virtue of its natural forces can make on the action of these external agencies, we could not establish the least general proposition in rational mechanics ; the more so, that this modification is far from being exactly known in the majority of cases.
Hence, it is only by commencing with a complete abstraction of them, so as to limit our thoughts to the reaction of the forces on each other, that it becomes possible to establish a science of abstract Mechanics. At a subsequent stage we pass from it to concrete Mechanics, by restoring to the bodies the active properties which are by nature inherent in them, but which at the outset we held as non-existent. It is this restitution which occasions our chief difficulty in passing from the abstract to the concrete in Mechanics, a difficulty which singularly limits in practice the important applications of this science, whose theoretical domain is, from its nature, necessarily indefinite. To give an idea of the extent of this fundamental obstacle, we may say, that in the present state of Mathematics there is but one natural and general property of bodies which we can conveniently take account of, that one being gravity, terrestrial and universal.
Hence the great applications of rational Mechanics have hitherto been really confined to celestial phenomena alone, and even to those of our own solar system; and here it is enough to consider only a general force of gravitation whose law is simple and well defined, and which, not withstanding, presents difficulties that we cannot yet overcome completely, when we would rigorously take into account all the secondary actions susceptible of appreciable effects. We may thus conceive how complex questions must become when we pass to terrestrial mechanics, where the greater part of the phenomena, even the simplest of them, probably never will allow, seeing the feebleness of our resources, of a purely rational and at the same time exact study of them according to the general laws of abstract Mechanics, although the knowledge of these laws (evidently indispensable on other accounts) can often lead to important indications.
As to the fundamental physical laws on which Rational Mechanics are founded, they are, according to Comte, three in number. They are generalised facts, the result of observation. They are the points from which the deductions of Science start, and are not themselves to be established ą priori, as metaphysicians believe. He exposes the insufficiency of the ą priori theory in each case, and the confusion of ideas which prevails in consequence of metaphysical conceptions on the subject.
The first of these laws is Kepler's law of inertia,a universal law, applicable to all bodies, animate and inanimate.
The second is Newton's law of action and reaction. The third is Galileo's discovery.
"This third fundamental law appears to me," he says, " to consist in what I propose to call the principles of independence or of coexistence of movements. It directly leads to what is popularly called the composition of forces. Galileo is, properly speaking, the real discoverer of this law, although he did not conceive it under the precise form which I have preferred giving it here. Considered under the simplest point of view, it comes to this general fact, that every movement strictly common to all the bodies of any system whatever, alters in no way the particular movements of those different bodies, as respects each other, these movements continuing to be the same as if the ensemble of the system were immoveable. In order to give the enunciation of this important principle a rigorous precision requiring no qualification, it is necessary to conceive that all the points of the system describe equal and parallel straight lines at the same time, and also that, whatever the velocity and direction of the general movement may be, it will not in the slightest degree affect the relative movements."
After discussing those three physical and fundamental laws of rational Mechanics, Comte gives an account of the chief divisions of the science.
The first and most important natural division of Me ehanics consists in distinguishing two orders of questions, according as the subject of inquiry is "the conditions of equilibrium," or "the laws of movement ;" whence Statics and Dynamics. A mere reference to this division suffices to make the necessity of it directly understood. Besides the real difference which evidently exists between these two fundamental classes of problems, it is easy to conceived priori that Statical questions, from their nature, must generally be much more easy to treat than questions of Dynamics. For in the first case, as has been justly said, we make abstraction of the time; that is to say, the phenomenon to be studied being necessarily instantaneous, we do not require to regard the variations which the forces of the system can undergo at different successive instants. It being, however, necessary to introduce the latter consideration into every dynamical question, it there forms a most fundamental element, and constitutes the principal difficulty.
It follows, from this radical difference, that when we treat Statics as a particular case of dynamics, the whole of the former corresponds only to by far the simplest part of the latter, to that, namely, which relates to the theory of uniform movements.
The importance of this division is very clearly verified by the general history of the actual development of the human mind. We see, in fact, that the ancients had acquired a knowledge of some fundamental and very essential truths relative to equilibrium, both as to solids and fluids, as may especially be seen in the beautiful researches of' Archimedes, although they were far from possessing a truly complete science of rational statics. Of Dynamics, on the contrary, they were entirely ignorant, even of the most elementary kind; the creation of this altogether modern science being due to Galileo.
After this fundamental division, the most important distinction to be made in Mechanics consists in the separation, both in Statics and Dynamics, of the study of solids from that of liquids. The discussion of this division, which Comte considers as subordinate to the other, occupies the remainder of this introductory lecture on rational Mechanics. The subject of the 16th lecture is Statics generally, of the 17th Dynamics, and the 18th is devoted to the consideration of the general theorems of rational Mechanics.
We must not follow this analysis into minuter detail. Indeed, only the extreme importance of Mathematics in its position in the hierarchy of the Sciences can warrant the length to which it has already extended.