The Positive Philosophy of Auguste Comte

"Yet this is the way our young physiologists proceed every day,—plunging into the study of living bodies, without any other preparation than a knowledge of a dead language or two, or... a superficial acquaintance with Physics and Chemistry, acquired without any philosophical method, or reference to any true point of departure in Natural philosophy."

"In the same way, with regard to Social phenomena, which are yet more complicated, what can be effected but by the rectification of the intellectual instrument, through an adequate study of the range of anterior phenomena? There are many who admit this: but they do not see how to set about the work, nor understand the Method itself, for want of the preparatory study; and thus, the admission remains barren, and social theories abide in the theological or metaphysical state, in spite of the efforts of those who believe themselves positive reformers."

"These, then, are the four points of view under which we have have recognized the importance of a Rational and Positive Classification."

"We have said nothing of Mathematical science. The omission was intentional; and the reason is no other than the vast importance of mathematics."

"In the present stage of our knowledge we must regard mathematics less as a constituent part of natural philosophy than as having been, since the time of Descartes and Newton, the true basis of the whole of natural philosophy; though it is... both the one and the other. To us it is of less value for the knowledge of which it consists, substantial and valuable as that knowledge is, than as being the most powerful instrument that the human mind can employ in the investigation of the laws of natural phenomena."

"Mathematics must be divided into two great sciences, quite distinct... Abstract Mathematics, or the Calculus (taking the word in its most extended sense), and Concrete Mathematics, which is composed of General Geometry and of Rational Mechanics."

"The Concrete part is necessarily founded on the Abstract, and it becomes in its turn the basis of all natural philosophy; all the phenomena of the universe being regarded, as far as possible, as geometrical or mechanical."

"The Abstract portion is the only one which is purely instrumental, it being simply an immense extension of natural logic to a certain order of deductions."

"Geometry and mechanics must, on the contrary, be regarded as true natural sciences, founded, like all others, on observation, though, by the extreme simplicity of their phenomena, they can be systematized to much greater perfection. It is this capacity which has caused the experimental character of their first principles to be too much lost sight of. But these two physical sciences have this peculiarity, that they are now, and will be more and more, employed rather as method than as doctrine."

"[I]n placing Mathematics at the head of Positive Philosophy, we are only extending the application of the principle which has governed our whole Classification. We are simply carrying back our principle to its first manifestation. Geometrical and Mechanical phenomena are the most general, the most simple, the most abstract of all,—the most irreducible to others, the most independent of them; serving, in fact, as a basis to all others."

"Therefore must Mathematics hold the first place in the hierarchy of the sciences, and be the point of departure of all Education, whether general or special."

"In an empirical way, this has hitherto been the custom,—a custom which arose from the great antiquity of mathematical science. We now see why it must be renewed on a rational foundation."

"We have now considered, in the form of a philosophical problem, the rational plan of the study of the Positive Philosophy. The order that results is this; an order which of all possible arrangements is the only one that accords with the natural manifestation of all phenomena. Mathematics, Astronomy, Physics, Chemistry, Physiology, Social Physics."

"[T]he direct measurement of a magnitude is often an impossible operation... We can rarely even measure a right line by another right line; and this is the simplest measurement of all."

"If there is difficulty about the measurement of lines, the embarrassment is much greater when we have to deal with surfaces, volumes, velocities, times, forces, etc., and in general with all other magnitudes susceptible of estimate, and, by their nature, difficult of direct measurement."

"[F]inding direct measurement so often impossible, we are compelled to devise means of doing it indirectly. Hence arose Mathematics."

"In observing a falling body, we are aware that two quantities are involved: the height from which the body falls, and the time occupied in its descent. ...In the language of mathematicians, they are functions of each other."

"We want to determine a distance not directly measurable. We shall conceive of it as making a part of some figure, or system of lines of some sort, of which the other parts are directly measurable; let us say a (for this is the simplest, and to it all others are reducible). ...The knowledge required is obtained by the mathematical labour of deducing the unknown distance from the observed elements, by means of the relation between them."

"Thus, when we once know the distance of any object, the observation, simple and always possible, of its apparent diameter, may disclose to us, with certainty, however indirectly, its real dimensions; and at length, by a series of analogous inquiries, its surface, its volume, even its weight, and a multitude of other qualities which might have seemed out of the reach of our knowledge for ever."

"It is by such labours that Man has learned to know, not only the distances of the planets from the earth and from each other, but their actual magnitude,—their true form, even to the inequalities on their surface, and... their respective masses, their mean densities ...etc."

"Through the power of mathematical theories, all this and very much more has been obtained by means of a very small number of straight lines, properly chosen, and a larger number of angles."

"We can... define Mathematical science... It has for its object the indirect measurement of magnitudes, and it proposes to determine magnitudes by each other, according to the precise relations which exist between them. Preceding definitions have given to Mathematics the character of an Art; this raises it at once to the rank of a true Science."

"[T]he spirit of Mathematics consists in regarding as mutually connected all the quantities which can be presented by any phenomenon whatsoever, in order to deduce all from each other."

"[T]here is evidently no phenomenon which may not be regarded as affording such considerations. Hence results the naturally indefinite extent, and the rigorous logical universality of Mathematical science. As for its actual practical extent, we shall see... hereafter."

"These explanations justify the name of Mathematics [from Greek: μάθημα, máthēma, 'knowledge, study, learning'] applied to the science we are considering. By itself it signifies Science [from Latin scientia 'knowledge']. The Greeks had no other, and we may call it the science; for its definition is neither more nor less (if we omit the specific notion of magnitudes) than the definition of all science whatsoever."

"All science consists in the co-ordination of facts; and no science could exist among isolated observations."

"It might even be said that Mathematics might enable us to dispense with all direct observation, by empowering us to deduce from the smallest possible number of immediate data the largest possible amount of results. Is not this the real use, both in speculation and in action, of the laws which we discover among natural phenomena?"

"If so, Mathematics merely urges to the ultimate degree, in its own way, researches which every real science pursues, in various inferior degrees, in its own sphere."

"Thus it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain. Any scientific education setting forth from any other point, is faulty in its basis."

"Every mathematical solution spontaneously separates into two parts. The inquiry being... the determination of unknown magnitudes, through their relation to the known..."

"[T]he student must... first... ascertain what these relations are... This... is the Concrete part of the inquiry. When it is accomplished, what remains is... the determination of unknown numbers, when we know by what relation they are connected with known numbers. This second operation is the Abstract part of the inquiry."

"The primary division of Mathematics is therefore into two great sciences:—Abstract Mathematics, and Concrete Mathematics. This division exists in all complete mathematical questions whatever, whether more or less simple."

"Recurring to the simplest case of a falling body, we must begin by learning the relation between the height from which it falls and the time occupied in falling. As Geometers say, we must find the ' which exists between them. Till this is done, there is no basis for a computation. This ascertainment may be extremely difficult, and it is incomparably the superior part of the problem."

"The true scientific spirit is so modern, that as far as we know, no one before Galileo had remarked the of velocity in a falling body, the natural supposition having been that the height was in uniform proportion to the time. This first inquiry issued in the discovery of the law of Galileo."

"The mathematical law may be easy to ascertain, and difficult to work; or it may be difficult to ascertain, and easy to work. In importance, in extent, and in difficulty, these two great sections of Mathematical Science will be seen hereafter to be equivalent."

"The Concrete must depend on the character of the objects examined, and must vary when new phenomena present themselves: whereas, the Abstract is wholly independent of the nature of the objects, and is concerned only with their numerical relations."

"The character of the Concrete is experimental, physical, phenomenal: while the Abstract is purely logical, rational."

"The s being once found... it is for the understanding, without external aid, to educe the results which these equations contain."

"[T]here are as yet only two great categories of phenomena whose equations are constantly known:—Geometrical and Mechanical... Thus, the Concrete part of Mathematics consists of Geometry and Rational Mechanics."

"Abstract Mathematics... is composed of what is called the Calculus, taking this word in its widest extension, which reaches from the simplest numerical operations to the highest combinations of transcendental analysis. Its proper object is to resolve all questions of numbers. Its starting-point is that which is the limit of Concrete Mathematics,—the knowledge of the precise relations—that is, the equations—between different magnitudes... considered simultaneously."

"The object of the Calculus, however indirect or complicated the relations may be, is to discover unknown quantities by the known. This science, though more advanced than any other, is... only at its beginning... but it is necessary, in order to define the nature of any science, to suppose it perfect."

"From an historical point of view, Mathematical Analysis appears to have arisen out of the contemplation of geometrical and mechanical facts; but it is... independent of these sciences, logically speaking."

"Analytical ideas are, above all others, universal, abstract, and simple; and geometrical and mechanical conceptions are necessarily founded on them. Mathematical Analysis is therefore the true rational basis of the whole system of our positive knowledge."

"If a single analytical question, brought to an abstract solution, involves the implicit solution of a multitude of physical questions, the mind is enabled to perceive relations between phenomena apparently isolated, and to extract from them the quality which they have in common."

"To the wonder of the student, unsuspected relations arise between problems which, instead of being, as they appeared before, wholly unconnected, turn out to be identical."

"There appears to be no connection between the determination of the direction of a curve at each of its points and that of the velocity of a body at each moment of its variable motion; yet, in the eyes of the geometer, these questions are but one."

"The perfection [of Mathematical Analysis] consists in the simplicity of the ideas contemplated; and not, as Condillac and others have supposed, to the conciseness and generality of the signs used as instruments of reasoning. The signs are of admirable use to work out the ideas, when once obtained; but, in fact, all the great analytical conceptions were formed without any essential aid from the signs."

"Subjects which are by their nature inferior in simplicity and generality cannot be raised to logical perfection by any artifice of scientific language."

"There is no inquiry which is not finally reducible to a question of Numbers..."

"[I]t becomes possible in geometry to treat curved lines as composed of an infinity of rectilinear elements, and curved surfaces as formed of plane elements; and, in mechanics, varied motions as an infinite series of uniform motions, succeeding each other at infinitely small intervals of time."