The Positive Philosophy of Auguste Comte

"Newton offered his conception under several different forms in succession. That... most commonly adopted... was called by himself, sometimes the Method of prime and ultimate Ratios, sometimes the Method of Limits... [W]e find nearly the equivalent of the facilities offered by the analysis of Leibnitz, which are merely considered from another point of view."

"Thus, curves will be regarded as the limits of a series of rectilinear polygons, and variable motions as the limits of an aggregate of uniform motions of continually nearer approximation, etc., etc."

"Such is... Newton's conception; or rather, that which Maclaurin and d'Alembert have offered as the most rational basis of the transcendental analysis, in the endeavour to fix and arrange Newton's ideas on the subject."

"Newton had another view... the Calculus of s and of fluents, founded on the general notion of velocities."

"Conceive of every curve as generated by a point affected by a motion varying according to any law whatever. The different quantities presented by the curve, the abscissa, the ordinate, the arc, the area, etc.,... regarded as simultaneously produced by successive degrees during this motion. The velocity with which each one will have been described will be called the fluxion of that quantity, which inversely would have been called its fluent. Henceforth, the transcendental analysis... forming directly the equations between the fluxions of the proposed quantities, to deduce from them afterwards, by a special Calculus, the equations between the fluents... any magnitudes... by the help of a suitable image... being produced by the motion of others."

"This method is... the same with that of limits complicated with the foreign idea of motion. ...[A] way of representing, by a comparison derived from mechanics, the method of prime and ultimate ratios, which alone is reducible to a calculus... without its being requisite for us to offer special proofs of this."

"Lagrange's conception consists, in its admirable simplicity, in considering the transcendental analysis to be a great algebraic artifice, by which... we must introduce, in the place of or with the primitive functions, their derived functions; that is... the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable."

"The Calculus of indirect functions... is destined here, as well as in... Leibnitz and Newton, to eliminate these derivatives, employed as auxiliaries, to deduce from their relations the corresponding equations between the primitive magnitudes."

"The transcendental analysis is then only a simple, but very considerable extension of ordinary analysis."

"It has long been a common practice with geometers to introduce... in the place of the magnitudes in question, their different powers, or their logarithms, or their sines, etc., in order to simplify the equations, and even to obtain them more easily. Successive derivation is a general artifice of the same nature, only of greater extent, and commanding, in consequence, much more important resources for this common object."

"Other theories have been proposed, such as Euler's Calculus of vanishing quantities: but they are merely modifications of the three just exhibited."

"Considering the three methods in regard to their destination... they all consist in the same general logical artifice... the introduction of a certain system of auxiliary magnitudes being substituted for the express object of facilitating the analytical expression of the mathematical laws of phenomena..."

"Whether these indirect equations are differential equations, according to Leibnitz, or equations of limits, according to Newton, or derived equations, according to Lagrange, the general procedure is evidently always the same. ...[T]he auxiliaries introduced are really identical, being only regarded from different points of view."

"The transcendental analysis... examined abstractly and in its principle, is always the same, whatever conception... and the processes... are necessarily identical in these different methods, which must therefore, under any application whatever, lead to rigorously uniform results."

"The method of Leibnitz has... the advantage in the rapidity and ease with which it effects the formation of equations between auxiliary magnitudes. We owe to its use the high perfection attained by all the general theories of geometry and mechanics."

"Lagrange... after having reconstructed the analysis on a new basis, rendered a candid and decisive homage to the conception of Leibnitz, by employing it exclusively in the whole system of his "Analytical Mechanics.""

"Yet... we... admit, with Lagrange, that the conception of Leibnitz is radically vicious in its logical relations. He himself declared the notion of infinitely small quantities to be a false idea: and it is... impossible to conceive of them clearly..."

"This false idea bears... the characteristic impress of the metaphysical age of its birth and tendencies of its originator. By the ingenious principle of the compensation of errors, we may... explain the necessary exactness of the processes... but it is a radical inconvenience to be obliged to indicate, in Mathematics, two clashes of reasonings so unlike, as that the one order are perfectly rigorous, while by the others we designedly commit errors which have to be afterwards compensated."

"[T]he infinitesimal method exhibits the very serious defect of breaking the unity of abstract mathematics by creating a transcendental analysis founded upon principles widely different from those which serve as a basis to ordinary analysis."

"Newton's conception is free from the logical objections imputable to that of Leibnitz. The notion of limits is in fact remarkable for its distinctness and precision. The equations are... regarded as exact from their origin; and the general rules of reasoning are as constantly observed as in ordinary analysis. But it is weak in resources, and embarrassing in operation, compared with the method. In its applications, the relative inferiority of this theory is very strongly marked. It also separates the ordinary and transcendental analysis, though not so conspicuously as the theory of Leibnitz."

"As Lagrange remarked, the idea of limits, though clear and exact, is not the less a foreign idea, on which analytical theories ought not to be dependent."

"This perfect unity of analysis, and a purely abstract character in the fundamental ideas, are found in the conception of Lagrange... alone. It is therefore the most philosophical..."

"Discarding every heterogeneous consideration, Lagrange reduced the transcendental analysis to its proper character,—that of presenting a very extensive class of analytical transformations, which facilitate... the expression of the conditions of the various problems. This exhibits the conception as a simple extension of ordinary analysis. It is a superior algebra. This philosophical superiority marks it for adoption as the final theory of transcendental analysis; but it presents too many difficulties in its application..."

"Lagrange... had great difficulty in rediscovering, by his own method, the principal results already obtained by the infinitesimal method, on general questions in geometry and mechanics. ...Though Lagrange... obtained results in some cases which other men would have despaired of... his conception has thus far remained... essentially unsuited to applications."

"In all the other departments of mathematical science, the consideration of different methods for a single class of questions may be useful, apart from the historical interest... but it is not indispensable. Here... it is strictly indispensable. Without it there can be no philosophical judgment of this admirable creation of the human mind; nor any success and facility in the use of this powerful instrument."

"[A]lmost all geometers employ the terms Differential Calculus and Integral Calculus established by Leibnitz. Newton... called the first the Calculus of Fluxions, and the second the Calculus of Fluents... According to the theory of Lagrange, the one... the Calculus of Derived Functions, and the other the Calculus of Primitive Functions."

"The differential calculus is... the rational basis of the . ...[T]en simple functions constitute the elements of our analysis. We cannot know how to integrate directly any other differential expressions than those produced by the differentiation of those ten... The art of integration consists therefore in bringing all the other cases, as far as possible, to depend wholly on this small number of simple functions."

"The entire system of the differential calculus is simple and perfect, while the integral calculus remains extremely imperfect."

"I will sketch the great impending philosophical regeneration from the four points of view... the scientific, or rather rational; the moral; the political; and finally, the aesthetic."

"The positive state will... be one of entire intellectual consistency, such as has never yet existed in an equal degree, among the best organized and most advanced minds."

"The kind of speculative unity which existed under the polytheistic system, when all human conceptions presented a uniformly religious aspect, was liable to perpetual disturbance from a spontaneous positivity of ideas..."

"In the scholastic period, the nearest approach to harmony was a precarious and incomplete equilibrium: and the present transition involves such contradiction that the highest minds are perpetually subject to three incompatible systems."

"[W]hen we shall habitually restrict our inquiries to the simplest affairs in life; and... to accessible subjects, and understand... the relative character of all human knowledge, our approximation towards the truth, which can never be completely attained by human faculties, will be thorough and satisfactory... and it will proceed as far as the state of human progress will admit."

"We have never experienced, and can... only imperfectly imagine, the state of unmingled conviction with which men will regard that natural order when all disturbing intrusions, such as... from lingering theological influences, shall have been cast out by the spontaneous certainty of the invariableness of natural laws."

"[T]he absolute tendencies of the old philosophies prevent our forming any adequate conception of the privilege of intellectual liberty which is secured by positive philosophy."

"Our existing state is so unlike all this, that we cannot... estimate the importance and rapidity of progress which will be thus secured; our only measure being the ground gained during the last three centuries, under and imperfect and even vicious system, which has occasioned the waste of the greater part of our intellectual labour."

"In abstract science, men will be spared the preliminary labour which has hitherto involved vast and various error, scientific and logical, and will be set forward far and firmly by the full establishment of the rational method."

"When the ascendancy of the sociological spirit shall have driven out that of the scientific, there will be an end of the vain struggle to connect every order of phenomena with one set of laws, and the desired unity will be seen to consist in the agreement of various orders of laws,—each set governing and actuating its own province; and thus will the free expansion of each kind of knowledge be provided for, while all are analogous in their method of treatment, and identical in their destination."

"In the final... positive state, the mind has given over the vain search after Absolute notions, the origin and destination of the universe, and the causes of phenomena, and applies itself to the study of their laws,—that is, their invariable relations of succession and resemblance."

"Reasoning and observation, duly combined, are the means of this knowledge."

"What is now understood when we speak of an explanation of facts is simply the establishment of a connection between single phenomena and some general facts, the number of which continually diminishes with the progress of science."

"There is no science which, having attained the positive stage, does not bear marks of having passed through the others. Some time since it was... composed... of metaphysical abstractions; and, further back... it took its form from theological conceptions. ...[O]ur most advanced sciences still bear very evident marks of the two earlier periods ..."

"The progress of the individual mind is not only an illustration, but an indirect evidence of that of the general mind. The point of departure of the individual and of the race being the same, the phases of the mind of a man correspond to the epochs of the mind of the race. Now, each of us is aware, if he looks back upon his own history, that he was a theologian in his childhood, a metaphysician in his youth, and a natural philosopher in his manhood. All men who are up to their age can verify this for themselves."

"[I]t is to the chimeras of astrology and alchemy that we owe the long series of observations and experiments on which our positive science is based. Kepler felt this on behalf of astronomy, and Berthollet on behalf of chemistry. Thus was a spontaneous philosophy, the theological, the only possible beginning, method, and provisional system, out of which the Positive philosophy could grow."

"M. Fourier, in his fine series of researches on , has given us all the most important and precise laws of the phenomena... and many large and new truths, without once inquiring into its nature, as his predecessors had done when they disputed about calorific matter and the action of an universal ether. In treating his subject in the Positive method, he finds inexhaustible material for all his activity of research, without betaking himself to insoluble questions."

"[I]f we must fix upon some marked period, to serve as a rallying point, it must be that,—about two centuries ago,—when the human mind was astir under the precepts of Bacon, the conceptions of Descartes, and the discoveries of Galileo. Then it was that the spirit of the Positive philosophy rose up in opposition to that of the superstitious and scholastic systems which had hitherto obscured the true character of all science. Since that date, the progress of the Positive philosophy, and the decline of the other two, have been so marked that no rational mind now doubts that the revolution is destined to go on to its completion,—every branch of knowledge being, sooner or later, brought within the operation of Positive philosophy."

"Now that the human mind has grasped celestial and terrestrial physics,—mechanical and chemical; organic physics, both vegetable and animal,—there remains one science, to fill up the series of sciences of observation,—Social physics. This is... the principal aim of the present work to establish."

"The purpose of this work is not to give an account of the Natural Sciences. Besides that it would be endless, and that it would require a scientific preparation such as no one man possesses, it would be apart from our object, which is to go through a course of not Positive Science, but Positive Philosophy. We have only to consider each fundamental science in its relation to the whole positive system... with regard to its methods and its chief results."

"In the primitive state of human knowledge there is no regular division of intellectual labour. Every student cultivates all the sciences. As knowledge accrues, the sciences part off; and students devote themselves each to some one branch. It is owing to this division... and concentration of whole minds upon a single department, that science has made so prodigious an advance... But... we cannot be blind to the eminent disadvantages which arise from... limitation... to a particular study. It is inevitable that each should be possessed with exclusive notions, and be therefore incapable of the general superiority of ancient students, who... owed that general superiority to the inferiority of their knowledge. ...[T]his is the weak side of the positive philosophy, by which it may yet be attacked, with some hope of success, by the adherents of the theological and metaphysical systems."

"As to the remedy... It lies in perfecting the division of employments itself,—in carrying it one degree higher,—in constituting one more speciality from the study of scientific generalities."