The Positive Philosophy of Auguste Comte

The Positive Philosophy of Auguste Comte, published in 1853, is Harriet Martineau's free translation and condensation of Auguste Comte's 7 volume Cours de philosophie positive, written 1830–1842.

August Comte, (1849)

QuotesEdit

from the 1896 edition in Bohn's Philosophical Library, unless otherwise noted: Vol. I, Vol. II, Vol. III.

Vol. 1, Introduction, Ch. 1Edit

Account of the Aim of this Work.—View of the Nature and Importance of the Positive Philosophy.
  • In order to understand the true value and character of the Positive Philosophy, we must take a brief general view of the progressive course of the human mind, regarded as a whole; for no conception can be understood otherwise than through its history.
  • 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 Heat, 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.
  • Let us have a new class of students... to take the respective sciences... determine the spirit of each, ascertain their relations and mutual connection, and reduce their respective principles to the smallest number of general principles...
  • [L]et other students be prepared for their special pursuit... so as to profit by the labours of the students of generalities, and so as to correct reciprocally, under that guidance, the results obtained by each.
  • [R]efer to a saying of M. de Blainille, in his work on Comparative Anatomy, that every active... being, may be regarded under two relations—the Statical and the Dynamical... apply this classification to the intellectual functions.
  • After two thousand years of psychological pursuit, no one proposition is established to the satisfaction of its followers. ...This interior observation gives birth to almost as many theories as there are observers.
  • The Positive Method can be judged of only in action. ...the work on which it is employed.
  • This... is the first great result of the Positive Philosophy—the manifestation by experiment of the laws which rule the Intellect in the investigation of truth; and, as a consequence the knowledge of the general rules suitable for that object.
  • The second effect of the Positive Philosophy... regenerate Education. [O]ur... education, still essentially theological, metaphysical, and literary, must be superseded by a Positive training... [E]verything yet done is inadequate to the object.
  • The present exclusive speciality of our pursuits, and the consequent isolation of the sciences, spoil our teaching.
  • If any student desires to form an idea of natural philosophy as a whole, he is compelled to go through each department as it is now taught, as if he were to be only an astronomer, or only a chemist... his training must remain very imperfect. And yet... he should obtain general positive conceptions of all the classes of natural phenomena.
  • In order to [attain] this regeneration of our intellectual system, it is necessary that the sciences, considered as branches from one trunk, should yield us, as a whole, their chief methods and their most important results.
  • The subject of our researches is one: we divide it for our convenience... to deal... more easily with its difficulties. But... sometimes... especially with the most important doctrines of each science... we need what we cannot obtain under the present isolation... a combination of several special points of view; and for want of this, very important problems wait for their solution much longer...
  • Descartes' grand conception with regard to analytical geometry... changed the whole aspect of mathematical science, and... it issued from the union of two sciences which had always before been separately regarded and pursued.
  • The case of pending questions is yet more impressive; as... in Chemistry, the doctrine of Definite Proportions. ...[I]n order to determine whether it is a law of nature that atoms should necessarily combine in fixed numbers... the chemical point of view should be united with the physiological. ...[P]hysiology must unite with chemistry to inform us whether azote is simple or compound, and to institute a new series of researches upon the relation between the composition of living bodies and their mode of alimentation.
  • The Positive Philosophy offers the only solid basis for that Social Reorganization which must succeed the critical condition in which the most civilized nations are now living.
  • Ideas govern the world, or throw it into chaos... [i.e.,] all social mechanism rests upon Opinions. The great political and moral crisis that societies are now undergoing... arise out of intellectual anarchy. ...[W]e are suffering under an utter disagreement which may be called universal.
  • Till a certain number of general ideas can be acknowledged as a rallying-point of social doctrine, the nations will remain in a revolutionary state...
  • [W]henever the necessary agreement on first principles can be obtained, appropriate institutions will issue from them, without shock or resistance; for the causes of disorder will have been arrested by the mere fact of the agreement.
  • [T]he existing disorder is... accounted for by the existence... of three incompatible philosophies,—the theological, the metaphysical, and the positive. Any one of these might alone secure... social order; but while the three co-exist, it is impossible... to understand one another...
  • [W]e have only to ascertain which of the philosophies must, in the nature of things, prevail...
  • [A]ll considerations... point to the Positive Philosophy as the one destined to prevail. It alone has been advancing during a course of centuries, throughout which the others have been declining. The fact is incontestable.
  • This general revolution of the human mind is nearly accomplished. We have only to complete the Positive Philosophy by bringing Social phenomena within its comprehension...
  • The marked preference which almost all minds, from the highest to the commonest, accord to positive knowledge over vague and mystical conceptions, is a pledge of what the reception of this philosophy will be...
  • When it has become complete, its supremacy will take place spontaneously, and will re-establish order throughout society.
  • It is time to complete the vast intellectual operation begun by Bacon, Descartes, and Galileo, by constructing the system of general ideas which must henceforth prevail among the human race. This is the way to put an end to the revolutionary crisis which is tormenting the civilized nations of the world.
  • [I]t must not be supposed that we are going to study this vast variety as proceeding from a single principle, and as subjected to a single law. There is something... chimerical in attempts at universal explanation by a single law...
  • Our intellectual resources are too narrow, and the universe is too complex, to leave any hope that it will ever be within our power to carry scientific perfection to its last degree of simplicity.
  • The only necessary unity is that of Method, which is already in great part established.
  • As for the doctrine, it need not be one; it is enough that it should be homogeneous.
  • While pursuing the philosophical aim of all science, the lessening of the number of general laws requisite for the explanation of natural phenomena, we shall regard as presumptuous every attempt, in all future time, to reduce them rigorously to one.

Vol. 1, Introduction, Ch. 2Edit

View of the Hierarchy of the Positive Sciences.
  • Classification of the Sciences... have failed, through one fault or another, to command assent: so that there are almost as many schemes as there are individuals to propose them.
  • [T]he distribution of the sciences, having become a somewhat discredited task, has of late been undertaken chiefly by persons who have no sound knowledge of any science at all.
  • A... reason... is, the want of homogeneousness in the different parts of the intellectual system,—some having successively become positive, while others remain theological or metaphysical.
  • Every attempt at a distribution has failed from this cause... the enterprise was premature; and... it was useless to undertake it till our principal scientific conceptions should all have become positive.
  • [T]he time has arrived for laying down a sound and durable system of scientific order.
  • We may derive encouragement from the example set by... botanists and zoologists... viz. that the classification must proceed from the study of the things to be classified, and must by no means be determined by à priori considerations.
  • [T]he mutual dependence of the sciences... resulting from that of the corresponding phenomena,—must determine the arrangement of the system of human knowledge.
  • Before proceeding to investigate this mutual dependence, we have only to determine the real bounds of the classification... [i.e.,] to settle what we mean by human knowledge...
  • The field of human knowledge is either speculation or action: and thus, we are accustomed to divide our knowledge into the theoretical and the practical. ...[I]n this inquiry, we have to do only with the theoretical.
  • [S]peculation is our material, and not the application of it,—except... to throw back light on its speculative origin.
  • This is probably what Bacon meant by that First Philosophy which he declared to be an extract from the whole of Science... so differently and... strangely interpreted by his metaphysical commentators.
  • Man's study of nature must furnish the only basis of his action upon nature; for it is only by knowing the laws of phenomena, and thus being able to foresee them, that we can, in active life, set them to modify one another for our advantage.
  • Our direct natural power over everything about us is extremely weak, and altogether disproportioned to our needs.
  • Whenever we effect anything great it is through a knowledge of natural laws...
  • The relation of science to art may be summed up in a brief expression:
    From Science comes Prevision: from Prevision comes Action.
  • We must not... fall into the error of our time, of regarding Science chiefly as a basis of Art.
  • However great may be the services rendered to Industry by science, however true may be the saying that Knowledge is Power, we must never forget that the sciences have a higher destination... [and] more direct;—that of satisfying the craving of our understanding to know the laws of phenomena.
  • This need of disposing facts in a comprehensible order (...the proper object of all scientific theories) is so inherent in our organization, that if we could not satisfy it by positive conceptions, we must inevitably return to those theological and metaphysical explanations which had their origin in this very fact of human nature.—It is this original tendency which acts as a preservative, in the minds of men of science, against the narrowness and incompleteness which the practical habits of our age are apt to produce. It is through this that we are able to maintain just and noble ideas of the importance and destination of the sciences; and if it were not thus, the human understanding would soon, as Condorcet has observed, come to a stand, even as to the practical applications for the sake of which higher things had been sacrificed; for...
  • [I]f the arts flow from science, the neglect of science must destroy the consequent arts.
  • Some of the most important arts are derived from speculations pursued during long ages with a purely scientific intention.
  • [T]he ancient Greek geometers delighted themselves with beautiful speculations on Conic Sections; those speculations wrought, after... generations, the renovation of astronomy; and out of this has the art of navigation attained a perfection which it never could have reached otherwise than through the speculative labours of Archimedes and Apollonius...
  • [T]o use Condorcet's illustration, "the sailor who is preserved from shipwreck by the exact observation of the longitude, owes his life to a theory conceived two thousand years before by men of genius who had in view simply geometrical speculations."
  • [A]n intermediate class is rising up, whose particular destination is to organize the relations of theory and practice; such as the engineers, who do not labour in the advancement of science, but who study it in its existing state, to apply it to practical purposes. Such classes are furnishing us with the elements of a future body of doctrine on the theories of the different arts.
  • If we remember that several sciences are implicated in every important art,—that...[e.g.,] a true theory of Agriculture requires a combination of physiological, chemical, mechanical, and even astronomical and mathematical science,—it will be evident that true theories of the arts must wait for a large and equable development of these constituent sciences.
  • We must distinguish between the two classes of Natural science;—the abstract or general, which have for their object the discovery of the laws which regulate phenomena in all conceivable cases: and the concrete, particular, or descriptive, which are sometimes called Natural sciences in a restricted sense, whose function it is to apply these laws to the actual history of existing beings.
  • The first are fundamental; and our business is with them alone, as the second are derived, and however important, not rising into the rank of our subjects of contemplation.
  • We may say of Concrete Physics, as these secondary sciences are called, the same thing that we said of theories of the arts,—that they require a preliminary knowledge of several sciences, and an advance of those sciences not yet achieved...
  • At a future time Concrete Physics will have made progress, according to the development of Abstract Physics, and will afford a mass of less incoherent materials than those which it now presents.
  • At present, too few of the students of these secondary sciences appear... aware that a due acquaintance with the primary sciences is requisite...
  • We have now considered,
    First, that science being composed of speculative knowledge and of practical knowledge, we have to deal only with the first; and
    Second, that theoretical knowledge, or science properly so called, being divided into general and particular, or abstract and concrete science, we have again to deal only with the first.
  • The classification of the sciences is not so easy a matter... However natural it may be, it will always involve something, if not arbitrary, at least artificial; and in so far, it will always involve imperfection.
  • It is impossible to fulfil... rigorously, the object of presenting the sciences in their natural connection, and according to their mutual dependence, so as to avoid... a vicious circle.
  • Every science may be exhibited under two methods or procedures, the Historical and the Dogmatic.
  • These are wholly distinct from each other, and any other method can be nothing but some combination of these two.
  • By the first method knowledge is presented in the same order in which it was actually obtained... together with the way in which it was obtained.
  • By the second, the system of ideas is presented as it might be conceived of at this day, by a mind which, duly prepared and placed at the right point of view, should begin to reconstitute the science as a whole.
  • A new science must be pursued historically, the only thing to be done being to study in chronological order the different works which have contributed to the progress...
  • But when such materials have become recast to form a general system, to meet the demand for a more natural logical order, it is because the science is too far advanced for the historical order to be practicable or suitable.
  • The more discoveries are made, the greater becomes the labour of the historical method of study, and the more effectual the dogmatic, because the new conceptions bring forward the earlier ones in a fresh light.
  • Thus, the education of an ancient geometer consisted simply in the study, in their due order, of the very small number of original treatises then existing on the different parts of geometry. The writings of Archimedes and Apollonius were, in fact, about all.
  • On the contrary, a modern geometer commonly finishes his education without having read a single original work dating further back than the most recent discoveries, which cannot be known by any other means.
  • Thus the Dogmatic Method is for ever superseding the Historical, as we advance to a higher position in science.
  • If every mind had to pass through all the stages that every predecessor in the study had gone through... however easy it is to learn rather than invent, it would be impossible to effect the purpose of education,—to place the student on the vantage-ground gained by the labours of all the men who have gone before. By the dogmatic method this is done, even though the living student may have only an ordinary intellect, and the dead may have been men of lofty genius.
  • By the dogmatic method, therefore, must every advanced science be attained, with so much of the historical combined with it as is rendered necessary...
  • The only objection to the preference of the Dogmatic method is that it does not show how the science was attained; but... this is the case also with the Historical method.
  • To pursue a science historically is quite a different thing from learning the history of its progress. This last pertains to the study of human history...
  • [A] science cannot be completely understood without a knowledge of how it arose; and... a dogmatic knowledge of any science is necessary to an understanding of its history; and therefore we shall notice, in treating of the fundamental sciences, the incidents of their origin, when distinct and illustrative; and we shall use their history, in a scientific sense, in our treatment of Social Physics; but the historical study, important, even essential, as it is, remains entirely distinct from the proper dogmatic study of science.
  • Great confusion would arise from any attempt to adhere strictly to historical order in our exposition of the sciences, for they have not all advanced at the same rate; and we must be for ever borrowing from each some fact to illustrate another, without regard to priority of origin.
  • Thus... in the system of the sciences, astronomy must come before physics, properly so called: and yet, several branches of physics, above all, optics, are indispensable to the complete exposition of astronomy.
  • In the main... our classification agrees with the history of science; the more general and simple sciences actually occurring first and advancing best... being followed by the more complex and restricted, though all were, since the earliest times, enlarging simultaneously.
  • Our problem is, then, to find the one rational order, among a host of possible systems. ...This order is determined by the degree of simplicity, or... [i.e.,] generality of their phenomena. Hence results their successive dependence, and the greater or lesser facility for being studied. ...à priori... the most simple phenomena must be the most general; for whatever is observed in the greatest number of cases is of course the most disengaged from the incidents of particular cases.
  • We must begin then with the study of the most general or simple phenomena, going on successively to the more particular or complex.
  • [T]he most general and simple phenomena are the furthest removed from Man's ordinary sphere, and must thereby be studied in a calmer and more rational frame of mind than those in which he is more nearly implicated; and this constitutes a new ground for the corresponding sciences being developed more rapidly.
  • We are first struck by the clear division of all natural phenomena into two classes—of inorganic and of organic bodies. The organized are evidently, in fact, more complex and less general than the inorganic, and depend upon them...
  • Therefore... physiological study should begin with inorganic phenomena...
  • We have not to investigate the nature of either; for the positive philosophy does not inquire into natures.
  • [T]he general laws of inorganic physics must be established before we can proceed with success to the examination of a dependent class...
  • Each of these great halves of natural philosophy has subdivisions. Inorganic physics must... be divided into two sections—of celestial and terrestrial phenomena. Thus we have Astronomy, geometrical and mechanical, and Terrestrial Physics.
  • Astronomical phenomena are the most general, simple, and abstract of all; and therefore the study of natural philosophy must... begin with them. ...[T]he laws to which they are subject influence all others whatsoever.
  • The general effects of gravitation preponderate, in all terrestrial phenomena, over all effects which may be peculiar to them, and modify the original ones.
  • It follows that the analysis of the simplest terrestrial phenomenon, not only chemical, but even purely mechanical, presents a greater complication than the most compound astronomical phenomenon.
  • The most difficult astronomical question involves less intricacy than the simple movement of even a solid body, when the determining circumstances are to be computed.
  • [W]e find a natural division of Terrestrial Physics into two, according as we regard bodies in their mechanical or their chemical character. Hence we have Physics... and Chemistry. Again, the second class must be studied through the first.
  • Chemical phenomena are more complicated than mechanical, and depend upon them, without influencing them in return. [A]ll chemical action is first submitted to the influence of weight, heat, electricity, etc., and presents moreover something which modifies all these. Thus, while it follows Physics, it presents itself as a distinct science.
  • An analogous division arises in the other half of Natural Philosophy—the science of organized bodies.
    • II. Organic
  • Here we find ourselves presented with two orders of phenomena; those which relate to the individual, and those which relate to the species, especially when it is gregarious. With regard to Man, especially, this distinction is fundamental.
  • [W]e have two great sections in organic physics—Physiology... and Social Physics, which is dependent on it.
  • In all Social phenomena we perceive the working of the physiological laws of the individual; and moreover something which modifies their effects, and which belongs to the influence of individuals over each other—singularly complicated in the case of the human race by the influence of generations on their successors.
  • [O]ur social science must issue from that which relates to the life of the individual. On the other hand, there is no occasion to suppose, as some eminent physiologists have done, that Social Physics is only an appendage to physiology. The phenomena of the two are not identical, though they are homogeneous; and it is of high importance to hold the two sciences separate.
  • As social conditions modify the operation of physiological laws, Social Physics must have a set of observations of its own.
  • It would be easy to make the divisions of the Organic... by dividing physiology into vegetable and animal, according to popular custom. But this distinction... hardly extends into those Abstract Physics... Vegetables and animals come alike... when our object is to learn the general laws of life— ...[i.e.,]to study physiology. ...[T]he distinction grows ever fainter and more dubious with new discoveries, it bears no relation to our plan of research...
  • Thus we have before us Five fundamental Sciences in successive dependence,—Astronomy, Physics, Chemistry, Physiology, and... Social Physics.
  • The first considers the most general, simple, abstract, and remote phenomena known to us, and those which affect all others without being affected by them. The last considers the most particular, compound, concrete phenomena, and those which are the most interesting to Man.
  • Between these two, the degrees of speciality, of complexity, and individuality are in regular proportion to the place of the respective sciences in the scale exhibited.
  • This—casting out everything arbitrary—we must regard as the true filiation of the sciences; and in it we find the plan of this work.
  • [W]e shall find that the same principle which gives this order to the whole body of science arranges the parts of each science; and its soundness will therefore be freshly attested us often as it presents itself afresh.
  • There is no refusing a principle which distributes the interior of each science after the same method with the aggregate sciences.
  • This gradation is in essential conformity with the order which has spontaneously taken place among the branches of natural philosophy, when pursued separately, and without any purpose of establishing such order.
  • Such an accordance is a strong presumption that the arrangement is natural. ...[I]t coincides with the actual development of natural philosophy.
  • If no leading science can be effectually pursued otherwise than through those which precede it in the scale, it is evident that no vast development of any science could take place prior to the great astronomical discoveries to which we owe the impulse given to the whole. The progression may since have been simultaneous; but it has taken place in the order we have recognized.
  • This consideration is so important that it is difficult to understand without it the history of the human mind. The general law which governs this history... cannot be verified, unless we combine it with the scientific gradation just laid down: for it is according to this gradation that the different human theories have attained in succession the theological state, the metaphysical, and finally the positive.
  • If we do not bear in mind the law which governs progression, we shall encounter insurmountable difficulties: for it is clear that the theological or metaphysical state of some fundamental theories must have temporarily coincided with the positive state of others which precede them in our established gradation, and actually have at times coincided with them; and this must involve the law itself in an obscurity which can be cleared up only by the classification we have proposed.
  • [T]his classification marks, with precision, the relative perfection of the different sciences, which consists in the degree of precision of knowledge, and in the relation of its different branches.
  • [T]he more general, simple, and abstract any phenomena are, the less they depend on others, and the more precise they are in themselves, and the more clear in their relations with each other.
  • Thus, organic phenomena are less exact and systematic than inorganic; and of these again terrestrial are less exact and systematic than those of astronomy.
  • This fact is completely accounted for by the gradation we have laid down; and we shall see... that the possibility of applying mathematical analysis to the study of phenomena is exactly in proportion to the rank which they hold in the scale of the whole.
  • We must beware of confounding the degree of precision which we are able to attain in regard to any science, with the certainty of the science itself.
  • The certainty of science, and our precision in the knowledge of it, are two very different things, which have been too often confounded; and are so still...
  • A very absurd proposition may be very precise; as if we should say, for instance, that the sum of the angles of a triangle is equal to three right angles; and a very certain proposition may be wanting in precision in our statement of it; as, for instance, when we assert that every man will die.
  • If the different sciences offer to us a varying degree of precision, it is from no want of certainty in themselves, but of our mastery of their phenomena.
  • The most interesting property of our formula of gradation is its effect on education, both general and scientific. This is its direct and unquestionable result.
  • [N]o science can be effectually pursued without the preparation of a competent knowledge of the anterior sciences on which it depends.
  • As such conditions are, as yet, rarely fulfilled, and as no organization exists for their fulfilment, there is amongst us, in fact, no rational scientific education.
  • To this may be attributed, in great part, the imperfection of even the most important sciences at this day.
  • If the fact is so in regard to scientific education, it is no less striking in regard to general education.
  • Our intellectual system cannot be renovated till the natural sciences are studied in their proper order.
  • Even the highest understandings are apt to associate their ideas according to the order in which they were received: and it is only an intellect here and there, in any age, which in its utmost vigour can, like Bacon, Descartes, and Leibnitz, make a clearance in their field of knowledge, so as to reconstruct from the foundation their system of ideas.
  • Such is the operation of our great law upon scientific education through its effect on Doctrine. We cannot appreciate it duly without seeing how it affects Method.
  • As the phenomena which are homogeneous have been classed under one science, while those which belong to other sciences are heterogeneous, it follows that the Positive Method must be constantly modified in an uniform manner in the Range of the same fundamental science, and will undergo modifications, different and more and more compound, in passing from one science to another.
  • [I]f... we cannot understand the positive method in the abstract, but only by its application... we can have no adequate conception of it but by studying it in its varieties of application. No one science... could exhibit it. Though the Method is always the same, its procedure is varied.
  • [I]t should be Observation with regard to one kind of phenomena, and Experiment with regard to another; and different kinds of experiment, according to the case. In the same way, a general precept, derived from one fundamental science, however applicable to another, must have its spirit preserved by a reference to its origin; as in the case of the theory of Classifications.
  • The best idea of the Positive Method would... be obtained by the study of the most primitive and exalted of the sciences, if we were confined to one; but this isolated view would give no idea of its capacity of application to others in a modified form.
  • Each science has its own proper advantages; and without some knowledge of them all, no conception can be formed of the power of the Method.
  • It is necessary, not only to have a general knowledge of all the sciences, but to study them in their order. What can come of a study of complicated phenomena, if the student have not learned, by the contemplation of the simpler, what a Law is, what it is to Observe; what a Positive conception is; and even what a chain of reasoning is?
  • 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.

Vol. 1, Book 1. Mathematics. Ch. 1.Edit

Mathematics, Abstract and Concrete.
  • [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 triangle (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 equation 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 acceleration 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 equations 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...
  • Nothing can appear less like a mathematical inquiry than the study of living bodies in a state of disease; yet, in studying the cure... we are endeavouring to ascertain the quantities of the different agents which are to modify the organism, in order to bring it to its natural state...
  • Kant has divided human ideas into the two categories of quantity and quality, which, if true, would destroy the universality of Mathematics; but Descartes' fundamental conception of the relation of the concrete to the abstract in Mathematics abolishes this division, and proves that all ideas of quality are reducible to ideas of quantity. He had in view geometrical phenomena... but his successors have included... first, mechanical phenomena, and, more recently, those of heat. There are now no geometers who do not consider it of universal application, and admit that every phenomenon may be as logically capable of being represented by an equation as a curve or a motion, if only we were always capable (which we are very far from being) of first discovering, and then resolving it.
  • The limitations of Mathematical science are not, then, in its nature. The limitations are in our intelligence: and by these we find the domain of the science remarkably restricted, in proportion as phenomeua, in becoming special, become complex.
  • [T]he reduction [to mathematics] cannot be made by us except in the case of the simplest and most general phenomena. ...[A]t the utmost, it is only the phenomena of the first three classes... of Inorganic Physics,—that we can even hope to subject to the process.
  • By the rapidity of their changes, and their incessant numerical variations, vital phenomena are, practically, placed in opposition to mathematical processes.
  • Social phenomena, being more complicated still, are even more out of the question, as subjects for mathematical analysis.
  • It is not that a mathematical basis does not exist in these cases... but that our faculties are too limited for the working of problems so intricate.
  • To the popular mind it may appear strange... that we know so much as we do about the planets. But... that class of phenomena is the most simple of all within our cognizance. The most complex problem... they present is the influence of a third body acting in the same way on two which are tending towards each other in virtue of gravitation; and this is a more simple question than any terrestrial problem... We have, however, attained only approximate solutions...
  • [T]he high perfection to which solar astronomy has been brought... is owing to our having profited by... facilities... accidental, which... our planetary system presents. The planets which compose it are few; their masses are very unequal, and much less than that of the sun; they are far distant from each other; their forms are nearly spherical; their orbits are nearly circular, and only slightly inclined in relation to each other; and so on.
  • Their perturbations are, in consequence, inconsiderable... and all we have to do is usually to take into the account, together with the influence of the sun on each planet, the influence of one other planet, capable, by its size and its nearness, of occasioning perceptible derangements.
  • If any of the conditions mentioned above had been different, though the law of gravitation had existed as it is, we might not... have discovered it.
  • The most difficult sciences must remain, for an indefinite time, in that preliminary state which prepares for the others the time when they too may become capable of mathematical treatment. Our business is to study phenomena... abstaining from introducing considerations of quantities, and mathematical laws... beyond our power to apply.
  • We owe to Mathematics both the origin of Positive Philosophy and its Method. When this method was introduced into the other sciences, it was natural that it should be urged too far. But each science modified the method by the operation of its own peculiar phenomena.
  • We must next pass in review the three great sciences of which [Mathematical Science] is composed,—the Calculus, Geometry, and Rational Mechanics.

Vol. 1, Book 1. Mathematics. Ch. 2.Edit

General View of Mathematical Analysis.
  • The historical development of the Abstract portion of Mathematical science has, since the time of Descartes, been for the most part determined by that of the Concrete. Yet the Calculus in all its principal branches must be understood before passing on to Geometry and Mechanics.
  • The Concrete portions of the science depend on the Abstract, which are wholly independent...
  • The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon... and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.
  • It is only by forming a true idea of an equation that we can lay down the real line of separation between the concrete and the abstract part of mathematics.
  • [I]t almost impossible to explain the difficulty we find in establishing the relation of the concrete to the abstract which meets us in every great mathematical question...
  • [F]unctions must... be divided into Abstract and Concrete; the first of which alone can enter into true equations.
  • Every equation is a relation of equality between two abstract functions of the magnitudes... including the primary magnitudes and all auxiliary magnitudes which may... facilitate the discovery of the equations...
  • This distinction may be established by both the à priori and à posteriori methods; by characterizing each kind of function, and by enumerating all the abstract functions yet known...
  • A priori; Abstract functions express a mode of dependence between magnitudes which may be conceived between numbers alone, without the need of pointing out any phenomena... while Concrete functions are those whose expression requires a specified... case of physics, geometry, mechanics, etc.
  • Most functions were concrete in their origin,—even those which are at present the most purely abstract; and the ancients discovered... through geometrical definitions elementary algebraic properties of functions, to which a numerical value was not attached till long afterwards, rendering abstract to us what was concrete to the old geometers.
  • [C]ircular functions, both direct and inverse... are still sometimes concrete, sometimes abstract, according to the point of view from which they are regarded.
  • A posteriori; the distinguishing character, abstract or concrete, of a function having been established, the question of any determinate function being abstract, and therefore able to enter into true analytical equations, becomes a simple question of fact, as we are acquainted with the elements which compose all the abstract functions at present known. We say we know them all, though analytical functions are infinite in number, because we are here speaking, it must be remembered,— of the elements— of the simple, not of the compound.
  • We have ten elementary formulas; and, few as they are, they may give rise to an infinite number of analytical combinations. There is no reason for supposing that there can never be more. We have more than Descartes had, and even Newton and Leibnitz; and our successors will doubtless introduce additions, though there is so much difficulty attending their augmentation, that we cannot hope that it will proceed very far.
    • Ref: Auguste Comte, "Enumeration of Functions, The Philosophy of Mathematics (1851) Book I. Analysis, Ch I. General View of Mathematical Analysis, p. 51, translated by W. M. Gillespie from the Cours de Philosophie Positive of Auguste Comte:1st couple(Sum, Difference); 2nd couple(Product, Quotient); 3rd couple(Power, Root), 4th couple(Exponential, Logarithmic), 5th couple(Direct Circular, Inverse Circular).
  • It is the insufficiency of this very small number of analytical elements which constitutes our difficulty in passing from the concrete to the abstract.
  • In order to establish the equations of phenomena, we must conceive of their mathematical laws by the aid of functions composed of these few elements.
  • [T]hese elements of our analysis have been supplied to us by the mathematical consideration of the simplest phenomena of a geometrical origin, which can afford us à priori no rational guarantee of their fitness to represent the mathematical laws of all other classes of phenomena.
  • [W]e have considered the Calculus as a whole. We must now consider its divisions... the Algebraic Calculus, and the Arithmetical Calculus, or Arithmetic, taking care to give them the most extended logical sense, and not the restricted one... usually received.
  • [E]very question of Mathematical Analysis presents two successive parts, perfectly distinct... The first stage is the transformation of the proposed equations, so as to exhibit the mode of formation of unknown quantities by the known. This constitutes the algebraic question. Then ensues the task of finding the values of the formulas thus obtained. ...this is the arithmetical question.
  • Thus the algebraic and the arithmetical calculus differ in their object. They differ also in their view of quantities,—Algebra considering quantities in regard to their relations, and Arithmetic in regard to their values.
  • In practice, it is not always possible... to separate the processes entirely in obtaining a solution; but the radical difference of the two operations should never be lost sight of.
  • Algebra... is the Calculus of Functions, and Arithmetic the Calculus of Values.
  • We have seen that the division of the Calculus is into two branches. It remains... to compare the two... to learn their respective extent, importance, and difficulty.
  • Functions being divided into simple and compound... when we become able to determine the value of simple functions, there will be no difficulty with the compound.
  • In the algebraic relation, a compound function plays a very different part from... the elementary functions which constitute it; and this is the source of our chief analytical difficulties. But it is quite otherwise with the Arithmetical Calculus.
  • [T]here can be no new arithmetical operations otherwise than by the creation of new analytical elements, which must... for ever be extremely small.
  • The domain of arithmetic then is, by its nature, narrowly restricted, while that of algebra is rigorously indefinite.
  • Still, the domain of arithmetic is... extensive... for there are many questions treated as incidental in the midst of a body of analytical researches, which... are... arithmetical. Of this kind are the construction of a table of logarithms, and the calculation of trigonometrical tables, and some distinct and higher procedures; in short, every operation which has for its object the determination of the values of functions.
  • [W]e must also include... the Theory of Numbers, the object of which is to discover the properties inherent in different numbers, in virtue of their values, independent of any particular system of numeration. It constitutes a sort of transcendental arithmetic.
  • Though the domain of arithmetic is thus larger than is commonly supposed, this Calculus of values will yet never be more than a point, as it were, in comparison with the calculus of functions, of which mathematical science essentially consists.
  • Determinations of values are, in fact, nothing else than real transformations of the functions to be valued. These transformations have a special end; but... are essentially of the same nature as all taught by analysis.
  • [T]he calculus of values might be regarded as a particular application of the calculus of functions, arithmetic thereby disappearing, as a distinct section, from the domain of abstract mathematics.
  • [W]e will now see how the establishment of the equations of phenomena has been achieved.
  • The first means of remedying the difficulty of the small number of analytical elements seems to be to create new ones. But... this resource is illusory.
  • [T]he introduction of another elementary abstract function into analysis supposes the simultaneous creation of a new arithmetical operation; which is certainly extremely difficult. ...We have... no idea how to proceed to create new elementary abstract functions. Yet, we must not... conclude that we have reached the limit... Special improvements in mathematical analysis have yielded us some partial substitutes, which have increased our resources: but... the augmentation... cannot proceed but with extreme slowness. It is not in this direction, then, that the human mind has found its means of facilitating the establishment of equations.
  • As it is impossible to find the equations directly, we must seek... corresponding ones between other auxiliary quantities, connected with the first according to a certain determinate law, and from the relation... ascend to that of the primitive magnitudes. This is the... transcendental analysis... our finest instrument for the mathematical exploration of natural phenomena.
  • [T]he auxiliary quantities... might be derived, according to any law whatever, from the immediate elements of the question.
  • [O]ur future improved analytical resources may perhaps be found in a new mode of derivation. But, at present, the only auxiliary quantities habitually substituted for the primitive quantities in transcendental analysis are what are called
    1st, infinitely small elements, the differentials of different orders of those quantities, if we conceive of this analysis in the manner of Leibnitz: or
    2nd, the fluxions, the limits of the ratios of the simultaneous increments of the primitive quantities, compared with one another; or, more briefly, the prime and ultimate ratios of these increments, if we adopt the conception of Newton: or
    3rd, the derivatives... of these quantities; that is, the coefficients of the different terms of their respective increments, according to the conception of Lagrange.
    These conceptions, and all others that have been proposed, are by their nature identical.
  • We now see that the Calculus of functions, or Algebra, must consist of two distinct branches.
  • To ordinary analysis I... give the name of Calculus of Direct Functions. To transcendental analysis, (...Infinitesimal Calculus, Calculus of fluxions and of fluents, Calculus of Vanishing quantities, the Differential and Integral Calculus, etc...) I shall give the title of Calculus of Indirect Functions.... by generalizing and giving precision to the ideas of Lagrange, and employ them to indicate the exact character of the two forms of analysis.
  • [A]nalysts first divide equations... into two principal classes, according as they contain functions of only the first three of the [five] couples, or as they include also either exponential or circular functions. Though the names of algebraic and transcendental functions given to these principal groups are inapt, the division between the corresponding equations is real enough, insofar as that the resolution of equations containing the transcendental functions is more difficult than that of algebraic equations. Hence the study of the first is extremely imperfect, and our analytical methods relate almost exclusively to the elaboration of the second.
  • [W]e must observe that, though [Algebraic equations] may often contain irrational functions of the unknown quantities, as well as rational functions, the first case can always be brought under the second, by transformations more or less easy...
  • The only question... of eminent importance... in its logical relations, would be the general resolution of algebraic equations of any degree whatever. But... we are led to suppose, with Lagrange, that it exceeds the scope of our understandings.
  • [I]f we had obtained the resolution of algebraic equations of any degree whatever, we should still have treated only a very small part of algebra... that is, of the calculus of direct functions, comprehending the resolution of all the equations that can be formed by the analytical functions known to us...
  • [B]y a law of our nature, we shall always remain below the difficulty of science, our means of conceiving of new questions being always more powerful than our resources for resolving them... [i.e.,] the human mind being more apt at imagining than at reasoning.
  • Thus, if we... resolved all the analytical equations now known, and if, to do this, we had found new analytical elements, these again would introduce classes of equations of which we now know nothing: and so, however great might be the increase of our knowledge, the imperfection of our algebraic science would be perpetually reproduced.
  • The methods that we have are, the complete resolution of the equations of the first four degrees; of any binomial equations; of certain special equations of the superior degrees; and of a very small number of exponential, logarithmic, and circular equations. These elements are very limited; but geometers have succeeded in treating with them a great number of important questions in an admirable manner.
  • The improvements introduced within a century into mathematical analysis have contributed more to render the little knowledge that we have immeasurably useful, than to increase it.
  • To fill up the vast gap in the resolution of algebraic equations of the higher degrees, analysts have had recourse to a new order of questions,—to... the numerical resolution of equations. Not being able to obtain the real algebraic formula, they have sought to determine at least the value of each unknown quantity for such or such a designated system of particular values attributed to the given quantities.
  • This operation is a mixture of algebraic with arithmetical questions; and it has been so cultivated as to be rendered possible in all cases, for equations of any degree and even of any form. The methods for this are now sufficiently general; and what remains is to simplify them so as to fit them for regular application.
  • [T]his is very imperfect algebra; and it is only isolated, or truly final questions (which are very few), that can be brought finally to depend upon only the numerical resolution of equations.
  • Most questions are only preparatory,—a first stage of the solution of other questions; and in these cases it is evidently not the value of the unknown quantity that we want to discover, but the formula which exhibits its derivation.
  • Even in the most simple questions, when this numerical resolution is strictly sufficient, it is... a very imperfect method. Because we cannot abstract and treat separately the algebraic part of the question, which is common to all the cases which result from the mere variation of the given numbers, we are obliged to go over again the whole series of operations for the slightest change that may take place in any one of the quantities concerned.
  • Thus is the calculus of direct functions at present divided into two parts, as it is employed for the algebraic or the numerical resolution of equations. The first, the only satisfactory one, is... very restricted, and there is little hope that it will ever be otherwise: the second, usually insufficient, has at least the advantage of a much greater generality. They must be carefully distinguished in our minds, on account of their different objects, and therefore of the different ways in which quantities are considered by them. Moreover, there is, in regard to their methods, an entirely different procedure in their rational distribution.
  • In the first part, we have nothing to do with the values of the unknown quantities, and the division must take place according to the nature of the equations which we are able to resolve; whereas in the second, we have nothing to do with the degrees of the equations, as the methods are applicable to equations of any degree whatever; but the concern is with the numerical character of the values of the unknown quantities.
  • These two parts, which constitute the immediate object of the Calculus of direct functions, are subordinated to a third, purely speculative, from which both derive their most effectual resources, ...designated by the general name of Theory of Equations, though it relates, as yet, only to algebraic equations. The numerical resolution of equations has, on account of its generality, special need of this rational foundation.
  • Two orders of questions divide this important department of algebra between them; first, those which relate to the composition of equations, and then those that relate to their transformation; the business of these last being to modify the roots of an equation without knowing them...
  • One more theory... to complete our rapid exhibition of the different essential parts of the calculus of direct functions... relates to the transformation of functions into series by the aid of the Method of indeterminate Coefficients... one of the most fertile and important in algebra. This... is one of the most remarkable discoveries of Descartes.
  • [I]nfinitesimal calculus, for which it might be... substituted in some respects, has... deprived it of some... importance; but the growing extension of the transcendental analysis has, while lessening its necessity, multiplied its applications and enlarged its resources; so by the useful combination of the two theories, the employment of the method of indeterminate coefficients has become much more extensive than... even before the formation of the calculus of indirect functions.
  • We must next pass on to the more important and extensive branch... the Calculus of Indirect Functions.
  • [T]he views of the transcendental analysis... by Leibnitz, Newton, and Lagrange... each... has advantages... all are finally equivalent, and... no method has yet been found which unites their respective characteristics.
  • [I]t is only by the use of them all that an adequate idea of the analysis and its applications can be formed.
  • The first germ of the infinitesimal method (which can be conceived of independently of the Calculus) may be recognized in the old Greek Method of Exhaustions, employed to pass from the properties of straight lines to those of curves. The method consisted in substituting for the curve the auxiliary consideration of a polygon, inscribed or circumscribed, by means of which the curve itself was reached... but there was in it no equivalent for our modern methods; for the ancients had no logical and general means for the determination of... limits, which was the chief difficulty of the question.
  • The task remaining for modern geometers was to generalize the conception of the ancients, and, considering it in an abstract manner, to reduce it to a system of calculation...
  • Lagrange justly ascribes to... Fermat the first idea in this new direction. Fermat... initiated the direct formation of transcendental analysis by his method for the determination of maxima and minima, and for the finding of tangents, in which process he introduced auxiliaries which he afterwards suppressed as null when the equations obtained had undergone... suitable transformations.
  • After some modifications of the ideas of Format in the intermediate time, Leibnitz stripped the process of some complications, and formed the analysis into a general and distinct calculus, having his own notation: and... is thus the creator of transcendental analysis, as we employ it now.
  • This pre-eminent discovery was so ripe, as all great conceptions are at the hour of their advent, that Newton had at the same time, or... earlier, discovered a method exactly equivalent, regarding the analysis from a different point of view, much more logical... but less adapted than that of Leibnitz to give all practicable extent and facility to the fundamental method.
  • Lagrange... discarding the heterogeneous considerations which had guided Leibnitz and Newton, reduced the analysis to a purely algebraic system, which only wants more aptitude for application.
  • The method of Leibnitz consists in introducing... in order to facilitate the establishment of equations, the infinitely small elements or differentials which are supposed to constitute the quantities whose relations we are seeking.
  • There are relations between these differentials which are simpler and more discoverable than those of the primitive quantities; and by these we may afterwards (through a special calculus employed to eliminate these auxiliary infinitesimals) recur to the equations sought, which it would usually have been impossible to obtain directly.
  • [W]hen there is too much difficulty in forming the equation between the differentials of the magnitudes under notice, a second application of the method is required, the differentials being now treated as new primitive quantities, and a relation being sought between their infinitely small elements, or second differentials, and so on... repeated any number of times...
  • [P]reliminary ideas being laid down, the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities; and generally, the infinitely small quantities of any order whatever in comparison with all those of an inferior order.
  • [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.
  • [T]he conception of transcendental analysis, as formed by Leibnitz... is... the loftiest idea ever yet attained by the human mind.
  • [T]his conception was necessary to complete the basis of mathematical science, by enabling us to establish... the relation of the concrete to the abstract. In this respect, we must regard it as the necessary complement of the great fundamental idea of Descartes on the general analytical representation of natural phenomena; an idea which could not be duly estimated or put to use till after the formation of the infinitesimal analysis.
  • The differential formulas exhibit an extreme generality, expressing in a single equation each determinate phenomenon, however varied may be the subjects to which it belongs.
  • Thus, one... equation gives the tangents of all curves, another their rectifications, a third their quadratures; and, in the same way, one invariable formula expresses the mathematical law of all variable motion; and one single equation represents the distribution of heat in any body, and for any case.
  • This remarkable generality is the basis of the loftiest views of the geometers.
  • Thus this analysis has not only furnished a general method for forming equations indirectly which could not have been directly discovered, but it has introduced a new order of more natural laws for our use in the mathematical study of natural phenomena, enabling us to rise at times to a perception of positive approximations between classes of wholly different phenomena, through the analogies presented by the differential expressions of their mathematical laws.
  • In virtue of this second property of the analysis, the entire system of an immense science, like geometry or mechanics, has submitted to a condensation into a small number of analytical formulas, from which the solution of all particular problems can be deduced, by invariable rules.
  • This beautiful method is, however, imperfect in its logical basis.
  • Leibnitz himself failed to justify his conception, giving, when urged, an answer which represented it as a mere approximative calculus, the successive operations of which might... admit an augmenting amount of error.
  • Some of his successors were satisfied with showing that its results accorded with those obtained by ordinary algebra, or the geometry of the ancients, reproducing... some solutions...
  • Some... demonstrated the conformity of the new conception with others; that of Newton especially, which was unquestionably exact. This afforded a practical justification: but... a logical justification is also required,—a direct proof of the necessary rationality of the infinitesimal method.
  • Carnot... furnished this at last, by showing that the method was founded on the principle of the necessary compensation of errors. We cannot say that all the logical scaffolding... may not have a merely provisional existence... but, in the present state of our knowledge, Carnot's principle... is of... importance, in legitimating the analysis of Leibnitz... His reasoning is founded on the conception of infinitesimal quantities indefinitely decreasing, while those from which they are derived are fixed. The infinitely small errors introduced with the auxiliaries cannot have occasioned other than infinitely small errors in all the equations... Carnot's theory is doubtless more subtle than solid; but it has no other radical logical vice than that of the infinitesimal method itself...
  • 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.
  • 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 infinitesimal 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 integral. ...[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.

Vol. 3, Book VI. Social Physics Ch. 15Edit

Estimate of the Final Action of the Positive Philosophy.
  • 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.
  • Then there will be an end to the efforts of the anterior sciences to absorb the more recent, and of the more recent to maintain their superiority by boasting of sanction from the old philosophies; and the positive spirit will decide the claims of each, without oppression or anarchy, and with the necessary assent of all.
  • The same unquestionable order will be established in the interior of each science; and every proved conception will be secured from such attacks as all are now liable to from the irregular ambition or empiricism of unqualified minds.
  • Though abstract science must hold the first place, as Bacon so plainly foresaw, the direct construction of concrete science is one of the chief offices of the new philosophical spirit, exercised under historical guidance, which can alone afford the necessary knowledge of the successive states of every thing that exists.
  • [T]here will be another result... the fixing,—not yet possible, but then certainly practicable,—of the general duration assigned by the whole economy to each of the chief kinds of existence; and, among others, to the rising condition of the human race.
  • [T]he collective organism is necessarily subject, like the individual, to a spontaneous decline, independently of changes in the medium. The one has no more tendency to rejuvenescence than the other; and the only difference in the two cases is in the immensity of duration and slow progression in the one, compared with the brief existence, so rapidly run through, of the other.
  • There is no reason why, because we decline the metaphysical notion of indefinite perfectibility, we should be discouraged in our efforts to ameliorate the social state; as the health of individuals is ministered to when destruction is certainly near at hand.
  • It is too soon in infancy to prepare for old age; and there would be less wisdom in such preparation in the collective than in the individual case.
  • Morality must become more practical than it ever could be under religious influences, because personal morality will be seen in its true relations,—withdrawn from all influences of personal prudence, and recognized as the basis of all morality whatever, and therefore as a matter of general concern and public rule. The ancients had some sense of this, which they could not carry out; and Catholicism lost it by introducing a selfish and imaginary aim.
  • [M]oral rules certainly hold the very first place, because they especially admit of the universal concurrence in which our chief power resides. If we are thus brought back from an immoderate regard to the future by a sense of the value of the present, this will equalize life by discouraging excessive economical preparation; while a sound appreciation of our nature, in which vicious or unregulated propensities originally abound, will render common and unanimous the obligation to discipline, and regulate our various inclinations.
  • [T]he scientific and moral conception of Man as the chief of the economy of nature will be a steady stimulus to the cultivation of the noble qualities, affective as well as intellectual, which place him at the head of the living hierarchy.
  • There can be no danger of apathy in a position like this,—with the genuine and just pride of such pre-eminence stirring within us; and above us the type of perfection, below which we must remain, but which will ever be inviting us upwards. The result will be a noble boldness in developing the greatness of Man in all directions, free from the oppression of any fear, and limited only by the conditions of life itself.
  • [P]rogression will develop more and more the natural differences on which such an economy is based, so that each element will tend towards the mode of existence most suitable to itself, and consonant with the general welfare.
  • The positive philosophy is the first that has ascertained the true point of view of social morality. The metaphysical philosophy sanctioned egotism; and the theological subordinated real life to an imaginary one; while the new philosophy takes social morality for the basis of its whole system.
  • We have yet to witness the moral superiority of a philosophy which connects each of us with the whole of human existence, in all times and places.
  • The restriction of our expectations to actual life must furnish new means of connecting our individual development with the universal progression, the growing regard to which will afford the only possible, and the utmost possible, satisfaction to our natural aspiration after eternity.
  • [T]he scrupulous respect for human life, which has always increased with our social progression, must strengthen more and more as the chimerical hope dies out which disparages the present life as merely accessory to the one in prospect.
  • The philosophical spirit being only an extension of good sense... it alone, in its spontaneous form, has for three centuries maintained any general agreement against the dogmatic disturbances occasioned or tolerated by the ancient philosophy, which would have overthrown the whole modern economy if popular wisdom had not restrained the social application of it.
  • [P]ositive morality will tend more and more to exhibit the happiness of the individual as depending on the complete expansion of benevolent acts and sympathetic emotions towards the whole of our race; and even beyond our race, by a gradual extension to all sentient beings below us, in proportion to their animal rank and their social utility.
  • Till the full rational establishment of positive morality has taken place, it is the business of true philosophers, ever the precursors of their race, to confirm it in the estimation of the world by the sustained superiority of their own conduct, personal, domestic, and social; giving the strongest conceivable evidence of the possibility of developing, on human grounds alone, a sense of general morality complete enough to inspire an invincible repugnance to moral offence, and an irresistible impulse to steady practical devotedness.
  • I have only to glance at the growth and application of the division between the spiritual or theoretical organism and the temporal or practical...
  • Catholicism afforded the suggestion of a double government of this kind, and that the Catholic institution of it shared the discredit of the philosophy to which it was attached: and... the Greek Utopia of a Reign of Mind (well called by Mr. Mill a Pedantocracy), transmitted to the modern metaphysical philosophy, gained ground till its disturbing influence rendered it a fit subject for our judgment and sentence.
  • The present state of things is that we have a deep and indestructible, though vague and imperfect, sense of the political requirements of existing civilization, which assigns a distinct province, in all affairs, to the material and the intellectual authority, the separation and co-ordination of which are reserved for the future.
  • The Catholic division was instituted on the ground of a mystical opposition between heavenly and earthly interests... and when the terrestrial view prevailed over the celestial, the principle of separation was seriously endangered, from there being no longer any logical basis which could sustain it against the extravagances of the revolutionary spirit.
  • The positive polity must... go back to the earliest period of the division, and re-establish it on evidence afforded by the whole human evolution; and, in its admission of the scientific and logical preponderance of the social point of view, it will not reject it in the case of morality, which must always allow its chief application, and in which everything must be referred, not to Man, but to Humanity.
  • Moral laws, like the intellectual, are much more appreciable in the collective than in the individual case; and, though the individual nature is the type of the general, all human advancement is much more completely characterized in the general than in the individual case; and thus morality will always, on both grounds, be connected with polity. Their separation will arise from that distinction between theory and practice which is indispensable to the common destination of both.
  • We may... sum up the ultimate conditions of positive polity by conceiving of its systematic wisdom as reconciling the opposing qualities of that spontaneous human wisdom successively manifested in antiquity and in the Middle Ages; for there was a social tendency involved in the ancient subordination of morality to policy, however carried to an extreme under polytheism; and the monotheistic system had the merit of asserting, though not very successfully, the legitimate independence, or rather, the superior dignity of morality.
  • Antiquity alone offered a complete and homogeneous political system; and the Middle Ages exhibit an attempt to reconcile the opposite qualities of two heterogeneous systems, the one of which claimed supreme authority for theory, and the other for practice.
  • Such a reconciliation will take place hereafter, on the ground of the systematic distinction between the claims of education and of action.
  • If the whole experience of modern progress has sanctioned the independence, amidst co-operation, of theory and practice, in the simplest cases, we must admit its imperative necessity, on analogous grounds, in the most complex.
  • Thus far, in complex affairs, practical wisdom has shown itself far superior to theoretical; but this is because much of the proudest theory has been ill-established. However, this evil may be diminished when social speculation becomes better founded, the general interest will always require the common preponderance of the practical or material authority, as long as it keeps within its proper limits, admitting the independence of the theoretical authority; and the necessity of including abstract indications among the elements of every concrete conclusion.
  • We have still to reap some of the bitter fruits of our intellectual and moral anarchy: and especially, in the quarrels between capitalists and labourers first, and afterwards in the unsettled rivalship between town and country. In short, whatever is now systematized must be destroyed; and whatever is not systematized, and therefore has vitality, must occasion collisions which we are not yet able accurately to foresee or adequately to restrain. This will be the test of the positive philosophy, and at the same time the stimulus to its social ascendancy.
  • The difficulties proper to the action of the new regime, the same in kind, will be far less in degree, and will disappear as the conditions of order and progress become more and more thoroughly reconciled.
  • [T]he advent of the positive economy will have been owing to the affinity between philosophical tendencies and popular impulses: and if so... that affinity must become the most powerful permanent support of the system.
  • For five centuries, society has been seeking an aesthetic constitution correspondent to its civilization. In the time to come... we may see how Art must eminently fulfil its chief service, of charming and improving the humblest and the loftiest minds, elevating the one, and soothing the other. For this service it must gain much by being fitly incorporated with the social economy, from which it has hitherto been essentially excluded.
  • The public life and military existence of antiquity are exhausted; but the laborious and pacific activity proper to modern civilization is scarcely yet instituted, and has never yet been aesthetically regarded; so that modern art, like modern science and industry, is so far from being worn out, that it is as yet only half formed. The most original and popular species of modern art, which forms a preparation for that which is to ensue, has treated of private life, for want of material in public life. But public life will be such as will admit of idealization: for the sense of the good and the true cannot be actively conspicuous without eliciting a sense of the beautiful; and the action of the positive philosophy is in the highest degree favourable to all the three.
  • The systematic regeneration of human conceptions must also furnish new philosophical means of aesthetic expansion, secure at once of a noble aim and a steady impulsion. There must certainly be an inexhaustible resource of poetic greatness in the positive conception of Man as the supreme head of the economy of Nature, which he modifies at will, in a spirit of boldness and freedom, within no other limits than those of natural law. This is yet an untouched wealth of idealization, as the action of Man upon Nature was hardly recognized as a subject of thought till art was declining from the exhaustion of the old philosophy.
  • The marvellous wisdom of Nature has been sung, in imitation of the ancients, and with great occasional exaggeration; and the conquests of Man over nature, with science for his instrument, and sociality for his atmosphere, remains, promising much more interest and beauty than the representation of an economy in which he has no share, and in which magnitude was the original object of admiration, and material grandeur continues to be most dwelt upon.
  • There is no anticipating what the popular enthusiasm will be when the representations of Art shall be in harmony with the noble instinct of human superiority, and with the collective rational convictions of the human mind.
  • To the philosophical eye it is plain that the universal reorganization will assign to modern Art at once inexhaustible material in the spectacle of human power and achievement, and a noble social destination in illustrating and endearing the final economy of human life.
  • What philosophy elaborates, Art will propagate and adapt for propagation, and will thus fulfil a higher social office than in its most glorious days of old.
  • I have here spoken of the first of the arts only,—of Poetry, which by its superior amplitude and generality has always superintended and led the development of them all: but the conditions which are favourable to one mode of expression are propitious to all, in their natural succession.
  • While the positive spirit remained in its first phase, the mathematical, it was reproached for its anti-aesthetic tendency: but we now see how, when it is systematized from a sociological centre, it becomes the basis of an aesthetic organization no less indispensable than the intellectual and social renovation from which it is inseparable.

Addendum to Miss Martineau's translationEdit

Translation of Comte's last 10 pages by Frederic Harrison
  • This summary estimate of the ultimate result of the Positive Philosophy brings me to the close of the long and arduous task... to carry forward the great impulse given to philosophy by Bacon and Descartes.
  • Their work was essentially occupied with the first canons of the positive method: it was entirely powerless to found any final reconstruction of human society, the need for which was hardly apparent in their age, but which is now so urgently required by the prospect of social anarchy and revolutionary agitation.
  • In the course of my labours... prolonged over twelve years, my own mind has spontaneously, but exactly, traversed the successive phases of our modern mental evolution.
  • As the new science of Sociology had to be created, it was not planned with the same precision... But even here I hope that... readers will admit that each science has been treated in the degree of its true philosophical importance.
  • Having thus worked through the... scale of the sciences... my mind has reached a.... positive condition, and has wholly disengaged itself from metaphysics as well as theology.
  • There are four essential works required... Two of the four contemplated works would be occupied with the more complete elaboration of the new system of philosophy; the other two works relate to the application of it to practice.
  • In the Treatise just completed, it was inevitable that each science in turn should be handled from the point of view of its actual condition... by a gradual and sure process of growth to the ultimate state which I had conceived from the first, but which I could only reach by passing through the successive stages of modern evolution in the same way as Descartes did in his famous formula.
  • [W]hatever may have been the advantages of this method of systematizing the sciences à posteriori, and without it I must have failed in my object, the consequence was, that the philosophy of each science, on which the general positive philosophy was founded, could not be presented in its definitive form.
  • This definitive form could only be secured by the reaction upon each science of the new philosophical synthesis.
  • Such an effect, which, duly completed, will be for abstract purposes the final state of the positive systematization, would properly require as many special philosophical treatises, each infused with the sociological spirit, as there are different substantive sciences.
  • It is... impossible that I could ever properly complete so vast a task... and I... restrict my... work to the first and the last of the sciences, which are the more decisive, and... those with which I am most familiar.
  • I... limit myself to Mathematics and Sociology, and... leave it to my successors or... colleagues to deal with the philosophy of the four intermediate sciences—astronomy, physics, chemistry, and biology.
  • When I composed the first part of this work twelve years ago, I... thought that the theories on the philosophy of mathematics there put forward would be sufficiently clear to be grasped. I underrated the extreme narrowness of the views now current in that science.
  • I feel it necessary to attempt to lay down a true philosophy of mathematical science, the base... of the whole scientific series.
  • The second work, on Sociology... will consist of... the methods of Sociology, the... Social Statics, the... Social Dynamics, the last... the practical application of the doctrine.
  • The philosophy of the Social Polity is the most important task that awaits me.
  • [T]he most direct mode of contributing to the general acceptance of the new positive philosophy must be found in promoting the normal completeness of the social science. ...[T]here are strong practical grounds for giving a special importance to Sociology.
  • With regard to... the practical application of the new system of philosophy, I propose... a Treatise upon the principles of positive education...
  • [E]ducation must always be the first step towards a political regeneration. Thus the third work I propose is a sequel to the present Treatise. Its important duty would be the reorganization of Morals on a positive basis. This will... be the principal part of education; and this alone will effectively dispel that theological philosophy, which, in its decline, is still powerful enough to embarrass the course both of intellectual and social progress.

Quotes about The Positive PhilosophyEdit

  • I have already read the noble preface and the excellent table of contents, as well as some decisive chapters. And I am convinced that you have displayed clearness of thought, truth, and sagacity in your long and diflicult task.
  • The important undertaking that you so happily conceived and have so worthily accomplished will give my 'Positive Philosophy' a competent audience greater than I could have hoped to find in my own lifetime.
    • August Comte, Letter to Harriet Martineau, as quoted by Frederick Harrison, Introduction, The Positive Philosophy of Auguste Comte (1896).
  • It is due to you, that the arduous study of my fundamental treatise is now indispensable only for the small number of those who purpose to become systematic students of philosophy. But the majority of readers, with whom theoretic training is only intended to provide them with practical good sense, may now prefer, and even ought to prefer for ordinary use, your admirable condensation... It realises a wish of mine that I formed ten years ago. And looking at it from the point of view of future generations, I feel sure that your name will be linked with mine, for you have executed the only one of those works that will survive amongst all those which my fundamental treatise has called forth.
  • Twelve years had passed... during which his life had been closed against any kind of distraction. No wish for premature publication was suffered to lead his mind off the conscientious completion of his task. No ambition of gaining popularity was allowed to modify a single line in conformity with the opinions of the time. With stern resolution, and deaf to all external distractions, he concentrated his whole soul upon his work. In the history of men who have devoted their lives to great thoughts, I know nothing nobler than that of these twelve years.
    • Émile Littré, "Auguste Comte, et la Philosopliie Positive" (1863) p. 188, as quoted by Frederick Harrison, Introduction, The Positive Philosophy of Auguste Comte (1896).

Introduction, The Positive Philosophy of Auguste ComteEdit

by Frederic Harrison
  • Auguste Comte wrote a few pieces for various periodicals in Paris, to which he attached but little importance. His first great philosophical work was a pamphlet... published in May, 1822... entitled a "Prospectus of the scientific works required for the reorganization of Society, by Auguste Comte, former pupil of the Ecole Polytechnique." He republished his pamphlet with some small modifications and additions in 1824, under the title "System of Positive Polity," and this is reprinted in vol. iv. of the " Politique Positive," 1854. This essay of 1822 contains a statement of the classification of the sciences, of the law of the three states, and the suggestion of a science of sociology. It is... the prospectus of that which for thirty years Comte continued to elaborate... and has always been treated by Comte and by his adherents as the the first sketch of the "Positive Philosophy."
  • At the fourth lecture the course was abruptly broken off. Intense mental strain, together with domestic misery, brought on an attack of insanity. He... was placed in an asylum by his friend Broussais.... [and] remained for seven months.
  • The devotion of his mother and his wife, who took him from the care of Dr. Esquirol whilst still suffering from the disease, succeeded in gradually restoring his reason. An epoch of profound despair followed, during which he threw himself into the Seine, but was rescued; and thenceforth he resolved to devote himself with patience and resignation to the work of his life, supporting himself with private lessons.
  • In January, 1829, he resumed his course of lectures on the Positive Philosophy, ...[with] the same eminent men amongst his audience, with the exception of Humboldt, who was no longer in France. On this occasion he completed the whole series of lectures, and in December, 1829, he repeated them in a public course at the Athénée.
  • The work of which these three volumes are a condensation was published were published from 1830 to 1842. The first volume, containing the Introduction and the philosophy of Mathematics, was published separately... A brief note described it as the result of the author's labours from the year 1816, and as a development of the new ideas put forth in his early essay of 1822, entitled a "System of Positive Polity."
  • The second volume, comprising Astronomy and Physics, did not appear until 1835, owing to the commercial disasters of the Revolution of July.
  • The third volume, comprising Chemistry and Biology, appeared in 1838.
  • The new science of Sociology, which was intended to he comprised in a single volume, ultimately extended to three volumes, published in 1839, 1841, and 1842.
  • The last volume... was introduced by a personal preface to explain the prolongation of the work over twelve years, and the grounds for devoting one half of the entire work to the new Social Science. And it contained in notes Comte's vehement repudiation of Saint-Simon, and his no less vehement condemnation of M. Arago and the official directors of the École Polytechnique.
  • His courses of lectures were all delivered without writing. When he commenced to prepare them for the press, he simply wrote them down from memory with great rapidity... without altering the proofs. ...[I]t had the obvious disadvantages of a certain multiplicity of phrase, a monotony, and that repetition which is only proper to oral exposition. ...[H]abitually and on system, he... uniformly cast his philosophic thoughts into a very formal, artificial, and undoubtedly cumbrous style...
  • Tedious and even repulsive as it is to the average reader, to the serious student of Positivism this method of exposition has rare and paramount advantages. It is unerringly precise, lucid, qualified, and suggestive. ...[H]is long-drawn and over-elaborated sentences never leave the student in doubt for a moment as to his meaning, as to his whole meaning, as to all that he wishes to express, and all that he means to disclaim or exclude.
  • The result is, that the general reader can hardly follow these crowded and closely welded paragraphs without the assistance of an expert, whilst the serious student of the Positive Philosophy finds some new light or some needful warning in everyone of these pregnant epithets and precise limitations.
  • Comte saw this clearly himself; and hence, in his "Popular Library," embodied in his later works, he inserts—not his own "Positive Philosophy" in six volumes—but Miss Martineau's condensed English version. Unfortunately not only the general reader, but the professed critics of Positivism have too often adopted his generous suggestion.
  • Sir David Brewster... a strong opponent of Positivism as a religious and social philosophy, reviewed the first two volumes in the "Edinburgh Review"... In this essay... Brewster pays homage to the depth and sagacity of Comte's mind, and he accepts in principle the law of the Three States, the Classification of the Sciences, and the ultimate extension of the methods of Science to Sociology.
    • Ref: Edinburgh Review (1838) No. 136, Vol. 67, p. 271.
  • Mr. Mill... in his "System of Logic," 1843... spoke of Auguste Comte as amongst the first of European thinkers, and by his institution of a new social science, as in some respects, the first.
  • In 1845-6, George Henry Lewes published his "Biographical History of Philosophy"... and... treated of Auguste Comte as "the greatest of modern thinkers," and as crowning the general history of philosophical evolution.
  • In 1853, Lewes published Comte's "Philosophy of the Sciences," a volume in Bohn's Philosophical Library. And in the same year Miss Martineau published the condensed translation which at once made Comte familiar to all English students.
  • It is a singular fact in literary history, and a striking testimony to the merit of Miss Martineau, that the work of a French philosopher should be studied in France in a French re-translation from his English translator—and that at his own formal desire and by his own special followers.
  • When Miss Martineau translated the " Philosophy," more than forty years ago, the later works of Comte were not before her; and... the later works of Comte are not referred to in her book at all. She carried this decision to the very extreme point of suppressing... the last ten pages of the sixth and concluding volume of the "Philosophy."
  • Now, from the point of view of the unity of Comte's career these ten pages are crucial, for they contain the entire scheme of Comte's future philosophical labours as he designed them in 1842, and as they were ultimately carried out in the "Polity," "Catechism," "Synthesis," etc... These important pages have been added by the present writer, in the condensed form adopted by Miss Martineau.
  • Miss Martineau seized the dominant idea of each sentence or rather paragraph—not without much sacrifice of the continuity of thought, no little loss in precision and accuracy of definition, sometimes a serious omission of important matter—but on the whole with an extraordinary gain to the freshness of impression on the general reader.

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