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The Philosophy of Mathematics: "A True Definition of Mathematics"试读：
The Philosophy of Mathematics
“A TRUE DEFINITION OF MATHEMATICS” [ANNOTATED] By Auguste Comte Translated by W. M. GillespieAnnotatedby Murat UkrayILLUSTRATED &PUBLISHED BYe-KİTAP PROJESİ & CHEAPEST BOOKS www.cheapestboooks.comwww.facebook.com/EKitapProjesi Copyright, 2016 by e-Kitap ProjesiIstanbulISBN: 978-605-9285-78-0
uguste Marie François Xavier Comte (1798 –1857), better known A as Auguste Comte was a French philosopher. He was a founder of the discipline of sociology and of thedoctrine of positivism. He is sometimes regarded as the first philosopher of science in the modern sense of the term.Influenced by the utopian socialist Henri Saint-Simon, Comte developed thepositive philosophy in an attempt to remedy the social malaise of the French Revolution, calling for a new social doctrine based on the sciences. Comte was a major influence on 19th-century thought, influencing the work of social thinkers such as Karl Marx, John Stuart Mill, and George Eliot. His concept of Sociologie and social evolutionism set the tone for early social theorists and anthropologists such as Harriet Martineau and Herbert Spencer, evolving into modern academic sociology presented by Émile Durkheim as practical and objective social research.Comte's social theories culminated in the "Religion of Humanity", which influenced the development of religious humanist and secular humanist organizations in the 19th century. Comte likewise coined the word altruisme (altruism).Auguste Comte was born in Montpellier, Hérault on 19 January 1798. After attending the Lycée Joffre and then theUniversity of Montpellier, Comte was admitted to the École Polytechnique in Paris. The École Polytechnique was notable for its adherence to the French ideals of republicanism and progress. The École closed in 1816 for reorganization, however, and Comte continued his studies at the medical school at Montpellier. When the École Polytechnique reopened, he did not request readmission.Following his return to Montpellier, Comte soon came to see unbridgeable differences with his Catholic and monarchist family and set off again for Paris, earning money by small jobs. In August 1817 he found an apartment at 36 rue Bonaparte in Paris (where he lived until 1822) and later that year he became a student and secretary to Henri de Saint-Simon, who brought Comte into contact with intellectual society and greatly influenced his thought therefrom. During that time Comte published his first essays in the various publications headed by Saint-Simon, L'Industrie, Le Politique, and L'Organisateur (Charles Dunoyerand Charles Comte's Le Censeur Européen), although he would not publish under his own name until 1819's "La séparation générale entre les opinions et les désirs" ("The general separation of opinions and desires"). In 1824, Comte left Saint-Simon, again because of unbridgeable differences. Comte published a Plan de travaux scientifiques nécessaires pour réorganiser la société (1822) (Plan of scientific studies necessary for the reorganization of society). But he failed to get an academic post. His day-to-day life depended on sponsors and financial help from friends. Debates rage as to how much Comte appropriated the work of Saint-Simon.Comte married Caroline Massin in 1825. In 1826, he was taken to a mental health hospital, but left without being cured – only stabilized by French alienist Jean-Étienne Dominique Esquirol – so that he could work again on his plan (he would later attempt suicide in 1827 by jumping off the Pont des Arts). In the time between this and their divorce in 1842, he published the six volumes of his Cours.Comte also developed a close friendship with John Stuart Mill. From 1844, he fell deeply in love with the Catholic Clotilde de Vaux, although because she was not divorced from her first husband their love was never consummated. After her death in 1846 this love became quasi-religious, and Comte, working closely with Mill (who was refining his own such system) developed a new "Religion of Humanity". John Kells Ingram, an adherent of Comte, visited him in Paris in 1855.He published four volumes of Système de politique positive (1851–1854). His final work, the first volume of "La Synthèse Subjective" ("The Subjective Synthesis"), was published in 1856.* * * * *
What is the Philosophy of Mathematics?
The philosophy of mathematics, mathematics employee Inclassification efforts to understand the philosophy is the branch.The main question is related to the source of the object that is the subject of mathematics and mathematics. In particular examine the characteristics of a true proposition:* What are the sources of mathematical subject matter?* What is about the meaning of a mathematical object?* What is the nature of a mathematical proposition?* What is the relationship between logic and mathematics?* What is the role of mathematics hermeneutic?* Mathematics played a role in the investigation which type* What is the subject of mathematical investigations?* What is the human traits behind mathematics?* What is mathematical beauty?* What is the nature and source of mathematical truth?* What is the relationship between mathematics and abstract material universe with the world?Another important issue is the reality of a mathematical theory. Mathematics (from the Natural Sciences as different) experimentally is sought reasons to find real specific mathematical theory can not be tested (see. Epistemology). Luitz that Brouwer 's laid the foundation for intuitionist mathematics of the representatives knew of this view. The logical mathematics is the approach of Bertrand Russell and Gottlob Frege was defended by David Hilbert, formalism is considered the representative of the current. Traditionalism logician the empiricist's (Rudolf Carnap, A. J. Ayer, Carl Hempel) were represented by one of the key issues in the philosophy of mathematics is also important to regard the certainty problem. Austrian logician Kurt Gödel's also work Mathematics and mathematics.M. Ukray THEPHILOSOPHYOFMATHEMATICS. TRANSLATED FROM THECOURS DE PHILOSOPHIE POSITIVEOFAUGUSTE COMTE,BYW. M. GILLESPIE,PROFESSOR OF CIVIL ENGINEERING & ADJ. PROF. OF MATHEMATICSIN UNION COLLEGE.NEW YORK: 1851.
he pleasure and profit which the translator has Treceived from the great work here presented, have induced him to lay it before his fellow-teachers and students of Mathematics in a more accessible form than that in which it has hitherto appeared. The want of a comprehensive map of the wide region of mathematical science—a bird's-eye view of its leading features, and of the true bearings and relations of all its parts—is felt by every thoughtful student. He is like the visitor to a great city, who gets no just idea of its extent and situation till he has seen it from some commanding eminence. To have a panoramic view of the whole district—presenting at one glance all the parts in due co-ordination, and the darkest nooks clearly shown—is invaluable to either traveller or student. It is this which has been most perfectly accomplished for mathematical science by the author whose work is here presented.Clearness and depth, comprehensiveness and precision, have never, perhaps, been so remarkably united as in Auguste Comte. He views his subject from an elevation which gives to each part of the complex whole its true position and value, while his telescopic glance loses none of the needful details, and not only itself pierces to the heart of the matter, but converts its opaqueness into such transparent crystal, that other eyes are enabled to see as deeply into it as his own.Any mathematician who peruses this volume will need no other justification of the high opinion here expressed; but others may appreciate the following endorsements of well-known authorities. Mill, in his "Logic," calls the work of M. Comte "by far the greatest yet produced on the Philosophy of the sciences;" and adds, "of this admirable work, one of the most admirable portions is that in which he may truly be said to have created the Philosophy of the higher Mathematics:" Morell, in his "Speculative Philosophy of Europe," says, "The classification given of the sciences at large, and their regular order of development, is unquestionably a master-piece of scientific thinking, as simple as it is comprehensive;" and Lewes, in his "Biographical History of Philosophy," names Comte "the Bacon of the nineteenth century," and says, "I unhesitatingly record my conviction that this is the greatest work of our age."The complete work of M. Comte—his "Cours de Philosophie Positive"—fills six large octavo volumes, of six or seven hundred pages each, two thirds of the first volume comprising the purely mathematical portion. The great bulk of the "Course" is the probable cause of the fewness of those to whom even this section of it is known. Its presentation in its present form is therefore felt by the translator to be a most useful contribution to mathematical progress in this country. The comprehensiveness of the style of the author—grasping all possible forms of an idea in one Briarean sentence, armed at all points against leaving any opening for mistake or forgetfulness—occasionally verges upon cumbersomeness and formality. The translator has, therefore, sometimes taken the liberty of breaking up or condensing a long sentence, and omitting a few passages not absolutely necessary, or referring to the peculiar "Positive philosophy" of the author; but he has generally aimed at a conscientious fidelity to the original. It has often been difficult to retain its fine shades and subtile distinctions of meaning, and, at the same time, replace the peculiarly appropriate French idioms by corresponding English ones. The attempt, however, has always been made, though, when the best course has been at all doubtful, the language of the original has been followed as closely as possible, and, when necessary, smoothness and grace have been unhesitatingly sacrificed to the higher attributes of clearness and precision.Some forms of expression may strike the reader as unusual, but they have been retained because they were characteristic, not of the mere language of the original, but of its spirit. When a great thinker has clothed his conceptions in phrases which are singular even in his own tongue, he who professes to translate him is bound faithfully to preserve such forms of speech, as far as is practicable; and this has been here done with respect to such peculiarities of expression as belong to the author, not as a foreigner, but as an individual—not because he writes in French, but because he is Auguste Comte.The young student of Mathematics should not attempt to read the whole of this volume at once, but should peruse each portion of it in connexion with the temporary subject of his special study: the first chapter of the first book, for example, while he is studying Algebra; the first chapter of the second book, when he has made some progress in Geometry; and so with the rest. Passages which are obscure at the first reading will brighten up at the second; and as his own studies cover a larger portion of the field of Mathematics, he will see more and more clearly their relations to one another, and to those which he is next to take up. For this end he is urgently recommended to obtain a perfect familiarity with the "Analytical Table of Contents," which maps out the whole subject, the grand divisions of which are also indicated in the Tabular View facing the title-page. Corresponding heads will be found in the body of the work, the principal divisions being in small capitals, and the subdivisions in Italics. For these details the translator alone is responsible.
lthough Mathematical Science is the most ancient and Athe most perfect of all, yet the general idea which we ought to form of it has not yet been clearly determined. Its definition and its principal divisions have remained till now vague and uncertain. Indeed the plural name—"The Mathematics"—by which we commonly designate it, would alone suffice to indicate the want of unity in the common conception of it.In truth, it was not till the commencement of the last century that the different fundamental conceptions which constitute this great science were each of them sufficiently developed to permit the true spirit of the whole to manifest itself with clearness. Since that epoch the attention of geometers has been too exclusively absorbed by the special perfecting of the different branches, and by the application which they have made of them to the most important laws of the universe, to allow them to give due attention to the general system of the science.But at the present time the progress of the special departments is no longer so rapid as to forbid the contemplation of the whole. The science of mathematics is now sufficiently developed, both in itself and as to its most essential application, to have arrived at that state of consistency in which we ought to strive to arrange its different parts in a single system, in order to prepare for new advances. We may even observe that the last important improvements of the science have directly paved the way for this important philosophical operation, by impressing on its principal parts a character of unity which did not previously exist.To form a just idea of the object of mathematical science, we may start from the indefinite and meaningless definition of it usually given, in calling it "The science of magnitudes," or, which is more definite, "The science which has for its object the measurement of magnitudes." Let us see how we can rise from this rough sketch (which is singularly deficient in precision and depth, though, at bottom, just) to a veritable definition, worthy of the importance, the extent, and the difficulty of the science.
THE OBJECT OF MATHEMATICS.
Measuring Magnitudes. The question of measuring a magnitude in itself presents to the mind no other idea than that of the simple direct comparison of this magnitude with another similar magnitude, supposed to be known, which it takes for the unit of comparison among all others of the same kind. According to this definition, then, the science of mathematics—vast and profound as it is with reason reputed to be—instead of being an immense concatenation of prolonged mental labours, which offer inexhaustible occupation to our intellectual activity, would seem to consist of a simple series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure them, by the aid of operations of similar character to the superposition of lines, as practiced by the carpenter with his rule.The error of this definition consists in presenting as direct an object which is almost always, on the contrary, very indirect. The direct measurement of a magnitude, by superposition or any similar process, is most frequently an operation quite impossible for us to perform; so that if we had no other means for determining magnitudes than direct comparisons, we should be obliged to renounce the knowledge of most of those which interest us.Difficulties. The force of this general observation will be understood if we limit ourselves to consider specially the particular case which evidently offers the most facility—that of the measurement of one straight line by another. This comparison, which is certainly the most simple which we can conceive, can nevertheless scarcely ever be effected directly. In reflecting on the whole of the conditions necessary to render a line susceptible of a direct measurement, we see that most frequently they cannot be all fulfilled at the same time. The first and the most palpable of these conditions—that of being able to pass over the line from one end of it to the other, in order to apply the unit of measurement to its whole length—evidently excludes at once by far the greater part of the distances which interest us the most; in the first place, all the distances between the celestial bodies, or from any one of them to the earth; and then, too, even the greater number of terrestrial distances, which are so frequently inaccessible. But even if this first condition be found to be fulfilled, it is still farther necessary that the length be neither too great nor too small, which would render a direct measurement equally impossible. The line must also be suitably situated; for let it be one which we could measure with the greatest facility, if it were horizontal, but conceive it to be turned up vertically, and it becomes impossible to measure it.The difficulties which we have indicated in reference to measuring lines, exist in a very much greater degree in the measurement of surfaces, volumes, velocities, times, forces, &c. It is this fact which makes necessary the formation of mathematical science, as we are going to see; for the human mind has been compelled to renounce, in almost all cases, the direct measurement of magnitudes, and to seek to determine them indirectly, and it is thus that it has been led to the creation of mathematics.General Method. The general method which is constantly employed, and evidently the only one conceivable, to ascertain magnitudes which do not admit of a direct measurement, consists in connecting them with others which are susceptible of being determined immediately, and by means of which we succeed in discovering the first through the relations which subsist between the two. Such is the precise object of mathematical science viewed as a whole. In order to form a sufficiently extended idea of it, we must consider that this indirect determination of magnitudes may be indirect in very different degrees. In a great number of cases, which are often the most important, the magnitudes, by means of which the principal magnitudes sought are to be determined, cannot themselves be measured directly, and must therefore, in their turn, become the subject of a similar question, and so on; so that on many occasions the human mind is obliged to establish a long series of intermediates between the system of unknown magnitudes which are the final objects of its researches, and the system of magnitudes susceptible of direct measurement, by whose means we finally determine the first, with which at first they appear to have no connexion.Illustrations. Some examples will make clear any thing which may seem too abstract in the preceding generalities.1. Falling Bodies. Let us consider, in the first place, a natural phenomenon, very simple, indeed, but which may nevertheless give rise to a mathematical question, really existing, and susceptible of actual applications—the phenomenon of the vertical fall of heavy bodies.The mind the most unused to mathematical conceptions, in observing this phenomenon, perceives at once that the two quantities which it presents—namely, the height from which a body has fallen, and the time of its fall—are necessarily connected with each other, since they vary together, and simultaneously remain fixed; or, in the language of geometers, that they are "functions" of each other. The phenomenon, considered under this point of view, gives rise then to a mathematical question, which consists in substituting for the direct measurement of one of these two magnitudes, when it is impossible, the measurement of the other. It is thus, for example, that we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling to its bottom, and by suitable procedures this inaccessible depth will be known with as much precision as if it was a horizontal line placed in the most favourable circumstances for easy and exact measurement. On other occasions it is the height from which a body has fallen which it will be easy to ascertain, while the time of the fall could not be observed directly; then the same phenomenon would give rise to the inverse question, namely, to determine the time from the height; as, for example, if we wished to ascertain what would be the duration of the vertical fall of a body falling from the moon to the earth.In this example the mathematical question is very simple, at least when we do not pay attention to the variation in the intensity of gravity, or the resistance of the fluid which the body passes through in its fall. But, to extend the question, we have only to consider the same phenomenon in its greatest generality, in supposing the fall oblique, and in taking into the account all the principal circumstances. Then, instead of offering simply two variable quantities connected with each other by a relation easy to follow, the phenomenon will present a much