education

"Every science consists in the co-ordination of facts; if the different observations were entirely isolated, there would be no science."

"[I]t is... necessary... to ascertain with precision the relations which exist between the quantities which we are considering. This first branch of inquiries constitutes that which I call the concrete part of the solution. When it is finished, the question changes... now reduced to a pure question of numbers, consisting simply in determining unknown numbers... This second branch of inquiries is what I call the abstract part of the solution."

"Taking up again... the vertical fall of a heavy body, and considering the simplest case... to succeed in determining, by means of one another, the height... fallen, and the duration... we must commence by discovering the exact relation of these two quantities, ...[i.e.,] the equation which exists between them."

"The first condition which is necessary in order that phenomena may admit of mathematical laws, susceptible of being discovered... is, that their different quantities should admit of being expressed by fixed numbers."

"It is this fact which makes necessary the formation of mathematical science... 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... led to the creation of mathematics."

"[C]alculation will become successively... more complicated, if the parts... supposed... known cannot themselves be determined (as is most frequently the case) except in an indirect manner, by the aid of new auxiliary systems, the number of which... becomes... considerable."

"General Method. The general method... 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."

"The difficulties... in reference to measuring lines, exist in a very much greater degree in the measurement of surfaces, volumes, velocities, times, forces, &c."

"[O]n many occasions the... 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... determine the first... which at first... appear to have no connexion."

"[T]he knowledge of the desired distance, instead of being obtained directly, will be the result of a mathematical calculation, which will consist in deducing it from the observed elements by means of the relation which connects it with them."

"The distance being once determined, the knowledge of it will frequently be sufficient for obtaining new quantities, which will become the subject of new mathematical questions. Thus, when we know at what distance any object is situated... its apparent diameter will... permit us to determine indirectly its real dimensions, however inaccessible it may be, and, by... analogous investigations, its surface... volume... weight, and a number of other properties... which seemed forbidden to us."

"According[ly]... the spirit of mathematics consists in... regarding all the quantities which any phenomenon can present, as connected and interwoven..."

"[I]f we did not fear to multiply calculations unnecessarily... the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles."

"We are now able to define mathematical science... by assigning... as its object the indirect measurement of magnitudes, and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them."

"The preceding explanations establish... the propriety of the name [Greek: μάθημα, máthēma, 'knowledge, study, learning'] employed to designate the science... This denomination... to-day... signifies simply science [Latin scientia 'knowledge'] in general. Such a designation, rigorously exact for the Greeks, who had no other real science, could be retained by the moderns only to indicate the mathematics as the science, beyond all others—the science of sciences."

"[T]here is... no phenomenon which cannot give rise to considerations of this kind; whence results the naturally indefinite extent and... rigorous logical universality of mathematical science. We shall seek... to circumscribe as exactly as possible its real extension."

"[S]cience is essentially destined to dispense, so far as the different phenomena permit it, with all direct observation, by enabling us to deduce from the smallest possible number of immediate data the greatest possible number of results. Is not this the real use, whether in speculation or in action, of the laws which we succeed in discovering among natural phenomena?"

"Mathematical science... pushes to the highest possible degree the same kind of researches which are pursued, in degrees more or less inferior, by every real science..."

"Hence follows the fundamental division of general mathematical science into two great sciences—Abstract Mathematics, and Concrete Mathematics."

"This science, although nearer perfection than any other, is really little advanced as yet, so that this object is rarely attained in a manner completely satisfactory."

"[O]ur conceptions having been so generalized and simplified that a single analytical question, abstractly resolved, contains the implicit solution of a great number of diverse physical questions..."

"[T]he human mind must necessarily acquire by these means a greater facility in perceiving relations between phenomena which at first appeared entirely distinct from one another."

"The superior perfection of the science of the calculus is due principally to the extreme simplicity of the ideas which it considers, by whatever signs they may be expressed; so that there is not the least hope, by any artifice of scientific language, of perfecting to the same degree theories which refer to more complex subjects, and which are necessarily condemned by their nature to a greater or less logical inferiority."

"Its Universality. ...[I]n the purely logical point of view, this science is... necessarily and rigorously universal; for there is no question... which may not be finally conceived as consisting in determining certain quantities from others by means of certain relations, and consequently as admitting of reduction... to a simple question of numbers."

"The fundamental idea of Descartes on the relation of the concrete to the abstract in mathematics, has proven, in opposition to the superficial distinction of metaphysics, that all ideas of quality may be reduced to those of quantity."

"This conception, established at first by its immortal author in relation to geometrical phenomena only, has since been... extended to mechanical phenomena, and in our days to those of heat."

"The Object of Mathematics. Measuring Magnitudes. According to this definition... the science of mathematics—vast and profound as it is... instead of being an immense concatenation of prolonged mental labours... [of] 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... by... operations... similar... to the superposition of lines, as practiced by the carpenter with his rule."

"Let us... rise from this rough sketch... to a veritable definition, worthy of the importance, the extent, and the difficulty of the science."

"The error of this definition consists in presenting as direct an object which is almost always, on the contrary, very indirect."

"[B]eing 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... excludes... the greater part of the distances which interest us... all the distances between the celestial bodies, or from any one of them to the earth; and... even the greater number of terrestrial distances... so frequently inaccessible."

"To form a just idea of the object of mathematical science... start from the indefinite and meaningless definition of it usually given, in calling it "The science of magnitudes," or... more definite, "The science which has for its object the measurement of magnitudes.""

"[T]his indirect determination of magnitudes may be indirect in very different degrees."

"Falling Bodies. ...The mind ...perceives that the two quantities which it presents— ...the height from which a body has fallen, and the time of its fall—are necessarily connected ...[I]n the language of geometers, that they are "functions" of each other. The phenomenon... gives rise then to a mathematical question... in substituting for the direct measurement of one... when it is impossible, the measurement of the other. ...[T]hus ...we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling ...On other occasions it is the height ...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 ..."

"Astronomical Facts. It is by such calculations that man has been able to ascertain, not only the distances from the planets to the earth, and, consequently, from each other, but their actual magnitude, their true figure... their respective masses, their mean densities, the principal circumstances of the fall of heavy bodies on the surface of each of them, &c."

"Although Mathematical Science is the most ancient and the most perfect... the general idea which we ought to form of it has not yet been clearly determined. Its definition and its principle divisions have remained till now vague and uncertain."

"[T]he plural name—"The Mathematics"—would alone suffice to indicate the want of unity in the common conception of it."

"[I]t was not till the commencement of the last century that the different fundamental conceptions which constitute this great science were each... 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"

"Inaccessible Distances. ...[T]o determine a distance which is not susceptible of direct measurement; it will be ...conceived as making part of a figure, or ...system of lines, chosen ...such ...that all its other parts may be observed directly; thus, in the case ...most simple, and to which all ...others may be ...reduced, the proposed distance will be considered as belonging to a triangle, in which we can determine directly either another side and two angles, or two sides and one angle."

"By the power of mathematical theories, all these different results, and many others... have required no other direct measurements than... a very small number of straight lines, suitably chosen, and of a greater number of angles."

"This inquiry... constitutes incomparably the greater part of the problem. The true scientific spirit is so modern, that no one, perhaps, before Galileo, had ever remarked the increase of velocity which a body experiences in its fall: a circumstance which excludes the hypothesis, towards which our mind (always involuntarily inclined to suppose in every phenomenon the most simple functions, without any other motive than its greater facility in conceiving them) would be naturally led, that the height was proportional to the time. In a word, this first inquiry terminated in the discovery of the law of Galileo."

"In this example the mathematical question is very simple... 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... But, to extend the question, we have only to consider the same phenomenon in its greatest generality..."

"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."

"[T]he whole of organic physics, and probably also the most complicated parts of inorganic physics, are necessarily inaccessible, by their nature, to our mathematical analysis, by reason of the extreme numerical variability of the corresponding phenomena."

"[O]btain 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."

"Passages which are obscure at the first reading will brighten up at the second; and as ...[the student's] studies cover a larger portion of... Mathematics, he will see more and more clearly their relations to one another, and to those which he is next to take up."

"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 Augusts 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... 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."

"It would be inconsistent with the scale of this work, and not necessary to its design, to carry the analysis of the truths and processes of algebra any further; which is moreover the less needful, as the task has been recently and thoroughly performed by other writers. Professor Peacock’s Algebra, and Mr. Whewell’s Doctrine of Limits, should be studied by every one who desires to comprehend the evidence of mathematical truths, and the meaning of the obscurer processes of the calculus; while, even after mastering these treatises, the student will have much to learn on the subject from M. Comte, of whose 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."

"Having thus exhibited the essential object and the principal composition of mathematical science, as well as its general relations with... natural philosophy, we have now to pass to... examination of the great sciences of which it is composed."

"John Stuart Mill, A System of Logic (1843) p. 369 of the 1846 edition."