The Philosophy of Mathematics (Comte)

"The abstract part of mathematics is then purely instrumental, and is only an immense and admirable extension of natural logic to a certain class of deductions."

"On the other hand, geometry and mechanics, which... constitute the concrete part, must be viewed as real natural sciences, founded on observation, like all the rest, although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization, which has sometimes caused a misconception of the experimental character of their first principles."

"We see, by this... comparison, how natural and profound is our fundamental division of mathematical science."

"Concrete Mathematics having for its object the discovery of the equations of phenomena... must be composed of as many distinct sciences as we find... distinct categories among natural phenomena. But... there are directly but two great general classes of phenomena, whose equations we constantly know... firstly, geometrical, and, secondly, mechanical phenomena."

"Thus... the concrete part of mathematics is composed of Geometry and Rational Mechanics."

"[I]f all the parts of the universe were conceived as immovable, we should... have only geometrical phenomena to observe, since all would be reduced to relations of form, magnitude, and position; then, having regard to the motions which take place in it, we would have also to consider mechanical phenomena."

"Hence the universe, in the statical point of view, presents only geometrical phenomena; and, considered dynamically, only mechanical phenomena."

"Thus geometry and mechanics constitute the two fundamental natural sciences, in this sense, that all natural effects may be conceived as simple necessary results, either of the laws of extension or of the laws of motion."

"But... the difficulty is... to effectually reduce each principal question of natural philosophy, for a certain determinate order of phenomena, to the question of geometry or mechanics... This transformation, which requires great progress... in the study of each class of phenomena, has thus far been... executed only for those of astronomy, and for a part of... terrestrial physics..."

"It is thus that astronomy, , optics, &c., have finally become applications of mathematical science to certain orders of observations."

"But these applications not being by their nature rigorously circumscribed, to confound them with the science would be to assign to it a vague and indefinite domain... [as] is done in the usual division, so faulty... of the mathematics into "Pure" and "Applied.""

"The nature of abstract mathematics... is composed of what is called the Calculus, taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis."

"The Calculus has the solution of all questions relating to numbers for its peculiar object. Its starting point is... necessarily, the knowledge of the precise relations, i.e., of the s, between the different magnitudes which are simultaneously considered; that which is... the stopping-point of concrete mathematics."

"[T]he final object of the calculus always is to obtain... the values of the unknown quantities by means of those which are known."

"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."

"Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. It constitutes the first and the most perfect of all the fundamental sciences. The ideas with which it occupies itself are the most universal, the most abstract, and the most simple which it is possible for us to conceive."

"[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."

"Could we... without the aid of analysis, perceive the least resemblance between the determination of the direction of a curve at each of its points and that of the velocity acquired by a body at every instant of its variable motion? and yet these questions, however different they may be, compose but one in the eyes of the geometer."

"The high relative perfection of mathematical analysis... is not due, as some have thought, to the nature of the signs [mathematical notation] which are employed as instruments of reasoning, eminently concise and general... [A]ll great analytical ideas have been formed without the algebraic signs having been of any essential aid, except for working them out after the mind had conceived them."

"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."

"Thus... the phenomena of living bodies, even when considered (to take the most complicated case) in the state of disease... is it not... that all the questions of therapeutics may be viewed as consisting in determining the quantities of the different agents which modify the organism... to bring it to its normal state ..?"

"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."

"As a result of this gradual generalization, there are now no geometers who do not consider it, in a purely theoretical sense, as capable of being applied to all our real ideas... so that every phenomenon is logically susceptible of being represented by an '... excepting the difficulty of discovering it, and then of resolving it, which may be, and oftentimes are, superior to the greatest powers of the human mind."

"Its Limitations. ...[I]t is no less indispensable to consider... the great... limitations which, through the feebleness of our intellect, narrow in... its... domain, in proportion as phenomena, in becoming special, become complicated. ...[I]t soon becomes insurmountable."

"[I]t is only in inorganic physics, at the most, that we can justly hope ever to obtain that high degree of scientific perfection."

"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."

"[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."

"Every precise idea of fixed numbers is truly out of place in the phenomena of living bodies... when we attach any importance to the exact relations of the values assigned."

"We ought not, however, on this account, to cease to conceive all phenomena as being necessarily subject to mathematical laws... The most complex phenomena of living bodies are doubtless essentially of no other special nature than the simplest phenomena of unorganized matter."

"There is a second reason... Even if we could ascertain the mathematical law which governs each agent, taken by itself, the combination of so great a number of conditions would render the corresponding mathematical problem so far above our feeble means, that the question would remain in most cases incapable of solution."

"[T]he very simple phenomenon of the flow of a fluid through a given orifice, by virtue of its gravity alone, has not as yet any complete mathematical solution, when we take into the account all the essential circumstances. It is the same even with the still more simple motion of a solid projectile in a resisting medium."

"Why has mathematical analysis been able to adapt itself with such admirable success to the most profound study of celestial phenomena? Because they are... much more simple than any others."

"The most complicated problem... of the modification produced in the motions of two bodies tending towards each other by virtue of their gravitation, by the influence of a third body acting on both of them in the same manner, is much less complex than the most simple terrestrial problem. And, nevertheless, even it presents difficulties so great that we yet possess only approximate solutions..."

"[T]he high perfection to which solar astronomy has been able to elevate itself... is... essentially due to... all the particular, and... accidental facilities presented by the peculiarly favourable constitution of our planetary system. The planets... are quite few in number, and their masses... very unequal, and much less than that of the sun; they are... very distant from one another; they have forms almost spherical; their orbits are nearly circular, and only slightly inclined to each other, and so on. It results from all these circumstances that the perturbations are generally inconsiderable, and that... it is usually sufficient to take into the account, in connexion with the action of the sun... the influence of only one other planet..."

"If... our solar system had been composed of a greater number of planets concentrated into a less space, and nearly equal in mass; if their orbits had presented very different inclinations, and considerable eccentricities; if these bodies had been of a more complicated form, such as very eccentric ellipsoids... supposing the same law of gravitation to exist, we should not yet have succeeded in subjecting the... celestial phenomena to our mathematical analysis, and probably we should not even have been able to disentangle the present principal law."

"Important as it was to render apparent the rigorous logical universality of mathematical science, it was equally so to indicate the conditions which limit for us its real extension, so as not to... lead the human mind astray from the true scientific direction in the study of the most complicated phenomena, by the chimerical search after an impossible perfection."

"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."

"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."

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

"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."

"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."

"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."

"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."

"[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."