The Philosophy of Mathematics (Comte)

The Philosophy of Mathematics, published in 1851, is W. M. Gillespie's translation of the mathematics discourse contained in Auguste Comte's 7 volume Cours De Philosophie Positive, written 1830–1842.

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The Science of Mathematics
frontispiece of Comte's
The Philosophy of Mathematics (1851)

Introduction

General Considerations

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

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.

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."

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

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.

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.

The difficulties... 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... 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.

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.

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

[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.

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 ...

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...

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.

[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.

[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.

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.

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.

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.

[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.

This enunciation, instead of giving the idea of only an art, as do... the ordinary definitions, characterizes... a true science, and shows it... to be composed of an immense chain of intellectual operations...

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

[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.

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.

[E]very true science has for its object the determination of certain phenomena by means of others, in accordance with the relations which exist between them.

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

[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...

We will... having determined above what is the general object of mathematical labours, now characterize... the principal different orders of inquiries, of which they are constantly composed.

Their different Objects. The complete solution of every mathematical question divides itself necessarily into two parts, of natures... distinct, and with relations... determinate.

[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.

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

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.

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.

When this concrete part is completed, the inquiry becomes one of... another nature. Knowing that the spaces passed through by the body in each successive second of its fall increase as the series of odd numbers, we have then a problem purely numerical and abstract; to deduce the height from the time, or the time from the height; and this consists in finding that the first of these two quantities... is a known multiple of the second power of the other; from which, finally, we have to calculate...

In this example the concrete question is more difficult than the abstract one. The reverse would be the case if we considered the same phenomenon in its greatest generality.

[T]he mathematical law of the phenomenon may be very simple, but very difficult to obtain, or it may be easy to discover, but very complicated; so that the two great sections of mathematical science, when we compare them as wholes, must be regarded as exactly equivalent in extent.. in difficulty... in importance.

Their different Natures. These two parts, essentially distinct in their object... are no less so with regard to the nature of the inquiries...

The first should be called concrete, since it... depends on the character of the phenomena... and must... vary when we examine new phenomena; while the second is... independent of the... objects examined, and is concerned with only the numerical relations... for which reason it should be called abstract.

The same relations may exist in a great number of different phenomena, which, in spite of their extreme diversity, will be viewed... as offering an analytical question susceptible, when studied by itself, of being resolved... for all.

Thus... the same law... between the space and the time of the vertical fall of a body in a vacuum, is found... in many other phenomena which offer no analogy with the first nor with each other; for it expresses the relation between the surface of a spherical body and the length of its diameter; it determines, in like manner, the decrease of the intensity of light or of heat in relation to the distance of the objects lighted or heated, &c.

[T]he concrete part will have necessarily to be again taken up for each question separately, without the solution of any one of them being able to give any direct aid, in that connexion, for the solution of the rest.

The abstract part of mathematics is, then, general in its nature; the concrete part, special.

[C]oncrete mathematics has a philosophical character, which is essentially experimental, physical, phenomenal; while that of abstract mathematics is purely logical, rational.

The concrete part of every mathematical question is... founded on the consideration of the external world, and could never be resolved by a simple series of intellectual combinations. The abstract part... when... completely separated, can consist only of a series of logical deductions, more or less prolonged; for if we have once found the equations of a phenomenon, the determination of the quantities, by means of one another, is a matter for reasoning only, whatever the difficulties may be.

It belongs to the understanding alone to deduce from these equations results... contained in them... without... occasion to consult anew the external world; the consideration of which, having become... foreign to the subject, ought... to be... set aside... to reduce the labour to its true peculiar difficulty.

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, acoustics, optics, &c., have finally become applications of mathematical science to certain orders of observations.
- Footnote: The investigation of the mathematical phenomena of the laws of heat by Baron Fourier has led to the establishment... of Thermological equations. This great discovery tends to elevate our philosophical hopes as to the future extensions of the legitimate applications of mathematical analysis, and renders it proper... to regard Thermology as a third principal branch of concrete mathematics.

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.
- Translator's Footnote: The translator has felt justified in employing this very convenient word [the Calculus] (for which our language has no precise equivalent) ...in its most extended sense, in spite of... being often popularly confounded with its Differential and Integral department.

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 equations, 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 equation... 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.

Quotes about The Philosophy of Mathematics

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.

Preface, The Philosophy of Mathematics

by W. M. Gillespie

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.

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