The Philosophy of Mathematics (Comte)

"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."