Thus it is not true that the historical succession of the divisions of mathematics has corresponded with the order of decreasing generality. It is not true that abstract mathematics was evolved antecedently to, and independently of, concrete mathematics. It is not true that of the subdivisions of abstract mathematics, the more general came before the more special. And it is not true that concrete mathematics, in either of its two sections, began with the most abstract and advanced to the less abstract truths.

It may be well to mention, parenthetically, that, in defending his alleged law of progression from the general to the special, M. Comte somewhere comments upon the two meanings of the word general , and the resulting liability to confusion. Without now discussing whether the asserted distinction exists in other cases, it is manifest that it does not exist here. In sundry of the instances above quoted, the endeavours made by M. Comte himself to disguise, or to explain away, the precedence of the special over the general, clearly indicate that the generality spoken of is of the kind meant by his formula. And it needs but a brief consideration of the matter to show that, even did he attempt it, he could not distinguish this generality which, as above proved, frequently comes last, from the generality which he says always comes first. For what is the nature of that mental process by which objects, dimensions, weights, times, and the rest, are found capable of having their relations expressed numerically? It is the formation of certain abstract conceptions of unity, duality, and multiplicity, which are applicable to all things alike. It is the invention of general symbols serving to express the numerical relations of entities, whatever be their special characters. And what is the nature of the mental process by which numbers are found capable of having their relations expressed algebraically? It is the same. It is the formation of certain abstract conceptions of numerical functions which are constant whatever be the magnitudes of the numbers. It is the invention of general symbols serving to express the relations between numbers, as numbers express the relations between things. Just as arithmetic deals with the common properties of lines, areas, bulks, forces, periods; so does algebra deal with the common properties of the numbers which arithmetic presents.

Having shown that M. Comte’s alleged law of progression does not hold among the several parts of the same science, let us see how it agrees with the facts when applied to the separate sciences. “Astronomy,” says M. Comte ( Positive Philosophy , Book III.), “was a positive science, in its geometrical aspect, from the earliest days of the school of Alexandria; but Physics, which we are now to consider, had no positive character at all till Galileo made his great discoveries on the fall of heavy bodies.” On this, our comment is simply that it is a misrepresentation based upon an arbitrary misuse of words – a mere verbal artifice. By choosing to exclude from terrestrial physics those laws of magnitude, motion, and position, which he includes in celestial physics, M. Comte makes it appear that the last owes nothing to the first. Not only is this unwarrantable, but it is radically inconsistent with his own scheme of divisions. At the outset he says – and as the point is important we quote from the original – “Pour la physique inorganique nous voyons d’abord, en nous conformant toujours à l’ordre de généralité et de dépendance des phénomènes, qu’elle doit être partagée en deux sections distinctes, suivant qu’elle considère les phénomènes généraux de l’univers, ou, en particulier, ceux que présentent les corps terrestres. D’où la physique céleste, ou l’astronomie, soit géométrique, soit mechanique; et la physique terrestre.” Here then we have inorganic physics clearly divided into celestial physics and terrestrial physics – the phenomena presented by the universe, and the phenomena presented by earthly bodies. If now celestial bodies and terrestrial bodies exhibit sundry leading phenomena in common, as they do, how can the generalization of these common phenomena be considered as pertaining to the one class rather than to the other? If inorganic physics includes geometry (which M. Comte has made it do by comprehending geometrical astronomy in its sub-section, celestial physics); and if its other sub-section, terrestrial physics, treats of things having geometrical properties; how can the laws of geometrical relations be excluded from terrestrial physics? Clearly if celestial physics includes the geometry of objects in the heavens, terrestrial physics includes the geometry of objects on the earth. And if terrestrial physics includes terrestrial geometry, while celestial physics includes celestial geometry, then the geometrical part of terrestrial physics precedes the geometrical part of celestial physics; seeing that geometry gained its first ideas from surrounding objects. Until men had learnt geometrical relations from bodies on the earth, it was impossible for them to understand the geometrical relations of bodies in the heavens. So, too, with celestial mechanics, which had terrestrial mechanics for its parent. The very conception of force , which underlies the whole of mechanical astronomy, is borrowed from our earthly experiences; and the leading laws of mechanical action as exhibited in scales, levers, projectiles, &c., had to be ascertained before the dynamics of the Solar System could be entered upon. What were the laws made use of by Newton in working out his grand discovery? The law of falling bodies disclosed by Galileo; that of the composition of forces also disclosed by Galileo; and that of centrifugal force found out by Huyghens – all of them generalizations of terrestrial physics. Yet, with facts like these before him, M. Comte places astronomy before physics in order of evolution! He does not compare the geometrical parts of the two together, and the mechanical parts of the two together; for this would by no means suit his hypothesis. But he compares the geometrical part of the one with the mechanical part of the other, and so gives a semblance of truth to his position. He is led away by a verbal illusion. Had he confined his attention to the things and disregarded the words, he would have seen that before mankind scientifically co-ordinated any one class of phenomena displayed in the heavens, they had previously co-ordinated a parallel class of phenomena displayed on the surface of the earth.

Were it needful we could fill a score pages with the incongruities of M. Comte’s scheme. But the foregoing samples will suffice. So far is his law of evolution of the sciences from being tenable, that, by following his example, and arbitrarily ignoring one class of facts, it would be possible to present, with great plausibility, just the opposite generalization to that which he enunciates. While he asserts that the rational order of the sciences, like the order of their historic development, “is determined by the degree of simplicity, or, what comes to the same thing, of generality of their phenomena;” it might contrariwise be asserted that, commencing with the complex and the special, mankind have progressed step by step to a knowledge of greater simplicity and wider generality. So much evidence is there of this as to have drawn from Whewell, in his History of the Inductive Sciences , the remark that “the reader has already seen repeatedly in the course of this history, complex and derivative principles presenting themselves to men’s minds before simple and elementary ones.” Even from M. Comte’s own work, numerous facts, admissions, and arguments, might be picked out, tending to show this. We have already quoted his words in proof that both abstract and concrete mathematics have progressed towards a higher degree of generality, and that he looks forward to a higher generality still. Just to strengthen this adverse hypothesis, let us take a further instance. From the particular case of the scales, the law of equilibrium of which was familiar to the earliest nations known, Archimedes advanced to the more general case of the lever of which the arms may or may not be equal; the law of equilibrium of which includes that of the scales. By the help of Galileo’s discovery concerning the composition of forces, D’Alembert “established, for the first time, the equations of equilibrium of any system of forces applied to the different points of a solid body” – equations which include all cases of levers and an infinity of cases besides. Clearly this is progress towards a higher generality – towards a knowledge more independent of special circumstances – towards a study of phenomena “the most disengaged from the incidents of particular cases;” which is M. Comte’s definition of “the most simple phenomena.” Does it not indeed follow from the admitted fact, that mental advance is from the concrete to the abstract, from the particular to the general, that the universal and therefore most simple truths are the last to be discovered? Should we ever succeed in reducing all orders of phenomena to some single law – say of atomic action, as M. Comte suggests – must not that law answer to his test of being independent of all others, and therefore most simple? And would not such a law generalize the phenomena of gravity, cohesion, atomic affinity, and electric repulsion, just as the laws of number generalize the quantitative phenomena of space, time and force?

The possibility of saying so much in support of an hypothesis the very reverse of M. Comte’s, at once proves that his generalization is only a half-truth. The fact is that neither proposition is correct by itself; and the actuality is expressed only by putting the two together. The progress of science is duplex. It is at once from the special to the general, and from the general to the special. It is analytical and synthetical at the same time.

M. Comte himself observes that the evolution of science has been accomplished by the division of labour; but he quite misstates the mode in which this division of labour has operated. As he describes it, it has been simply an arrangement of phenomena into classes, and the study of each class by itself. He does not recognize the effect of progress in each class upon all other classes: he recognizes only the effect on the class succeeding it in his hierarchical scale. Or if he occasionally admits collateral influences and intercommunications, he does it so grudgingly, and so quickly puts the admissions out of sight and forgets them, as to leave the impression that, with but trifling exceptions, the sciences aid one another only in the order of their alleged succession. The fact is, however, that the division of labour in science, like the division of labour in society, and like the “physiological division of labour” in individual organisms, has been not only a specialization of functions, but a continuous helping of each division by all the others, and of all by each. Every particular class of inquirers has, as it were, secreted its own particular order of truths from the general mass of material which observation accumulates; and all other classes of inquirers have made use of these truths as fast as they were elaborated, with the effect of enabling them the better to elaborate each its own order of truths. It was thus in sundry of the cases we have quoted as at variance with M. Comte’s doctrine. It was thus with the application of Huyghens’s optical discovery to astronomical observation by Galileo. It was thus with the application of the isochronism of the pendulum to the making of instruments for measuring intervals, astronomical and other. It was thus when the discovery that the refraction and dispersion of light did not follow the same law of variation, affected both astronomy and physiology by giving us achromatic telescopes and microscopes. It was thus when Bradley’s discovery of the aberration of light enabled him to make the first step towards ascertaining the motions of the stars. It was thus when Cavendish’s torsion-balance experiment determined the specific gravity of the Earth, and so gave a datum for calculating the specific gravities of the Sun and Planets. It was thus when tables of atmospheric refraction enabled observers to write down the real places of the heavenly bodies instead of their apparent places. It was thus when the discovery of the different expansibilities of metals by heat, gave us the means of correcting our chronometrical measurements of astronomical periods. It was thus when the lines of the prismatic spectrum were used to distinguish the heavenly bodies that are of like nature with the sun from those which are not. It was thus when, as recently, an electro-telegraphic instrument was invented for the more accurate registration of meridional transits. It was thus when the difference in the rates of a clock at the equator, and nearer the poles, gave data for calculating the oblateness of the earth, and accounting for the precession of the equinoxes. It was thus – but it is needless to continue. Here, within our own limited knowledge of its history, we have named ten additional cases in which the single science of astronomy has owed its advance to sciences coming after it in M. Comte’s series. Not only its minor changes, but its greatest revolutions have been thus determined. Kepler could not have discovered his celebrated laws had it not been for Tycho Brahe’s accurate observations; and it was only after some progress in physical and chemical science that the improved instruments with which those observations were made, became possible. The heliocentric theory of the Solar System had to wait until the invention of the telescope before it could be finally established. Nay, even the grand discovery of all – the law of gravitation – depended for its proof upon an operation of physical science, the measurement of a degree on the Earth’s surface. So completely, indeed, did it thus depend, that Newton had actually abandoned his hypothesis because the length of a degree, as then stated, brought out wrong results; and it was only after Picart’s more exact measurement was published, that he returned to his calculations and proved his great generalization. Now this constant intercommunion which, for brevity’s sake, we have illustrated in the case of one science only, has been taking place with all the sciences. Throughout the whole course of their evolution there has been a continuous consensus of the sciences – a consensus exhibiting a general correspondence with the consensus of the faculties in each phase of mental development; the one being an objective registry of the subjective state of the other.

From our present point of view, then, it becomes obvious that the conception of a serial arrangement of the sciences is a vicious one. It is not simply that, as M. Comte admits, such a classification “will always involve something, if not arbitrary, at least artificial;” it is not, as he would have us believe, that, neglecting minor imperfections such a classification may be substantially true; but it is that any grouping of the sciences in a succession gives a radically erroneous idea of their genesis and their dependencies. There is no “one rational order among a host of possible systems.” There is no “true filiation of the sciences.” The whole hypothesis is fundamentally false. Indeed, it needs but a glance at its origin to see at once how baseless it is. Why a series? What reason have we to suppose that the sciences admit of a linear arrangement? Where is our warrant for assuming that there is some succession in which they can be placed? There is no reason; no warrant. Whence then has arisen the supposition? To use M. Comte’s own phraseology, we should say, it is a metaphysical conception. It adds another to the cases constantly occurring, of the human mind being made the measure of Nature. We are obliged to think in sequence; it is a law of our minds that we must consider subjects separately, one after another: therefore Nature must be serial – therefore the sciences must be classifiable in a succession. See here the birth of the notion, and the sole evidence of its truth. Men have been obliged when arranging in books their schemes of education and systems of knowledge, to choose some order or other. And from inquiring what is the best order, have fallen into the belief that there is an order which truly represents the facts – have persevered in seeking such an order; quite overlooking the previous question whether it is likely that Nature has consulted the convenience of book-making. For German philosophers, who hold that Nature is “petrified intelligence,” and that logical forms are the foundations of all things, it is a consistent hypothesis that as thought is serial, Nature is serial; but that M. Comte, who is so bitter an opponent of all anthropomorphism, even in its most evanescent shapes, should have committed the mistake of imposing upon the external world an arrangement which so obviously springs from a limitation of the human consciousness, is somewhat strange. And it is the more strange when we call to mind how, at the outset, M. Comte remarks that in the beginning “