The perfection of the positive system, towards which it unceasingly tends, although very probably it may never reach it, would be to represent all observable phenomena as particular cases of a single general fact, such as, for example, that of gravitation. The fundamental identity of phenomena, the reduction of particular laws to a supreme law; this is an ideal which we are free to entertain. Comte, after d’Alembert and Saint-Simon, has formulated it himself at the beginning of the _Cours de philosophie positive_.[46] Unfortunately this ideal is not realisable. We apply a very weak intellect to a very complicated world.[47] The unity which, scorning experience, we might establish, would naturally be valueless. For the several categories of phenomena proposed to us seem irreducible. If this[48] be the case, the pursuit after scientific unity is “irrational.” Comte ended by treating it as an “absurd utopia.”[49] However, this utopia is forever reappearing; for the human mind is secretly attached to it. It is because, on the one hand, unity pleases it above all things, and on the other hand because there is here an illusion produced and maintained by a philosophy born of mathematical inspiration. Descartes’ discovery which allowed questions of geometry to be dealt with by algebra has been the occasion of a grave error. It gave rise to the thought that differences of quality could be reduced to differences of quantity. Hence the idea of “reducing” the various categories of phenomena to one another. But this was a wrong interpretation of the principle of analytical geometry. Even there, we have a translation, not reduction, “The geometrical ideas of form and of situation,” says Comte--and Mr. Renouvier will repeat it after him--“are not naturally more like numerical notions than the other real conceptions. Every phenomenon, even social, would certainly have its equation, as a figure or a motion if its law were known to us with sufficient precision. Analysis is therefore but an instrument of incomparable power for the study of phenomena. But, from the fact that we can make use of it, it does not in the least follow that the phenomena may be all brought back to an identical type. Quality is in no way by this means reduced to quantity, which is something entirely abstract, and this no more takes place in the case of geometrical quality than in the case of any other. Neither can the geometrical quality be reduced to pure analysis, nor the physical to the geometrical, nor the living to the inorganic, nor the social to the biological. At every stage something qualitatively new appears. Whether or no we can formulate the relations of phenomena in the form of an equation, their heterogeneity subsists always irreducible.