oys this mode of outlook without hesitation. For terrestrial mechanics the mass of the earth is regarded As infinitely large, just as for astronomy terrestrial masses and the meteorites corresponding to them are regarded as infinitely small, and just as the distances and masses of the planets of the solar system dwindle to nothing as soon as astronomy investigates the constitution of our stellar system extending beyond the nearest fixed stars. As soon, however, as the mathematicians withdraw into their impregnable fortress of abstraction, so-called pure mathematics, all these analogies are forgotten, infinity becomes something totally mysterious, and the manner in which operations are carried out with it in analysis appears as something absolutely incomprehensible, contradicting all experience and all reason. The stupidities and absurdities by which mathematicians have rather excused than explained their mode of procedure, which remarkably enough always leads to correct results, exceed the worst apparent and real fantasies, e.g., of the Hegelian philosophy of nature, about which mathematicians and natural scientists can never adequately express their horror. What they charge Hegel with doing, viz., pushing abstractions to the extreme limit, they do themselves on a far greater scale. They forget that the whole of so-called pure mathematics is concerned with abstractions, that all its magnitudes, strictly speaking, are imaginary, and that all abstractions when pushed to extremes are transformed into nonsense or into their opposite. Mathematical infinity is taken from reality, although unconsciously, and therefore can only be explained from reality and not from itself, from mathematical abstraction. And, as we have seen, if we investigate reality in this regard we come also upon the real relations from which the mathematical relation of infinity is taken, and even the natural analogies of the mathematical way in which this relation operates. And thereby the matter is explained.
(Haeckel’s bad reproduction of the identity of thinking and being. But also the contradiction between continuous and discrete matter; see Hegel.)
The differential calculus for the first time makes it possible for natural science to represent mathematically processes and not only states: motion.
Application of mathematics: in the mechanics of solid bodies it is absolute, in that of gases approximate, in that of fluids already more difficult; in physics more tentative and relative; in chemistry, simple equations of the first order and of the simplest nature; in biology=0
213. In the former case, Engels has in mind Hegel’s remark that in arithmetic, thought moves in “thoughtlessness” (Science of Logic, Book I, Section II, Chapter 2, Observation on the employment of numerical determinations to express philosophic concepts); in the latter case, Hegel’s statement that “already the natural numerical system exemplifies a nodal line of qualitative moments, which manifest themselves in the merely external progression,” etc. (ibid., Section III, Chapter 2, Observation on examples of nodal lines of measure-relations; natura non facit saltum).
214. This expression occurs in the book by Bossut, referred to by Engels in the fragment “Straight and Curved.” In the chapter on “Integral Calculation with Finite Differences,” Bossut examines first of all the following problem: “To integrate or sum the whole-number steps of a variable magnitude x.” Bossut assumes that the difference Dx is constant and he denotes it by the Greek letter w. Since the sum of Dx or of w is equal to x, the sum of w×1 or of wx0 is also equal to x. Bossut writes this equation in the form Swx0=x. Bossut then takes out the constant w and puts it before the summation sign, obtaining the expression wSx0 = x, from x/w which he obtains the equation Sx0= w/w. This last equation is then to used by Bossut to find the magnitudes Sx, Sx2, Sx3, etc., for solving other problems. See Bossut, Traités de Calcul differentiel et de Calcul integral, t. 1, Paris, 1798, p. 38.
215. Ch. Bossut, Traités de Calcul diffe