[Mathematics]¶
Core Argument¶
Mathematics¶
Engels’s treatment of mathematics is a sustained dialectical assault on the metaphysical conception of the discipline as a realm of self-evident, a priori truths. The core argument is that mathematical concepts are not pure products of thought but thought-determinations abstracted from reality, whose apparent axioms are in fact provable dialectically and far from tautological. The rigid distinctions of elementary arithmetic—between addition and multiplication, subtraction and division—dissolve under dialectical scrutiny: multiplication is merely abbreviated addition, division abbreviated subtraction, and algebraic manipulation allows any operation to be presented as its opposite. This capacity for transformation, Engels argues, is a powerful lever of mathematical science.
The decisive historical turning point was Descartes’s introduction of variable magnitude, which brought motion—and therefore dialectics—into mathematics. This necessarily led to the differential and integral calculus. Number, seemingly the purest quantitative determination, is in fact “chock-full of qualitative differences”: prime numbers, powers, and roots introduce definite qualities; the infinitely large and small generate a qualitative opposition to the point of incommensurability. Even zero is not empty but a definite borderline between positive and negative, richer in content than any other number. The differential calculus treats straight and curved as equal, while trigonometry dialectically connects the triangle to the circle.
Engels’s most striking polemic is his insistence that the infinitesimal calculus, often regarded as a pure mental creation, has real prototypes in nature. Terrestrial mechanics treats the Earth’s mass as infinitely large relative to falling bodies; stellar distances render solar system distances infinitely small. Molecules are differentials relative to mechanical masses, and nature operates with them exactly as mathematics does with its abstract differentials. A growing sulphur crystal illustrates differentiation; the condensation of vapour, integration. Chemical equations are differential equations integrated by atomic weights. Matter is divided into groups—star systems, solar system, terrestrial masses, molecules, atoms, ether particles—each standing in an infinite or infinitesimal ratio to the next. Pure mathematics forgets these analogies, making infinity mysterious, but mathematical infinity is taken from reality and can only be explained from it. The unity of thought and being is the unconscious premise of all theoretical thought; mathematical axioms appear self-evident only because they are the sediment of inherited, accumulated experience.