Dialectics of Nature. Frederick Engels 1883
Source: Dialectics of Nature, pp. 257-272;
First Published: by Progress Publishers, 1934, 6th printing 1974;
Translated: from the German by Clemens Dutt;
Transcribed: by Andy Blunden, 2006.
The so-called axioms of mathematics are the few thought determinations which mathematics needs for its point of departure. Mathematics is the science of magnitudes; its point of departure is the concept of magnitude. It defines this lamely and then adds the other elementary determinations of magnitude, not contained in the definition, from outside as axioms, so that they appear as unproved, and naturally also as mathematically unprovable. The analysis of magnitude would yield all these axiom determinations as necessary determinations of magnitude. Spencer is right in as much as what thus appears to us to be the self-evidence of these axioms is inherited. They are provable dialectically, in so far as they are not pure tautologies.
Mathematics. Nothing appears more solidly based than the difference between the four species of arithmetical operations, the elements of all mathematics. Yet right at the outset multiplication is seen to be an abbreviated addition, and division an abbreviated subtraction, of a definite number of equal numerical magnitudes; and in one case – when the divisor is a fraction – division is even carried out by multiplying by the inverted fraction. In algebraic calculation the thing is carried much further. Every subtraction (a-b) can be represented as an addition (-b+a), every division a/b as a multiplication a×1/b. In calculations with powers of magnitudes one goes much further still. All rigid differences between the kinds of calculation disappear, everything can be presented in the opposite form. A power can be put as a root (x2=√x4;), a root as a power (√x=x½). Unity divided by a power or root can be put as a power of the denominator (1/√x =x-½; 1/x3=x-3)
Multiplication or division of the powers of a magnitude becomes converted into addition or subtraction of their exponents. Any number can be conceived and expressed as the power of any other number (logarithms, y =ax). And this transformation of one form into the opposite one is no idle trifling, it is one of the most powerful levers of mathematical science, without which today hardly any of the more difficult calculations are carried out. If negative and fractional powers alone were abolished from mathematics, how far could one get?
(—.— =+, —/— =+, √-1, etc., to be expounded earlier.)
The turning point in mathematics was Descartes’ variable magnitude. With that came motion and hence dialectics in mathematics, and at once, too, of necessity the differential and integral calculus, which moreover immediately begins, and which on the whole was completed by Newton and Leibniz, not discovered by them.
Quantity and quality. Number is the purest quantitative determination that we know. But it is chock-full of qualitative differences. 1. Hegel, number and unity, multiplication, division, raising to a higher power, extraction of roots. Thereby, and this is not shown in Hegel, qualitative differences already make their appearance: prime numbers and products, simple roots and powers. 16 is not merely the sum of 16 ones, it is also the square of 4, the fourth power of 2. Still more. Prime numbers communicate new, definitely determined qualities to numbers derived from them by multiplication with other numbers; only even numbers are divisible by 2, and there is a similar determination in the case of 4 and 8. For 3 there is the rule of the sum of the figures, and the same thing for 9 and also for 6, in the last case in combination with the even number. For 7 there is a special rule. These form the basis for tricks with numbers which seem incomprehensible to the uninitiated. Hence what Hegel says (Quantity, p. 237) on the absence of thought in arithmetic is incorrect. Compare, however, Measure.
When mathematics speaks of the infinitely large and infinitely small, it introduces a qualitative difference which even takes the form of 