Engels, and before him Hegel, pointed to the numerous contradictions that abound in mathematics. This was always the case, despite the claims of perfection and almost papal infallibility made by mathematicians for their “sublime science”. This fashion was started by the Pythagoreans, with their mystical conception of Number, and the harmony of the universe. Very soon, however, they found out that their harmonious and orderly mathematical universe was plagued with contradictions, the solution of which drove them to despair. For example, they found that it was impossible to express the length of the diagonal of a square in numbers.

The later Pythagoreans discovered that there were many numbers, like the square root of two, which could not be expressed in numbers. It is an “irrational number”. But although the square root of two cannot be expressed as a fraction, it is useful to find the length of the side of a triangle. Present-day mathematics contains a veritable menagerie of such strange animals, still untamed, despite all efforts to domesticate them, but which, once accepted for what they are, render valuable services. Thus we have irrational numbers, imaginary numbers, transcendental numbers, transfinite numbers, all displaying strange and contradictory features, and all indispensable to the workings of modern science.

The mysterious Pi was well known to the ancient Greeks, and generations of schoolchildren have learned to identify it as the ratio between the circumference and diameter of a circle. Yet, strangely, its exact value cannot be found. Archimedes calculated its approximate value by a method known as “exhaustion”. It was between 3.14085 and 3.14286. But if we try to write down the exact value, we get a strange result:

Pi = 3.14159265358979323846264338327950…and so on ad infinitum.

Pi is now known as a transcendental number, is absolutely necessary to find the circumference of a circle, but cannot be expressed as the solution to an algebraic equation. Then we have the square root of minus one, which is not an arithmetical number at all. Mathematicians refer to it as an “imaginary number”, since no real number, when multiplied by itself, can give the result of minus one, because two minuses give a plus. A most peculiar creature, this – but not a figment of the imagination, despite its name. In Anti-Dühring, Engels points out that:

It is a contradiction that a negative magnitude should be the square of anything, for every negative magnitude multiplied by itself gives a positive square. The square root of minus one is therefore not only a contradiction, but even an absurd contradiction, a real absurdity. And yet the square root of minus 1 is in many cases a necessary result of correct mathematical operations. Furthermore, where would mathematics – lower or higher – be, if it were prohibited from operating with the square root of minus 1?[4]

Engels’ remark is even more true today. This contradictory combination of plus and minus plays an absolutely crucial role in quantum mechanics, where it appears in a whole host of equations, which are fundamental to modern science.

That this mathematics involves startling contradictions is not open to doubt. Here is what Hoffman has to say about it:

That such a formula should have any connection with that world of strict experiment which is the world of physics is in itself difficult to believe. That it was to be the deep foundation of the new physics, and that it should actually probe more profoundly than anything before towards the very core of science and metaphysics is as incredible as must once have seemed the doctrine that the earth is round.[5]

Nowadays, the use of the so-called imaginary numbers is taken for granted. The square root of minus one is used for a whole range of necessary operations, such as the construction of electrical circuits. Transfinite numbers, in turn, are needed to understand the nature of time and space. Modern science, and particularly quantum mechanics, could not manage without the use of mathematical concepts, which are frankly contradictory in character. Paul Dirac, one of the founders of quantum mechanics, discovered the “Q” numbers, which defy the laws of ordinary mathematics which state that a multiplied by b is the same thing as b multiplied by a.

  1. Engels, F. Anti-Dühring, p. 154.

  2. Hoffman, B. The Strange Story of the Quantum, p. 95.