Many of the so-called axioms of classical Greek mathematics were already undermined by the discovery of the differential and integral calculus, the greatest breakthrough in mathematics since the Middle Ages. It is an axiom of geometry that straight and curved are absolute opposites, and that the two are incommensurable, that is, the one cannot be expressed in terms of the other. Yet, in the last analysis, straight and curved in the differential calculus are regarded as equal. As Engels points out, the basis for this was laid a long time before it was elaborated by Leibniz and Newton:

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.[7]

The discovery of the calculus opened up a whole new horizon for mathematics and science in general. Once the old taboos and prohibitions were lifted, mathematicians were free to investigate entirely new areas. But they made use of infinitely large and small numbers uncritically, without considering their logical and conceptual implications. The use of infinitely small and great quantities was regarded as a kind of “useful fiction”, which, for some incomprehensible reason, always gave the correct result. In the section on Quantity in the first volume of Science of Logic, Hegel points out that, while the introduction of the mathematical infinite opened up new horizons for mathematics, and led to important results, it remained unexplained, because it clashed with the existing traditions and methods:

But in the method of the mathematical infinite mathematics finds a radical contradiction to that very method which is characteristic of itself, and on which it rests as a science. For the calculation of the infinite admits of, and demands, modes of procedure which mathematics, when it operates with finite magnitudes, must altogether reject, and at the same time it treats these infinite magnitudes as finite Quanta, seeking to apply to the former those same methods which are valid for the latter.[8]

The result was a long period of controversy concerning the validity of the calculus. Berkeley denounced it as in open contradiction to the laws of logic. Newton, who made use of the new method in his Principia, felt obliged to conceal the fact from the public, for fear of an adverse reaction. In the early 18th century, Bernard Fontenelle finally had the courage to state categorically that inasmuch as there are infinitely many natural numbers, an infinite number exists as truly as do finite numbers, and that the reciprocal of infinity is an infinitesimal. However, he was contradicted by Georges de Buffon, who rejected the infinity as an illusion. Even the great intellect of Jean le Rond d’Alembert was incapable of accepting this idea. In the article in his Encyclopaedia on the Differential, he denied the existence of infinity, except in the negative sense of a limit on finite quantities.

The concept of “limit” was in fact introduced in an attempt to get round the contradiction inherent in infinity. This was especially popular in the 19th century, when mathematicians were no longer prepared simply to accept the calculus unthinkingly, as the earlier generation had been content to do. The differential calculus postulated the existence of infinitesimally small magnitudes of varying orders – a first differential, a second differential, and so on to infinity. By introducing the concept of “limit” they at least created the appearance that an actual infinity was not involved. The intention was to make the idea of infinity seem subjective, to deny it objectivity. The variables were said to be potentially infinitely small, in that they become less than any given quantity, as potentially infinite, in that they become larger than any preassigned magnitude. In other words, “as big or small as you like!” This sleight of hand did not remove the difficulty, but only provided a fig leaf to cover up the logical contradictions involved in the calculus.

The great German mathematician Karl Frederick Gauss (1777-1855) was prepared to accept the mathematical infinite, but expressed horror at the idea of real infinity. However, his contemporary Bernhard Bolzano, setting out from Galileo’s paradox, began a serious study of the paradoxes implicit in the idea of a “completed infinite”. This work was further developed by Richard Dedekind (1813-1914) who characterised the infinite as something positive, and pointed out that, in fact, the positive set of numbers can be regarded as negative (that is, as one that is not infinite). Finally, Georg Cantor (1845-1918) went far beyond the definition of infinite sets and developed an entirely new arithmetic of “transfinite numbers”. Cantor’s papers, beginning in 1870, are a review of the whole history of the infinite, beginning with Democritus. Out of this, there developed a whole new branch of mathematics, based on the theory of sets.

Cantor showed that the points in an area, however large, or in a volume or a continuum of still higher dimension, can be matched against the points on a line or a segment, no matter how small it may be. Just as there can be no last finite number, so there can be no last transfinite number. Thus, after Cantor, there can be no argument about the central place of the infinite in mathematics. Moreover, his work revealed a series of paradoxes that have plagued modern mathematics, and have yet to be resolved.

All modern scientific analysis relies on the concept of continuity, that is to say, that between two points in space, there is an infinite number of other points, and also that, between any two points in time there is an infinite number of other moments. Without making these assumptions, modern mathematics simply could not function. Yet such contradictory concepts would have been indignantly rejected, or at least regarded with suspicion, by earlier generations. Only the dialectical genius of Hegel (a great mathematician incidentally) was capable of anticipating all this in his analysis of finite and infinite, space, time and motion.

Yet despite all the evidence, many modern mathematicians persist in denying the objectivity of infinity, while accepting its validity as a phenomenon of “pure” mathematics. Such a division makes no sense at all. For unless mathematics was able to reflect the real, objective world, what use would it be? There is a certain tendency in modern mathematics (and, by extension, incredibly, in theoretical physics) to revert to idealism in its most mystical form, alleging that the validity of an equation is purely a question of its aesthetic value, with no reference to the material world.

The very fact that mathematical operations can be applied to the real world and get meaningful results indicates that there is an affinity between the two. Otherwise, mathematics would have no practical application, which is clearly not the case. The reason why infinity can be used, and must be used, in modern mathematics is because it corresponds to the existence of infinity in nature itself, which has imposed itself upon mathematics, like an uninvited guest, despite all the attempts to bar the door against it.

The reason why it took so long for mathematics to accept infinity was explained very well by Engels:

It is clear that an infinity which has an end but no beginning is neither more nor less infinite than one with a beginning but no end. The slightest dialectical insight should have told Herr Dühring that beginning and end necessarily belong together, like the North Pole and the South Pole, and that if the end is left out, the beginning just becomes the end – the one end which the series has; and vice versa. The whole deception would be impossible but for the mathematical usage of working with infinite series. Because in mathematics it is necessary to start from determinate, finite terms in order to reach the indeterminate, the infinite, all mathematical series, positive and negative, must start with 1, or they cannot be used for calculation. But the logical need of the mathematician is far from being a compulsory law for the real world.[9]

  1. Engels, F. The Dialectics of Nature, pp. 341-2.

  2. Hegel, G. Science of Logic, Vol. 1, p. 257.

  3. Engels, F. Anti-Dühring, p. 63.