The idea of the transformation of quantity into quality is implicit in modern mathematics in the study of continuity and discontinuity. This was already present in the new branch of geometry, topology, invented in the early years of the 20th century by the great French mathematician, Jules Henri Poincaré (1854-1912). Topology is the mathematics of continuity. As Ian Stewart explains it:

Continuity is the study of smooth, gradual changes, the science of the unbroken. Discontinuities are sudden, dramatic: places where a tiny change in cause produces an enormous change in effect.[11]

The standard textbook mathematics gives a wrong impression of how the world actually is, how nature really works. “The mathematical intuition so developed,” wrote Robert May, “ill equips the student to confront the bizarre behaviour exhibited by the simplest non-linear systems.”[12] Whereas elementary school geometry teaches us to regard squares, circles, triangles and parallelograms as entirely separate things, in topology (“rubber-sheet geometry”), they are treated as the same. Traditional geometry teaches that the circle cannot be squared, however in topology this is not the case. The rigid lines of demarcation are broken down: a square can be turned (“deformed”) into a circle. Despite the spectacular advances of 20th century science, it is surprising to note that a large number of what would seem to be quite simple phenomena are not properly understood and cannot be expressed in mathematical terms, for example, the weather, the flow of liquids, turbulence. The shapes of classical geometry are inadequate to express the extremely complex and irregular surfaces found in nature, as Gleick points out:

Topology studies the properties that remain unchanged when shapes are deformed by twisting or stretching or squeezing. Whether a shape is square or round, large or small, is irrelevant in topology, because stretching can change those properties. Topologists ask whether a shape is connected, whether it has holes, whether it is knotted. They imagine surfaces not just in the one-, two-, and three-dimensional universes of Euclid, but in spaces of many dimensions, impossible to visualise. Topology is geometry on rubber sheets. It concerns the qualitative rather than the quantitative.[13]

Differential equations deal with the rate of change of position. This is more difficult and complex than what may appear at first sight. Many differential equations cannot be solved at all. These equations are able to describe motion, but only as a smooth change of position, from one point to another, with no sudden leaps or interruptions. However, in nature, change does not only occur in this way. Periods of slow, gradual, uninterrupted change are punctuated by sharp turns, breaks in continuity, explosions and catastrophes. This fact can be illustrated by innumerable examples from organic and inorganic nature, the history of society and of human thought. In a differential equation, time is assumed to be divided into a series of very small “time-steps”. This gives an approximation of reality, but in fact there are no such “steps”. As Heraclitus expressed it, “everything flows”.

The inability of traditional mathematics to deal with qualitative as opposed to merely quantitative change represents a severe limitation. Within certain limits, it can suffice. But when gradual quantitative change suddenly breaks down, and becomes “chaotic”, to use the current expression, the linear equations of classical mathematics no longer suffice. This is the starting point for the new non-linear mathematics, pioneered by Benoit Mandelbrot, Edward Lorenz and Mitchell Feigenbaum. Without realising it, they were following in the footsteps of Hegel, whose nodal line of measurement expresses the very same idea, which is central to dialectics.

The new attitude to mathematics developed as a reaction against the dead end of the existing schools of mathematics. Mandelbrot had been a member of the French school of mathematical Formalism known as the Bourbaki group, which advocated a purely abstract approach, proceeding from first principles and deducing everything from them. They were actually proud of the fact that their work had nothing to do with science or the real world. But the advent of the computer introduced an entirely new element into the situation. This is yet another example of how the development of technique conditions that of science. The vast number of computations which could be made at the press of a button made it possible to discover patterns and lawfulness where previously only random and chaotic phenomena appeared to exist.

Mandelbrot began by investigating unexplained phenomena of the natural world, like apparently random bursts of interference in radio transmissions, the flooding of the Nile, and crises of the stock exchange. He realised that the traditional mathematics could not deal adequately with such phenomena. In investigating infinity in the 19th century, Georg Cantor invented the set that is named after him. This involves a line, which is divided into an infinite number of points (Cantor “dust”) the total length of which is 0. Such a manifest contradiction disturbed many 19th century mathematicians, yet it served as the starting point for Mandelbrot’s new theory of fractal mathematics, which played a key role in chaos theory. Gleick explains:

Discontinuity, bursts of noise, Cantor dusts, phenomena like these had no place in the geometries of the past 2,000 years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Aristotle found an ideal beauty in them. But for understanding complexity, they turn out to be the wrong kind of abstraction.[14]

All science involves a degree of abstraction from the world of reality. The problem with classical Euclidean measurement, dealing with length, depth and thickness, is that it failed to capture the essence of irregular shapes that are found in the real world. The science of mathematics is the science of magnitude. The abstractions of Euclidean geometry therefore leave aside all but the quantitative side of things. Reality is reduced to planes, lines and points. However, the abstractions of mathematics, despite the exaggerated claims made for them, remain only a rough approximation to the real world, with its irregular shapes and constant and abrupt changes. In the words of the Roman poet Horace, “You may drive out nature with a pitch-fork, yet she’ll be constantly running back.” James Gleick describes the difference between classical mathematics and chaos theory in the following way:

Clouds are not spheres, Mandelbrot is fond of saying. Mountains are not cones. Lightning does not travel in a straight line. The new geometry mirrors a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined. The understanding of nature’s complexity awaited a suspicion that the complexity was not just random, not just accident. It required a faith that the interesting feature of a lightning bolt’s path, for example, was not its direction, but rather the distribution of zigs and zags. Mandelbrot’s work made a claim about the world, and the claim was that such odd shapes carry meaning. The pits and tangles are more than blemishes distorting the classic shapes of Euclidean geometry. They are often the keys to the essence of a thing.[15]

These things were seen as monstrous aberrations by traditional mathematicians. But to a dialectician, they suggest that the unity of finite and infinite, as in the infinite divisibility of matter, can also be expressed in mathematical terms. Infinity exists in nature. The universe is infinitely large. Matter can be divided into infinitely small particles. Thus, all talk about the “beginning of the universe” and the search after the “bricks of matter” and the “ultimate particle” are based on entirely wrong assumptions. The existence of the mathematical infinite is merely a reflection of this fact. At the same time, it is a dialectical contradiction that this infinite universe consists of finite bodies. Thus, finite and infinite form a dialectical unity of opposites. The one cannot exist without the other. The question is therefore not whether the universe is finite or infinite. It is both finite and infinite as Hegel explained long ago.

The advances of modern science have permitted us to penetrate deeper and deeper into the world of matter. At each stage, an attempt has been made to “call a halt”, to erect a barrier, beyond which it was allegedly impossible to go. But at each stage, the limit was overcome, revealing startling new phenomena. Ever new and more powerful particle accelerators have uncovered new and smaller particles, existing in ever-tinier time scales. There is no reason to suppose that the situation will be any different in relation to the quarks, which at present are being represented as the last of the particles.

Similarly, the attempt to establish the beginning of the universe and “time” will turn out to be a wild goose chase. There is no limit to the material universe, and all efforts to impose one will inevitably fail. The most encouraging thing about the new mathematics of chaos theory is that it represents a rejection of sterile abstractions and ivory-tower reductionism, and an attempt to move back towards nature and the world of everyday experience. And to the degree that mathematics reflects nature, it must begin to lose its one-sided character and acquire a whole new dimension which expresses the dynamic, contradictory, in a word, dialectical character of the real world.

  1. Stewart, I. op. cit., p. 63.

  2. Quoted in Gleick, J. op. cit., p. 80.

  3. Gleick, J. op. cit., p. 46.

  4. Ibid p. 94.

  5. Ibid p. 94.