The law of the transformation of quantity into quality has an extremely wide range of applications, from the smallest particles of matter at the subatomic level to the largest phenomena known to man. It can be seen in all kinds of manifestations, and at many levels. Yet this very important law has yet to receive the recognition it deserves. This dialectical law forces itself to our attention at every turn. The transformation of quantity into quality was already known to the Megaran Greeks, who used it to demonstrate certain paradoxes, sometimes in the form of jokes. For example, the “bald head” and the “heap of grain” – does one hair less mean a bald head, or one grain of corn a heap? The answer is no. Nor one more? The answer is still no. The question is then repeated until there is a heap of corn and a bald head. We are faced with the contradiction that the individual small changes, which are powerless to effect a qualitative change, at a certain point do exactly that: quantity changes into quality.

The idea that, under certain conditions, even small things can cause big changes finds its expression in all kinds of sayings and proverbs. For instance: “The straw that broke the camel’s back”, “many hands make light work”, “constant dripping wears away the stone”, and so on. In many ways, the law of the transformation of quantity into quality has penetrated the popular consciousness, as Trotsky wittily pointed out:

Every individual is a dialectician to some extent or other, in most cases, unconsciously. A housewife knows that a certain amount of salt flavours soup agreeably, but that added salt makes the soup unpalatable. Consequently, an illiterate peasant woman guides herself in cooking soup by the Hegelian law of the transformation of quantity into quality. Similar examples from daily life could be cited without end. Even animals arrive at their practical conclusions not only on the basis of the Aristotelian syllogism but also on the basis of the Hegelian dialectic. Thus a fox is aware that quadrupeds and birds are nutritious and tasty. On sighting a hare, a rabbit, or a hen, a fox concludes: this particular creature belongs to the tasty and nutritive type, and – chases after the prey. We have here a complete syllogism, although the fox, we may suppose, never read Aristotle. When the same fox, however, encounters the first animal that exceeds it in size, for example, a wolf, it quickly concludes that quantity passes into quality, and turns to flee. Clearly, the legs of a fox are equipped with Hegelian tendencies, even if not fully conscious ones.

All this demonstrates, in passing, that our methods of thought, both formal logic and the dialectic, are not arbitrary constructions of our reason but rather expressions of the actual inter-relationships in nature itself. In this sense, the universe throughout is permeated with ‘unconscious’ dialectics. But nature did not stop there. No little development occurred before nature’s inner relationships were converted into the language of the consciousness of foxes and men, and man was then enabled to generalise these forms of consciousness and transform them into logical (dialectical) categories, thus creating the possibility for probing more deeply into the world about us.[24]

Despite the apparently trivial character of these examples, they do reveal a profound truth about the way the world works. Take the example of the heap of corn. Some of the most recent investigations related to chaos theory have centred on the critical point where a series of small variations produces a massive change of state. (In the modern terminology, this is called “the edge of chaos”.) The work of the Danish-born physicist Per Bak (1947-2003) and others on “self-organised criticality” used precisely the example of a sand-heap to illustrate profound processes which occur at many levels of nature and which correspond precisely to the law of the transformation of quantity into quality.

One of the examples of this is that of a pile of sand – a precise analogy with the heap of grain of the Megarans. We drop grains of sand one by one on a flat surface. The experiment has been conducted many times, both with real sand heaped on tables, and in computer simulations. For a time they will just pile up on top of each other until they make a little pyramid. Once this point is reached, any additional grains will either find a resting place on the pile, or will unbalance one side of it just enough to cause some of the other grains to fall in an avalanche. Depending on how the other grains are poised, the avalanche could be very small, or devastating, dragging a large number of grains with it. When the pile reaches this critical point, even a single grain would be capable of dramatically affecting all around it. This seemingly trivial example provides an excellent “edge-of-chaos model”, with a wide range of applications, from earthquakes to evolution; from stock exchange crises to wars.

The pile of sand grows bigger, with excess sand slipping from the sides. When all the excess sand has fallen off, the resulting sand-pile is said to be “self-organised.” In other words, no one has consciously shaped it in this way. It “organises itself”, according to its own inherent laws, until it reaches a state of criticality, in which the sand grains on its surface are barely stable. In this critical condition, even the addition of a single grain of sand can cause unpredictable results. It may just cause a further tiny shift, or it may trigger a chain reaction resulting in a catastrophic landslide and the destruction of the pile.

According to Per Bak, the phenomenon can be given a mathematical expression, according to which the average frequency of a given size of avalanche is inversely proportional to some power of its size. He also points out that this “power-law” behaviour is extremely common in nature, as in the critical mass of plutonium, at which the chain reaction is on the point of running away into a nuclear explosion. At the sub-critical level, the chain reaction within the plutonium mass will die out, whereas a supercritical mass will explode. A similar phenomenon can be seen in earthquakes, where the rocks on two sides of a fault in the earth’s crust reach a point where they are ready to slip past each other. The fault experiences a series of little slips and bigger slips, which maintain the tension at the critical point for some time until it finally collapses into an earthquake.

Although the proponents of chaos theory seem unaware of it, these examples are all cases of the law of the transformation of quantity into quality. Hegel invented the nodal line of measure relations, in which small quantitative changes at a certain point give rise to a qualitative leap. The example is often given of water, which boils at 100°C at normal atmospheric pressure. As the temperature nears boiling point, the increase in heat does not immediately cause the water molecules to fly apart. Until it reaches boiling point, the water keeps its volume. It remains water, because of the attraction of the molecules for each other. However, the steady change in temperature has the effect of increasing the motion of the molecules. The volume between the atoms is gradually increased, to the point where the force of attraction is insufficient to hold the molecules together. At precisely 100°C, any increase in heat energy will cause the molecules to fly apart, producing steam.

The same process can be seen in reverse. When water is cooled from 100°C to 0°C, it does not gradually congeal, passing from a paste, through a jelly, to a solid state. The motion of the atoms is gradually slowed as heat energy is removed until, at 0°C, a critical point is reached, at which the molecules will lock into a certain pattern, which is ice. The qualitative difference between a solid and a liquid can be readily understood by anyone. Water can be used for certain purposes, like washing and quenching one’s thirst, which ice cannot. Technically speaking, the difference is that, in a solid, the atoms are arranged in a crystalline array. They do not have a random position at long distances, so that the position of the atoms on one side of the crystal is determined by the atoms on the other side. That is why we can move our hand freely through water, whereas ice is rigid and offers resistance. Here we are describing a qualitative change, a change of state, which arises from an accumulation of quantitative changes. A water molecule is a relatively simple affair, one oxygen atom attached to two hydrogen atoms governed by well-understood equations of atomic physics. However, when a very large number of these molecules are combined, they acquire a property which none of them possesses in isolation – liquidity. Such a property is not implied in the equations. In the language of complexity, liquidity is an “emergent” phenomenon.

As Mitchell Waldrop says:

Cool those liquid water molecules down a bit, for example, and at 32°F they will suddenly quit tumbling over one another at random. Instead they will undergo a ‘phase transition’, locking themselves into the orderly crystalline array known as ice. Or if you were to go the other direction and heat the liquid, those same tumbling water molecules will suddenly fly apart and undergo a phase transition into water vapour. Neither phase transition would have any meaning for one molecule alone.[25]

The phrase “phase transition” is neither more nor less than a qualitative leap. Similar processes can be seen in phenomena as varied as the weather, DNA molecules, and the mind itself. This quality of liquidity is well known on the basis of our daily experience. In physics, too, the behaviour of liquids is well understood and perfectly predictable – up to a point. The laws of motion of fluids (gases and liquids) clearly distinguish between smooth laminar flow, which is well defined and predictable, and turbulent flow, which can be expressed, at best, approximately. The movement of water around a pier in a river can be accurately predicted from the normal equations for fluids, provided it is moving slowly. Even if we increase the speed of the flow, causing eddies and vortices, we can still predict their behaviour. But if the speed is increased beyond a certain point, it becomes impossible to predict where the eddies will form, or, indeed, to say anything about the behaviour of the water at all. It has become chaotic.

  1. Trotsky, L. In Defence of Marxism, pp. 106-7.

  2. Waldrop, M. Complexity, The Emerging Science at the Edge of Order and Chaos, p. 82.