The idea of the infinite seems difficult to grasp, because, at first sight, it is beyond all human experience. The human mind is accustomed to dealing with finite things, reflected in finite ideas. Everything has a beginning and an end. This is a familiar thought. But what is familiar is not necessarily true. The history of mathematical thought has some highly instructive lessons on this score. For a long time, mathematicians, at least in Europe, sought to banish the concept of infinity. Their reasons for so doing are obvious enough. Apart from the evident difficulty in conceptualising infinity, in purely mathematical terms it involves a contradiction. Mathematics deals with definite magnitudes. Infinity by its very nature cannot be counted or measured. This means that there is a real conflict between the two. For that reason, the great mathematicians of ancient Greece avoided infinity like the plague. Despite this, from the beginnings of philosophy, men speculated about infinity. Anaximander (610-547 B.C.) took it as the basis of his philosophy.

The paradoxes of Zeno (c. 450 B.C.) point to the difficulty inherent in the idea of infinitesimal quantity as a constituent of continuous magnitudes by attempting to prove that movement is an illusion. Zeno “disproved” motion in different ways. He argued that a body in motion, before reaching a given point, must first have travelled half the distance. But before this, it must have travelled half of that half, and so on ad infinitum. Thus, when two bodies are moving in the same direction, and the one behind at a fixed distance from the one in front is moving faster, we assume that it will overtake the other. Not so, says Zeno. “The slower one can never be overtaken by the quicker.” This is the famous paradox of Achilles the Swift. Imagine a race between Achilles and a tortoise. Suppose that Achilles can run ten times faster than the tortoise, which has 1000 metres start. By the time Achilles has covered 1000 metres, the tortoise will be 100 metres ahead; when Achilles has covered that 100 metres, the tortoise will be one metre ahead; when he covers that distance, the tortoise will be one tenth of a metre ahead, and so on to infinity.

Zeno’s paradoxes do not prove that movement is an illusion, or that Achilles, in practice, will not overtake the tortoise, but they do reveal brilliantly the limitations of the kind of thinking now known as formal logic. The attempt to eliminate all contradiction from reality, as the Eleatics did, inevitably leads to this kind of insoluble paradox, or antinomy, as Kant later called it. In order to prove that a line could not consist of an infinite number of points, Zeno claimed that, if it were really so, then Achilles would never overtake the tortoise. There really is a logical problem here. As Alfred Hooper explains:

This paradox still perplexes even those who know that it is possible to find the sum of an infinite series of numbers forming a geometrical progression whose common ratio is less than 1, and whose terms consequently become smaller and smaller and thus ‘converge’ on some limiting value.[6]

In fact, Zeno had uncovered a contradiction in mathematical thought that would have to wait two thousand years for a solution. The contradiction relates to the use of the infinite. From Pythagoras right up to the discovery of the differential and integral calculus in the 17th century, mathematicians went to great lengths to avoid the use of the concept of infinity. Only the great genius Archimedes approached the subject, but still avoided it by using a roundabout method. The early atomists, starting with Leukippus, who may have been a pupil of Zeno, stated that the atoms “indivisible and infinite in number, move about ceaselessly in empty space, of infinite extent.”

Modern physics accepts that the number of instants between two seconds is infinite, just as the number of instants in a span of time with neither beginning nor end. The universe itself consists of an infinite chain of cause and effect, ceaselessly changing, moving and developing. This has nothing in common with the crude and one-sided notion of infinity contained in the infinite series of numbers in simple arithmetic, in which “infinity” always “starts” with the number one! This is what Hegel called “Bad Infinity”.

The greatest of Greek mathematicians, Archimedes (287-212 B.C.) made effective use of indivisibles in geometry, but considered the idea of infinitely large and small as without logical foundation. Likewise, Aristotle argued that, since a body must have form, it must be bounded, and therefore cannot be infinite. While accepting that there were two kinds of “potential” infinities – successive addition in arithmetic (infinitely large), and successive subdivision in geometry (infinitely small) – he nevertheless polemicised against geometers who held that a line segment is composed of infinitely many fixed infinitesimals, or indivisibles.

This denial of the infinite constituted a real barrier to the development of classical Greek mathematics. By contrast, the Indian mathematicians had no such scruples and made great advances, which, via the Arabs, later entered Europe. The attempt to banish contradiction from thought, in accordance with the rigid schemas of formal logic held back the development of mathematics. But the adventurous spirit of the Renaissance opened men’s minds to new possibilities, which were, in truth, infinite. In his book The New Science (1638), Galileo pointed out that every integer (whole number) has only one perfect square, and every perfect square is the square of only one positive integer. Thus, in a sense, there are just as many perfect squares as there are positive integers. This immediately leads us into a logical contradiction. It contradicts the axiom that the whole is greater than any of its parts, inasmuch as not all the positive integers are perfect squares, and all the perfect squares form part of all the positive integers.

This is only one of the numerous paradoxes that have plagued mathematics ever since the Renaissance when men began to subject their thoughts and assumptions to a critical analysis. As a result of this, slowly, and in the teeth of stubborn resistance from conservative minds, one by one the supposedly unassailable axioms and “eternal truths” of mathematics have been overthrown. We arrive at the point where the entire edifice has been shown to be unsound and in need of a thoroughgoing reconstruction on more solid, yet more flexible foundations, which are already in the process of being laid, and which will inevitably have a dialectical character.

  1. Hooper, A. Makers of Mathematics, p. 237.