Brian Clegg

INFINITY

The quest to think the unthinkable

Brian Clegg traces the notion of infinity from the earliest times to the latest cosmological speculations, with a quick side-glance at theology. I get the impression that Clegg is not much interested in the theological aspects of his subject, however, and these are touched on in just a short chapter, which seems almost an afterthought. Most of the book is concerned with mathematics

We begin with Zeno and his famous paradoxes, notably that of Achilles and the tortoise. The Greeks were much interested in mathematics; Clegg is good on the differences between the ancient Greek approach to mathematics and our own. The Greeks used a predominantly visual, geometric system for conceptualizing fractions, although this did not prevent Archimedes from making numerical speculations. In one of his late works he calculated the number of sand grains that would be needed to fill the universe. His purpose, Clegg thinks, was to show that loose remarks about the number of sand grains on a beach being infinite were unfounded.

Infinity is linked with irrational numbers, such as π, which can be calculated indefinitely without ever reaching a final value. Clegg gives a formula for making this calculation which was discovered by John Wallis in the seventeenth century, although he does not tell us how Wallis came up with this, which is a pity. Wallis also invented the famous infinity symbol, like an 8 on its side.

The concept of infinity was implicit in the branch of mathematics known as calculus that was developed independently by Newton and Leibniz (and which gave rise to a notorious quarrel about precedence between the two philosophers). Newton avoided getting involved in the difficult question of the nature of infinity itself, but this was tackled later by others, notably Georg Cantor and Kurt Gödel. Both of these were apparently actually driven to insanity by their pursuit of the infinite.

Cantor introduced the Aleph symbol to denote infinity, and it is at this point that the paradoxes really take off. For example, Cantor showed that you can have bigger numbers than infinity; in fact, Clegg has a chapter with the title "An infinity of infinities". He manages to make these ideas at least approximately comprehensible; and he also implies that all this is, in a sense, an elaborate game, played according to predetermined rules.

But is it more than a game? Does the concept of infinity have any application in the real world? This question has been a recurrent subject of debate in mathematics and philosophy. David Hume did not accept that infinity could be more than an idea, but Bernard Bolzano, a mathematician whose important contribution to the subject was published posthumously in 1848, believed in a "real" infinity. In the twentieth century, however, the great German mathematician David Hilbert said that there could not be such a thing as true infinity but only very big or very small dimensions.

Clegg takes this question up in his penultimate chapter, and finds that it is still undecided. Most though not all cosmologists think the universe is probably not infinite but whether or not time could be infinite is less certain. We also cannot be sure that it is impossible to divide up space or time into an infinite number of infinitesimal portions. The modern branch of mathematics that deals with fractals suggests the possibility of endless subdivisions. It is however the likely construction of quantum computers that may finally make infinity a practical reality.

Clegg has done a good job of making an abstruse subject (mostly) accessible to the non-mathematical reader.