Methods of Infinity
By Reviel Netz of Stanford University

The most fundamental intellectual question in all of mathematics is the nature of infinity. In many ways, this is what mathematics is about. Take the most ordinary objects, look at them through the infinitely powered magnifying glass that mathematics is – and you come to see infinity in one of its many (indeed, infinite) forms. Draw a circle – what is the ratio of its circumference to its diameter? Right: in order to express this, you need a certain kind of infinity. (To simplify things a bit, this is because the number Pi has infinitely many, endlessly varying digits). Draw now a square, presumably a simpler object. But then again: draw the diagonal to that square. What is the ratio of the side to the diagonal? Once again, one needs infinity, of a somewhat different kind, in order to express this. (To simply again: this involves the surprisingly complex nature of the square root of the number two). And now draw just a line, any line. How many points does it contain? Right again – infinitely many. Which kind of infinity? Curiously enough, mathematicians do not know the answer to this. The simplest line, with the simplest question, already brings us to an enigma of infinity that mathematics still cannot solve.

The enigma of infinity has fed into all mathematical progress. Given that there are infinitely many prime numbers, are there ways in which we can characterize how many precisely they are? Much of number theory emerges out of this question. And even more fundamentally: given that curved figures involve infinitely complex ratios (akin to the nature of the number Pi), are there ways by which we can characterize, in general, curved lines and measure them as simply as we do straight lines? The attempt to answer this question animates most of Archimedes' works. He measures spiral lines, spheroids, conoids, spheres and parabolas, again and again extending the ways by which measurement may be affected on curves. Out of this endeavour would emerge the central line of development in the history of mathematics – from Archimedes' measurements, through early modern attempts to generalize his approach into a more systematic tool of calculation, giving rise, in the hands of Newton and Leibniz, to the Differential and Integral and Calculus, the foundation of modern mathematics.

We may divide all of infinity into two major types, one to be called “potential infinity”, the other to be called “actual infinity”. By potential infinity something such as the following is meant. Suppose you go into an auction, representing a buyer who mysteriously guarantees you that he has limitless funds, that is, he guarantees you that, however much the opponents are offering, you are allowed to beat their offers. (According to one rumor, such was the authorization given by the current owner of the Archimedes Palimpsest, going into the Christie's auction). They offer one million – and you offer two. They offer a billion – and you a trillion. However much they put up, you may put up more. You are the representative, then, of potential infinity. Why is this infinity merely potential? Because you are always making a finite commitment. In other words, you know that, if the auction is ever to end, then it will end with a finite commitment – however large that finite commitment may be. It may be a million, a trillion, or it may be the number 1 followed by a trillion zeroes, but it will always be some finite number.

Potential infinity, then, may be viewed as an endlessly extendible, and yet forever finite, magnitude. Not so actual infinity. By actual infinity we mean something such as, say, the number of points on a line. If your buyer would have authorized you to offer as your price, in that auction, a dollar for each point there is on a line, then he would have authorized you to make an actually infinite bid. This is a very high bid indeed. After all, even the number 1, followed by a trillion zeroes, is still dwarfed by actual infinity. It is a curious bid, too. If it takes you, say, a minute to pass a suitcase with a million dollars, than it will take you all of eternity to pay your bid actually infinite bid. Actual infinity actually never ends. And for this reason many philosophers, throughout the ages, have doubted its very existence.

The established historical wisdom on infinity and the history of mathematics, then, used to go something like this. At first came the Greeks, who preferred above all to have precise, rigorous proofs. For this reason they have completely avoided the concept of actual infinity, concentrating instead on potential infinity (whose rigorous treatment was perfected by Archimedes in his measurement of curves). Next came the early modern mathematicians, who wanted to get results come what may. To obtain general result concerning curves, they have brought in the notion of actual infinity – paying the price that their mathematics was less rigorous and precise compared to that of the Greeks. Along came the mathematicians of the 19th and 20th centuries, slowly and laboriously building up a new kind of mathematics, where the concept of actual infinity is meticulously built up so as to have the same kind of rigorous foundations that the Greeks have provided for potential infinity.

The most important discovery made through the Palimpsest over the last few years involved some 12 lines of Greek, in an otherwise neglected proposition of Archimedes' Method. They have changed no less than our understanding of how western mathematicians came to handle infinity. To be sure, the Method was always considered, ever since the discovery of the Palimpsest in 1906, as the most significant contribution made by the Palimpsest to our knowledge of the history of mathematics. It is generally considered to be Archimedes' most interesting treatise. And it survives on the Palimpsest alone.

In it, Archimedes sets out to solve many separate problems using a variety of techniques, most often one involving a striking combination of physics and of mathematics. For instance, two geometrical objects are considered simultaneously, say a triangle and a parabolic segment. Then Archimedes shows how each line in the triangle balances a line in the parabolic segment around a given fulcrum, so that it follows that the triangle as a whole, and the parabolic segment as a whole, balance as well around that fulcrum. Since the property of “balancing” has a precise geometrical correlate (objects balance if and only if their magnitudes are reciprocally proportional to their distances), the discovery of the fulcrum allows us to deduce the magnitudes of the triangle as well as the parabolic segment – in other words, we have gained, in a surprising fashion, the measurement of a curve – the parabolic segment.

So much was known since 1906. This was striking indeed – but even more so was the discovery made in 2001. In proposition 14 of the Method, it turns out, as Archimedes is measuring the volume of a cylindrical segment, he makes systematic reference – which previous readers could not decipher, in part because this was so unexpected – to actual infinity itself. This pushes the mathematical use of actual infinity nearly some 2,000 years back in time.

Archimedes crucially needs to argue that the number of triangles inside a prism, as well as the number of lines inside a rectangle, are equal to each other – making a statement concerning an infinity which is not just potential but is, precisely, that kind of infinity of “the number of points in a line”. Archimedes even speaks of the “number” (plethos) of such objects, making him the first person ever to consider an infinite numerosity.

The curious twist is this: while we know today that Archimedes' statement is correct – the two sets of objects, both infinite, are indeed of the same kind of infinity – we are still no better than Archimedes in saying which kind of infinity this is. This – the number of points on a line – is still the big unsolved riddle of mathematics. The enigma of infinity still haunts us, staring us, as it were, at the face even from the pages of the Palimpsest.