Aftermath

In the later part of the century, attitudes changed. When the 1998 film Good Will Hunting featured a famous mathematician at the Massachusetts Institute of Technology who had won a Fields Medal for combinatorics, many found this somewhat unbelievable. 2 However, life followed art in this case when, later in the same year, Fields Medals were awarded to Richard Borcherds and Tim Gowers for work much of which was in combinatorics.

A more remarkable instance of life following art involves Stanisław Lem’s 1968 novel His Master’s Voice [29]. The narrator, a mathematician, describes how he single-mindedly attacked his rival’s work: I do not think I ever finished any larger paper in all my younger work without imagining Dill’s eyes on the manuscript. What effort it cost me to prove that the Dill variable combinatorics was only a rough approximation of an ergodic theorem! Not before or since, I daresay, did I polish a thing so carefully; and it is even possible that the whole concept of groups later called Hogarth groups came out of that quiet, constant passion with which I plowed Dill’s axioms under.

In 1975, Szemerédi [42] published his remarkable combinatorial proof that a set of natural numbers with positive density contains arbitrarily long arithmetic progressions; in 1977, Furstenberg [12] gave a proof based on ergodic theory! (This is not to suggest that Furstenberg’s attitude to Szemerédi parallels Hogarth’s to Dill in the novel.)

In this chapter, I have attempted to tease apart some of the interrelated reasons for this change, and perhaps to throw some light on present trends and future directions. I have divided the causes into four groups: the influence of the computer; the growing sophistication of combinatorics; its strengthening links with the rest of mathematics; and wider changes in society. I have told the story mostly through examples.

Even before computers were built, pioneers such as Babbage and Turing realised that they would be designed on discrete principles, and would raise theoretical issues which led to important mathematics.

Kurt Gödel [15] showed that there are true statements about the natural numbers which cannot be deduced from the axioms of a standard system such as Peano’s. This result was highly significant for the foundations of mathematics, but Gödel’s unprovable statement itself had no mathematical significance. The first example of a natural mathematical statement which is unprovable in Peano arithmetic was discovered by Paris and Harrington [32], and is a theorem in combinatorics (it is a slight strengthening of Ramsey’s theorem). It is unprovable from the axioms because the corresponding ‘Paris-Harrington function’ grows faster than any provably computable function. Several further examples of this phenomenon have been discovered, mostly combinatorial in nature. 3More recently, attention has turned from computability to computational complexity: given that something can be computed, what resources (time, memory, etc.) are required for the computation. A class of problems is said to be polynomialtime computable, or in P, if any instance can be solved in a number of steps bounded by a polynomial in the input size. A class is in NP if the same assertion holds if we are allowed to make a number of lucky guesses (or, what amounts to the same thing, if a proposed solution can be checked in a polynomial number of steps). The great unsolved problem of complexity theory asks:

On 24 May 2000, the Clay Mathematical Institute announced a list of seven unsolved problems, for each of which a prize of one million dollars was offered. The P = NP problem was the first on the list [8].

This problem is particularly important for combinatorics since many intractable combinatorial problems (including the existence of a Hamiltonian cycle in a graph) are known to be in NP. In the unlikely event of an affirmative solution, ‘fast’ algorithms would exist for all these problems. Now we turn to the practical use of computers.

Computer systems such as GAP [13] have been developed, which can treat algebraic or combinatorial objects, such as a group or a graph, in a way similar to the handling of complex numbers or matrices in more traditional systems. These give the mathematician a very powerful tool for exploring structures and testing (or even formulating) conjectures.

But what has caught the public eye is the use of computers to prove theorems. This was dramatically the case in 1976 when Kenneth Appel and Wolfgang Haken [1] announced that they had proved the Four-Colour Theorem by computer. Their announcement started a wide discussion over whether a computer proof is really a ‘proof’ at all: see, for example, Swart [41] and Tymoczko [46] for contemporary responses. An even more massive computation by Clement Lam and his co

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