Whither Mathematics? Brian Davies Introduction that agreement about its solution is not imminent During most of the twentieth century there was re- [5], [6], [25], [26]. markable agreement about the right way to present Mathematicians as amateur philosophers are results in (pure) mathematics. The subject con- no more agreed about the status of their subject sisted of a list of theorems, each of which was than are philosophers. As representatives of many proved from an underlying set of axioms using others we cite Roger Penrose as a committed real- what were called rigorous arguments. In a few ist (i.e., Platonist) [20], [21] and Paul Cohen as an cases, such as Peano arithmetic, the truth of the ax- anti-realist [12], [13]. Einstein was clear that math- ioms seemed self-evident, but in many cases they ematics was a product of human thought and that, simply defined the domain of discourse. For math- as far as the propositions of mathematics are cer- ematicians, talking as mathematicians rather than tain, they do not refer to reality [16]. The author as amateur philosophers, philosophical distinc- of the present article has always been critical of Pla- tions between the invention and discovery of new tonism [14]; he now fully accepts the existence of concepts did not affect the way they practised mathematical entities, but only in the Carnapian their subject. sense [15]. This allows mathematical theories to be In this paper we will argue that developments products of the human imagination, but never- of the classical Greek view of mathematics do not theless to have definite properties just as chess and adequately represent current trends in the sub- Roman law do; it also allows numbers to exist in ject. It proved remarkably successful for many cen- the same sense as the black king does in chess. For- turies, but three crises in the twentieth century tunately mathematicians as mathematicians do not force us to reconsider the status of an increasing need to refer to their philosophical beliefs, and amount of current mathematical research. hence can achieve a large degree of agreement The apparent consensus among mathematicians amongst themselves. This agreement is, however, as mathematicians stands in stark contrast to the not total: constructivists adopt a strict, algorithmic disagreements between those studying the phi- notion of existence that is more acceptable to ap- losophy of mathematics. This subject has been plied mathematicians, numerical analysts, and lo- dominated by a single issue. This concerns the pe- gicians than it is to most pure mathematicians [7], culiar status of mathematical objects: if one main- [8], [9], [15]. tains that they exist in some Platonic realm, it Kurt Gödel’s astonishing insights in the 1930s seems impossible to give any account of how we, created the first of the three crises to which we as creatures embedded in space and time, can refer. He demonstrated that within any sufficiently come to know about them. The argument that we rich axiomatic system there must exist certain may have no access to these objects but can nev- statements that cannot be proved or disproved. He ertheless work out what they are like by the use of also established that the consistency of arithmetic our reasoning powers is unconvincing for the fol- was not provable. There have been many discus- lowing reason (among others): we could appar- sions of his work, but these frequently involve im- ently follow exactly the same lines of reasoning plicit philosophical assumptions on the part of the about the properties of and relationships between writer. For example, the belief of Gödel himself mathematical entities even if the Platonic realm did that the continuum hypothesis must be either true not exist. Whole books have been devoted to the or false independently of whether we can prove this discussion of the relationship between ontology and fact reveal his wholehearted commitment to Pla- epistemology in mathematics, but it is fair to say tonism in mathematics. Gödel’s theorems are tech- nical in nature and do not establish that there is a Brian Davies is professor of mathematics at King’s College fundamental distinction between truth and prov- London. His email address is E.Brian.Davies@ ability in mathematics without the insertion of kcl.ac.uk. extra philosophical assumptions. 1350 NOTICES OF THE AMS VOLUME 52, NUMBER 11 It might be thought that Gödel’s attitude to- relevant. The crises may simply be the analogy of wards his own results must be of great signifi- realizing that human beings will never be able to cance, but he was a somewhat eccentric figure; his construct buildings a thousand kilometres high argument that one can have the same confidence and that imagining what such buildings might “re- in mathematical intuition as in sense perception ally” be like is simply indulging in fantasies. does not sit happily with the consensus of psy- chologists that sense perception is heavily depen- Computer-Assisted Proofs dent on constructions within the human mind [11], The first example of a major mathematical theo- [14, p. 38]. Other giants in the field have taken rem that depended on computer assistance was the quite different attitudes. For example, Paul Cohen, four-colour theorem, proved by Appel and Haken who eventually proved the independence of the in 1976 [1], [2]. It caused great uneasiness among continuum hypothesis, did not share Gödel’s views, some mathematicians for two reasons. One was that believing that set theory was no more than an it was considered that one could not be certain that axiomatic structure: it was not the partial de- a machine had performed a calculation correctly scription of an external entity [12], [13]. if one could not check every line of the proof by In spite of the enormous literature emphasizing hand. At that time “proper” theorems had proofs the importance of Gödel’s work for the foundations that were agreed to be unassailable. Mistakes might and philosophy of mathematics, it had very little occasionally occur, but they could and would be rec- effect within mathematics itself for several decades, tified with the passage of time. The other issue was excepting logic, regarded as one among many fields that some mathematicians considered that they of mathematics. Its relevance within mainstream were not interested in whether theorems were true mathematics only emerged when it was discov- but why they were true. A proof that did not gen- ered that the word problem and the isomorphism erate understanding was of no interest to them. problem for finitely presented groups were algo- The four-colour theorem did not have any very rithmically insoluble and, as a consequence, the important applications, and for a considerable time homeomorphism problem for 4-manifolds was also it was possible to regard it as an aberration. Per- insoluble. Gradually more and more such issues haps it was not really very interesting after all and have been revealed, but in spite of this, most math- had only acquired fame because it was easily stated. ematicians ply their trade exactly as they would However, as time has passed, and computers have have done if Gödel had never existed. become more available, the number of computer- Since 1970 two other crises have arisen in math- assisted proofs has slowly grown. It would serve ematics, neither of which was anticipated, just as no useful purpose to enumerate all such cases, so Gödel’s work had not been. Both involve the issue we turn to the most recent example. of complexity: proofs that are too long and com- The Kepler problem is to determine the best plex for anyone to be able to assert with total con- way of packing identical solid spheres in three- fidence that the theorems claimed are certainly dimensional space, so as to maximize their aver- true. These crises have not been discussed much age density. The expected solution has been known in the philosophical literature, even though both for many years, and involves packing the spheres are starting to have more impact on the way that exactly as oranges are displayed in every grocer’s mathematicians think about their subject than shop. In 1998 Tom Hales announced the rigorous Gödel’s work ever has. In October 2004 the Royal solution of this problem using a combination of Society held a two-day discussion meeting in Lon- geometrical analysis and heavy computer calcula- don on “The Nature of Mathematical Proof” to dis- tions. Annals of Mathematics solicited his paper and cuss possible ways of responding to them; see [10]. set up a team of twenty of the top experts in the The meeting provided a variety of insights into field to referee the work. They started by holding the issues involved but no solutions. There was ev- a conference in Princeton to decide their strategy. idence of a serious communication problem be- As the years passed referees gradually left the tween the mathematicians and computer scien- team, and early in 2004 the effort of refereeing the tists present. paper had to be discontinued. The Annals editors At first sight it seems obvious that the “crises decided to publish the “theoretical part” of the of complexity” that we will describe are epistemo- paper and send the computer-based part to a more logical in character and say nothing about the on- appropriate journal for publication. One of the An- tology of mathematics. On the other hand some nals editors, Robert MacPherson, admitted that the mathematicians prefer to think of mathematics as (unpublished) policy of the Annals editors for such involving a process of creation rather than dis- papers had failed; see [18].