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]. covery, just as in architecture. One is free to pur- At the Royal Society meeting there were lively sue many different ideas as long as one follows discussions about whether formal proofs of the cor- certain basic rules and need not accept that dis- rectness of programs could have made a contri- tinctions between ontology and epistemology are bution to the refereeing process. According to

DECEMBER 2005 NOTICES OF THE AMS 1351 MacPherson the panel did not have any member proofs of the existence of solutions have been pro- who understood the technology of program cor- vided; [22] and [23] provide typical examples. These rectness proofs, so this way of increasing confi- use interval arithmetic to control the rounding er- dence in the computer-assisted part of the proof rors in calculations that are conceptually com- was not considered. The programs had not been pletely conventional. The key is to provide a written with the possibility of formal verification rigourous proof of an inequality that is then used in mind, and it is generally recognized that this as a vital ingredient in the proof of the theorem. greatly impeded any attempt to apply such meth- In principle the calculations could be done by hand, ods. but in practice this would be quite impossible. Another possibility would be to write a totally new program that implemented the ideas in the the- Formal Verification of Proofs oretical part of the proof. This was dismissed as Anyone who has written even short computer pro- being too much to demand of any group of refer- grams knows that they are much less forgiving ees, a statement that shows how little mathemati- than mathematics. Tiny errors of syntax are caught cians appreciate the labour involved carrying by the compiler and stop the program completely. through projects to completion in other areas of The multiple uses of variable labels do not stop the science, for example the Cassini space probe to Sat- program running, but they are usually easily de- urn. Also relevant is the fact that as the refereeing tected by the fact that the output is rubbish. Math- process continued it became apparent that the ematical errors are often detected by running the computations were so specific to the particular program on a very simple problem of the same type, problem that they provided few insights that could to which the solution is already known. Varying the be applied to other similar problems. parameters of the problem allows one to check The Kepler problem is closely related to finding that the effects are as expected. Possible errors or the ground state energy of a large assembly of inaccuracies in standard routines built into a soft- bodies, which may have a variety of shapes and ware package are more difficult to detect, since the ways of interacting with each other. There is a effects are likely to be small or infrequent. Never- huge number of similar minimization problems, theless programs of only a few hundred lines in and it is infeasible to understand the field by solv- length can be extremely powerful aids to mathe- ing them one at a time by highly specific compu- maticians, and experience shows that they can be tations. If there is no other way perhaps most of made to function as expected after some debug- these problems are not so interesting after all. ging. The real problems occur with much bigger pro- However, the Kepler problem itself has connec- grams and are a major problem: the British Civil tions with several other issues of known impor- Service recently had to resolve a flawed software tance, including the theory of error-correcting upgrade that stopped the work of an entire de- codes. partment for almost a week. On the positive side I must mention the steadily increasing use of computers, which are trans- The formal verification of software packages is forming the work of pure mathematicians. Here are simultaneously an area of applied logic and a busi- a few randomly chosen examples, which fall into ness. The increased reliability of Windows XP has several different categories. Computer algebra can been achieved with the aid of powerful program transform hopelessly lengthy calculations and has analysis tools, which are themselves based on the been used extensively in various fields. The inves- mathematics of program correctness which was tigation of chaotic dynamical systems could not originally explored with the goal of formal verifi- have progressed without the possibility of nu- cation. However, in some respects the problem merical experimentation; it is true that the existence faced by computer scientists is quite unlike that of chaotic phenomena was discovered by Henri faced by mathematicians. The specification of some Poincaré at the end of the nineteenth century, but software, such as Java, may run to more than a hun- progress in understanding the subject had to wait dred pages, far longer than would be acceptable for for the development of computers. The enormous the statement of a theorem. It is not clear in some differences between the spectral behaviour of self- cases whether unexpected behaviour of a software adjoint and non-selfadjoint matrices came to light package should be called a bug or a feature. as a result of numerical experiments and has Crashes, often caused by buffer overflows, are spawned the new field of pseudospectra, which is clearly the consequences of design faults, but one now being studied as an area of rigorous mathe- cannot say the same of the refusal of LATEX to allow matics in its own right [28]. the user to do something that the designers never Controlled numerical calculations are also play- thought of. Inadequate specifications of large soft- ing an essential role as intrinsic parts of papers in ware projects are a much more common cause of various areas of pure mathematics. In some areas commercial disasters than incorrect implementa- of nonlinear PDE, rigourous computer-assisted tions of the specifications.

1352 NOTICES OF THE AMS VOLUME 52, NUMBER 11 The proven value of formal proofs of correctness that it would be obtained by extracting the core of in the software context has encouraged some com- Mercer’s argument. The student ended up satisfied. puter scientists to try to apply the same methods It seems that mathematics is carried in people’s to mathematics, but this is, at present, an imma- heads, and that it is malleable in the sense that ex- ture field. The following comments indicate that perts “know” almost instinctively whether it is pos- there are likely to be serious difficulties in imple- sible to modify standard theorems to fit the con- menting formal proofs of correctness in my area text being discussed; perhaps this is the definition of analysis. They may well not be so relevant to of an expert. Every now and again someone sum- other fields, such as logic or algebra, but I leave such mons up the energy to write out a fairly compre- judgements to others. I give some details in order hensive account of a field as a monograph. This pro- to provide some feeling for the issues, but these vides a huge service, by giving a systematic account are not essential. Almost every proof of a theorem of a field to which one can then refer. Very fre- in analysis alludes to external facts that are fre- quently it also misrepresents the literature some- quently not spelled out because they are assumed what, because an author is almost bound to adopt to be a part of the background of the reader. A a particular, uniform context in his monograph, and paper might well start by stating that it intends to many of the theorems that he proves will be true study the spectral theory of the Laplacian on a under weaker conditions. bounded Euclidean region subject to Dirichlet Finite Simple Groups boundary conditions. There are hundreds, possi- bly thousands, of papers even on this tiny subject, The third crisis that we discuss is also one con- and the writer will assume a familiarity with a sub- cerning complexity, but it is in some ways more se- stantial part of the literature. On some occasions rious. Since it does not involve computers, we can- he will refer to papers containing recent results that not dismiss it simply by declaring computer- assisted proofs illegitimate, i.e., not a part of what he considers the reader might not know about, but we call pure mathematics. In addition, the exam- in many cases he will use older results without ref- ple that I will describe involves one of the most cen- erence, confident that almost everyone who is well tral concepts in mathematics: group theory. enough educated to want to read the paper will al- During the 1970s more than a hundred group ready know these. theorists came together in a consortium devoted There are real traps into which one can fall, and to classifying all finite simple groups. The task people sometimes do fall into them. When using a was a massive one and provided what is still the particular result it is possible to forget that there only example of industrial scale pure mathemat- are often many versions of a theorem in analysis, ics. Under the leadership of Daniel Gorenstein the with similar conclusions, but depending on dif- problem was broken up into smaller packages that ferent technical hypotheses. Monographs often were entrusted to various groups around the world. make standing hypotheses, which are mentioned Intensive work over ten years led to a complete list at the start of some section or chapter, but not any- of all finite simple groups: three infinite families, where near the statement of the theorem being together with twenty-six sporadic (i.e., exceptional) quoted. groups. The existence of the largest of these, the It is commonplace to justify a step in a proof by so-called Monster, was only proved with the aid of reference to some classical result for which no ref- a computer. Fortunately we can discuss the crisis erence is given. I was challenged recently by one surrounding this problem without knowing what of my students in relation to Mercer’s theorem. Mer- the classification is, and without even knowing cer’s original version referred to kernels on a one- what a finite simple group is. dimension interval, but I was using a more general What happened after 1980 has been as inter- version of the theorem without explanation. When esting as the classification itself. One positive de- he asked me to justify my comment I was unable velopment in this period was the discovery of a to find a statement of the theorem in the literature method of avoiding the use of computers in the that was sufficiently general to cover the applica- proof of the existence of the Monster. It was ap- tion that I was making. After looking through a half preciated that the work of the different groups dozen books I eventually decided to write out the needed to be integrated into a single coherent ac- proof. It was obvious to me, and would have been count, but attempts to do this led to the discovery to anyone who had read the original proof in suf- of many gaps in the proofs. Many of these were ficient detail, that the classical restriction to an in- patched up, but one seemed very serious, and in terval was unnecessary, but it nevertheless took me 1990 claims that the classification was complete four pages to describe and prove a sufficiently had to be reconsidered. Eventually this gap was also general form of the result. I did not regard this as filled by Aschbacher and Smith and, once again, it a serious gap, in the sense that I was confident seems likely that the proof is sound [3]. However throughout that the result needed was correct, and only about five out of the twelve volumes of the

DECEMBER 2005 NOTICES OF THE AMS 1353 final proof have been published, almost twenty-five It is of course possible that a much simpler ap- years after the theorem was “proved”; see [3], [27] proach to this particular classification problem for details. Michael Aschbacher, one of the people will one day be discovered, but it is equally possi- most heavily involved in the project, admits the pos- ble that it will not. Aschbacher is pessimistic about sibility that a new finite simple group might one the existence of a moderately simple proof, ob- day be discovered. If that group has characteristics serving that the estimated overall length of the sufficiently similar to the others, this might not be (still unwritten) proof has not decreased much too disturbing, but he accepts that the discovery over the last quarter century. It follows from Tur- of a new finite simple group quite different from ing’s work that there are theorems whose proofs the others would throw the problem wide open are far longer than their statements: indeed the ratio again; see [4]. Note that Jean-Pierre Serre is also very of the two lengths can be arbitrarily large. Ac- cautious about accepting the proof [24]. cording to Cohen “the vast majority of even ele- Aschbacher has noted that the proof seems to mentary questions in number theory, of reasonable be robust. By this he means that every gap so far complexity, are beyond the reach of any reasoning” discovered can be plugged with only a moderate [13]. So we have to anticipate that more and more amount of extra work, leaving the main lines of the such results will be discovered as time passes. proof unaffected. Unfortunately, this does not imply that the result is correct. A chain is as strong The Consistency of Arithmetic as its weakest link, and the fact that every faulty In this section we argue that the existence of sim- link has so far been replaced by a sound one pro- ple statements that have extraordinarily long proofs vides no guarantee that it will remain so. If one may be of great importance. Gödel taught us that thinks that the proof is more like a web, in which it is not possible to prove that Peano arithmetic is flaws in many threads would not jeopardize the in- consistent, but everyone has taken it for granted tegrity of the whole, then it is possible that the web that in fact it is indeed consistent. contains a large enough hole for a fly to escape Platonistically-inclined mathematicians would through it. Most flies might be caught by the web, deny the possibility that Peano arithmetic could be but not necessarily all. flawed. From Kronecker onwards many consider The idea of comparing mathematical knowledge that they have a direct insight into the natural to a web of interrelated facts de-emphasizes the role numbers, which guarantees their existence. If the of linear logic in favour of the confidence associ- natural numbers exist and Peano’s axioms describe ated with a highly redundant structure. This is not properties that they possess then, since the ax- a new idea, but it has not been emphasized by ioms can be instantiated, they must be consistent. mathematicians much until recently. Aschbacher Often this is dressed up with references to the ex- uses a related analogy in [4], invoking the paradigm pected or intended model of Peano’s axioms, but of biology as an information-rich subject in which expectations or intentions do not by themselves set- there is an overabundance of different ways of or- tle anything. ganizing the data, and contrasting this with “clas- When we delve into history we see many reasons sical mathematics”. for doubting claims for certainty, even in mathe- The completion of the classification project (in matics. For many centuries it was thought self- the sense of the publication of a connected ac- evident that Euclidean geometry necessarily pro- count of the entire calculation) is threatened by the vided the correct description of space, but even- attrition of the leading players by death and re- tually Riemann and then Einstein proved this wrong. tirement. Within ten years most of them may have The status of the axiom of choice is usually re- stopped working, and there may well be too few garded as unproblematical nowadays, but there left with the necessary deep understanding of the was a vigorous debate early in the twentieth cen- subject to complete the task. Even if the project is tury about its acceptability. Even its inventor, Zer- brought to a conclusion, it is likely that fewer than melo, eventually agreed that the most compelling a dozen mathematicians will be able to claim a reason to accept it was the fact that without it reasonably comprehensive understanding of the mathematicians could not prove large numbers of main lines of the proof. results that they needed; see Maddy [19, p. 56]. We have thus arrived at the following situation. These doubts have not been resolved, but merely A problem that can be formulated in a few sen- forgotten, by most of the community. We finally tences has a solution more than ten thousand mention that Hilbert’s confidence about the pos- pages long. The proof has never been written down sibility of resolving all mathematical problems was in its entirety, may never be written down, and as shared by most of his contemporaries, until Gödel presently envisaged would not be comprehensible showed that it was unfounded. to any single individual. The result is important, and It is, in fact, logically possible that Peano arith- has been used in a wide variety of other problems metic is internally inconsistent. There is no evidence in group theory, but it might not be correct. for this, and we do not claim that it is likely to be

1354 NOTICES OF THE AMS VOLUME 52, NUMBER 11 inconsistent, only that it is possible. To investigate than they are for the classification of finite simple this idea further we consider an example from groups. One day the programs may be rewritten in group theory. Consider the following list of ax- a form that permits a formal proof of the correct- ioms. ness of Hales’ theorem. In the Royal Society meet- (1) G is the set of elements considered, and it ing some mathematicians repeated the well-known is supposed that the elements obey the group ax- argument that this would still not be satisfactory, ioms. because computer programs are fallible, computer (2) G is supposed to be finite but not isomor- hardware is fallible, and anyway the computer phic to any of the known list of finite simple groups. might be hit by a cosmic ray during the computa- (3) G is supposed to be simple. In other words, tion. These statements are obviously correct, but if N is a subset that has a certain list of properties it would be absurd to think that similar criticisms (those of a normal subgroup other than the trivial do not apply to human-generated proofs, particu- subgroup), then N = G. larly in the light of the finite simple group experi- These axioms can be compared to those of Peano ence. All one can ask of the formal computer ver- arithmetic. The last is similar in form to the in- ification of proofs is that they perform better than duction axiom (or axiom schema in first order human beings, in the sense that they find mistakes logic) in that it refers to an unspecified set with cer- in proofs that humans have missed and that hu- tain properties, and concludes that it is equal to G mans recognize once they are pointed out. In the (we assume that one can switch back and forth be- field of software and chip design verification this tween subsets and predicates). Although G is as- has already happened, and it is to be expected that sumed to be finite, its size is not specified, so one it will become more common in mathematics itself. cannot simply enumerate all objects of the above A number of mathematicians are very concerned type, however long the time given: the only way of about where this revolution is leading us. If the goal understanding the axiom system is via proofs. of mathematics is understanding, then one cannot The fact that an axiom scheme so similar to deny that computer-assisted proofs do not supply Peano arithmetic might require such a long proof it in full measure. But neither does the proof of the of its inconsistency (if indeed it is inconsistent, as classification of finite simple groups. In both cases most group theorists believe) provides a reason why the proofs are only locally checkable, and this pro- we cannot be absolutely sure of the consistency of vides no guarantee of global correctness. Many Peano arithmetic itself. Perhaps the shortest proof mathematicians find the prospect of losing this un- of an inconsistency in Peano arithmetic is one hun- derstanding abhorrent, and their best remedy is to dred million pages long, and we will never dis- stick to fields in which such methods are not yet cover it. If we were never led into a contradiction, needed. Fortunately there are vast swathes of the would the inconsistency matter? We could con- subject that remain ripe for development by tra- tinue to prove theorems and derive interesting in- ditional methods, so they need not worry too much terconnections between ideas without ever sus- that their contribution will become unnecessary pecting the awful truth. within the foreseeable future. Such a situation need not imply that our efforts Taking an historical perspective, we can see that were worthless. There are many examples in the once the number of mathematicians became large past in which contradictions in axiom systems, or enough, they were almost bound to start produc- counterexamples to theorems, once pointed out, ing a quantity of mathematics that could only be have been rectified. A famous book of Imre Lakatos validated at a collective level. Combine this with the is a celebration of the ability of mathematicians to development of ever more sophisticated computer respond to counterexamples to a sequence of software, and the possibility of individuals being flawed statements of Euler’s theorem [17]. The able to understand all aspects of a complex proof most famous inconsistency was in Frege’s foun- was certain to vanish. The twentieth century pro- dations of mathematics, to which Bertrand Russell vided both of these conditions for the decisive and found a paradox. Within twenty years the ZFC set irreversible change in the nature of mathematical theory removed these particular problems, although research. Pure mathematics will remain more reli- at some cost in terms of elegance. Interesting math- able than most other forms of knowledge, but its ematics (certainly in the field of analysis) is re- claim to a unique status will no longer be sustain- markably tolerant of changes in the axiomatic able. It will be seen as the creation of finite human framework, and can often be rescued from technical beings, liable to error in the same way as all other errors, possibly after changing or increasing the activities in which we indulge. Just as in engineer- number of assumptions. ing, mathematicians will have to declare their de- gree of confidence that certain results are reliable, Discussion rather than being able to declare flatly that the It seems to the author that the prospects for a proofs are correct. Hilbert’s goal of achieving per- complete proof of the Kepler problem are better fect certainty by the laying of firm foundations

DECEMBER 2005 NOTICES OF THE AMS 1355 died with Gödel’s work, but the problem of com- [4] ——— , Highly complex proofs and implications of plexity would have killed his dreams with equal fi- such proofs, in [10]. nality fifty years later. [5] J. AZZOUNI, Deflating Existential Consequence, Oxford We finally ask if there are further crises still to Univ. Press, Oxford, 2004. be faced. One possibility is the discovery of a con- [6] M. BALAGUER, Platonism and Anti-Platonism in Mathe- tradiction in a mathematical argument whose com- matics, Oxford Univ. Press, Oxford, 1998. [7] E. BISHOP, Foundations of Constructive Analysis, plexity is beyond any yet contemplated. One might McGraw-Hill, 1967. imagine that the contradiction is the result of a mis- [8] ——— , Schizophrenia in contemporary mathematics (Er- take that is too deep for us to be able to locate it, rett Bishop: Reflections on him and his research), even with the aid of computers. This may seem far- M. Rosenblatt, ed., Contemporary Mathematics, vol. 39, fetched, but a somewhat similar problem has al- Amer. Math. Soc., Providence, RI, 1985, pp. 1–32. ready arisen in computer chess programs, which [9] E. BISHOP and D. BRIDGES, Constructive Analysis, occasionally make moves for which the best chess Grundlehren der math. Wiss. vol. 279, Springer- grandmasters can find no rationale. The computer Verlag, Heidelberg, 1985. can, of course, only declare that the said move [10] A. BUNDY, D. MACKENZIE, M. ATIYAH, and A. MACINTYRE, yielded the highest score out of billions of combi- eds., The nature of mathematical proof, Proceedings nations that it had considered. This does not imply of a Royal Society discussion meeting, Phil. Trans. R. that the move is indeed the best in the given po- Soc. A, 363 (2005), to appear. [11] C. S. CHIHARA, Constructibility and Mathematical Ex- sition, because the method of scoring positions is istence, Clarendon Press, Oxford, 1990. derived from human advice. If such a scenario ma- [12] P. J. COHEN, Comments on the foundations of set the- terializes, we may finally have to admit to limits ory, in Axiomatic Set Theory, Proc. Symp. Pure Math. on what our species can aspire to in the mental vol. XIII, Part I, Amer. Math. Soc., Providence, RI, 1967, realm, as well as in other types of activity. pp. 9–15. Whether or not these prognostications prove [13] ——— , Skolem and pessimism about proofs in math- correct, the future of pure mathematics is certain ematics, in [10]. to be very different from its past. In 1875 every suf- [14] E. B. DAVIES, Science in the Looking Glass, Oxford ficiently able mathematician could fully absorb the Univ. Press, 2003. proof of most theorems that existed within a few [15] ——— , A defence of pluralism in mathematics, Phil. months. By 1975, a year before the four-colour Math. to appear. theorem was proved, this was not even close to [16] A. EINSTEIN, Lecture delivered to the Prussian Acad- being true, but it was still the case that some math- emy of Sciences, January, 1921, Ideas and Opinions, Crown Publ. Inc., New York, 1982, p. 233. ematicians fully understood the proof of any known [17] I. LAKATOS, Proofs and Refutations: The Logic of Math- theorem. By 2075 many fields of pure mathemat- ematical Discovery, Cambridge Univ. Press, Cambridge, ics will depend upon theorems that no mathe- 1976. matician could fully understand, whether individ- [18] R. MACPHERSON, Machine computation and proof, in ually or collectively. Many mathematicians will still [10]. prove theorems by traditional methods, but these [19] P. MADDY, Naturalism in Mathematics, Clarendon will stand out as landmarks in a much broader Press, Oxford, 1997. subject. Formal verifications of complex proofs [20] R. PENROSE, The Emperor’s New Mind, Oxford Univ. will be commonplace, but there will also be many Press, Oxford, 1989. results whose acceptance will owe as much to so- [21] ——— , Shadows of the Mind, Oxford Univ. Press, Ox- cial consensus as to rigorous proof. Perhaps by then ford, 1994. the differences between mathematics and other [22] M. PLUM, Computer-assisted enclosure methods for disciplines will be so much reduced that philo- elliptic differential equations, Lin. Alg. Appl. 324 (2001), 147–187. sophical discussions of the unique status of math- [23] M. PLUM and C. WIENERS, New solutions of the Gelfand ematical entities will no longer seem relevant. problem, J. Math. Anal. Appl. 269 (2002), 588–606. Acknowledgements [24] M. RAUSSEN and C. SKAU, Interview with Jean-Pierre I should like to thank M. Aschbacher and C. A. R. Serre, Notices Amer. Math. Soc. 51 (2004), 210–214. Hoare for valuable advice. Reprinted from European Mathematical Society Newsletter, September 2004, pp. 18–20. 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