5. Dramatizing the History of Mathematics:
Dennis Guedj has used the nice metaphor of "the drama of axiomatics" to describe the fact that in an axiomatized mathematical theory, the contents of a theorem are implicit in the axioms and that in the derivation of a theorem from the axioms there is an inexorability of the kind that characterizes a drama. Perhaps you may wonder about the details of the way from the axioms to the theorem (i.e., the details of the plot), but there is no escape from the one possible denouement of this story. It its highly problematic, however, to attempt to extend the scope of this metaphor so that, rather than being limited to the logical status of a theorem vis-à-vis the axioms, it will become a style of historical writing and a historiographical conception about mathematics (or science in general). In order to explain this point I would like to pledge to the Gelfand-Gowers principle and start with an illuminating, and well known example: Simon Singh's Fermat's Engima.
Fermat's Engima is possibly the best-selling and most widely known among a relatively large group of popularization books on mathematics that appeared over the last ten years. As such, I think it is fair to say that it has done a greater service to the recent public perception of mathematics than any other individual text that I can think of. In order to write his book on FLT, Singh certainly needed to invest great efforts in order to gather and digest an enormous amount of relevant mathematical material, and to present it in a more or less, popularized version. This is by all means a difficult and laudable task, and in order to accomplish it, Singh relied on a far reaching dramatic structure to support a narrative specifically designed to keep the attention of the readers throughout. In doing so, however, the book brings to bear a great amount of misconceptions about the history of mathematics, not only concerning specific details, but also concerning broader issues. The over-dramatization of the history of mathematics is among the latter. For better and worse, Fermat's Engima has played over the last decade a role similar to that played several decades ago by Eric Temple Bell's Men of Mathematics.[*]
The over-dramatized approach is evident even before starting to read the book, as the publishers (at least in some editions) make clear that this is "the epic quest to solve the world's great mathematical problem". The cause is supported by no less a scientist than Sir Roger Penrose who is quoted as saying: "An excellent account of one of the most dramatic and moving events of the century." No less than that. And then in the dust jacket we read the following:
FLT became the Holy Grail of mathematics. Whole and colorful lives were devoted, and even sacrificed, to finding a proof. Leonhard Euler, the greatest mathematician of the eighteenth century, had to admit defeat. Sophie Germain took on the identity of a man to do research in a field forbidden to females, and made the most significant breakthrough of the nineteenth century. Evariste Galois scribbled down the results of his research deep into the night before venturing out into a duel in 1832. Yutaka Taniyama, whose insights would ultimately lead to the solution, tragically killed himself in 1958. On the other hand, Paul Wolfskehl, a famous German industrialist, claimed Fermat had saved him from suicide and established a rich prize for the first person to prove the theorem.
Lives “devoted, and even sacrificed” in the pursuit of an abstruse mathematical question is definitely a story worth of attention, but in closer analysis, every sentence in this description turns out to be a dramatic overstatement at best.[*] This spirit of over-dramatization dominates a great part of the book. The preface, for instance, opens with the following passage:
The story of Fermat’s Last Theorem is inextricably linked with the history of mathematics, touching on all the major themes of number theory. … The Last Theorem is at the heart of an intriguing saga of courage, skullduggery, cunning, and tragedy, involving all the greatest heroes of mathematics.
In this way the dramatizing effect comes to be closely connected with the "royal-road-to-X" approach mentioned in the first part of this article. Not only many intriguing episodes in the history of mathematics are harnessed on behalf of the denouement of the drama even if they have nothing or very little to do with FLT (Galois is a case in point), but many significant, and highly interesting mathematical developments that were at the heart of the attempts to prove Fermat's conjecture are totally ignored just because in the final account they did not become part of the triumphant party.[*] Moreover, the many indications of the minimal or nil efforts devoted by leading number-theorists to the conjecture throughout history are systematically ignored.[*]
Within the entire story of FLT, the episode involving Wiles and his lifetime interest in FLT is perhaps the one that comes closer to a real personal drama of the kind implied by Singh’s account. But then, the triumphant party in this account includes anyone who was even remotely associated with the line of developments leading to Wiles' proof (or even not associated at all, like Galois). The apotheosis, embodied in Wiles' Cambridge lecture of 1993, appears right at the opening sentence of the first chapter. Singh says:
It was the most important mathematics lecture of the century. Two hundred mathematicians were transfixed. Only a quarter of them fully understood the dense mixture of Greek symbols and algebra that covered the blackboard.
The skeptical mathematician --used as she is to sit among somewhat baffled colleagues in even remotely less demanding colloquium talks-- will wonder about the count of 'only a quarter' that fully understood what was said. 'As much as a quarter' would probably still sound to her as an over-overstatement.[*] Thus, in the bottom line, not only a balanced historical picture is abandoned in the book in favor of the dramatized account of FLT and the efforts to solve it, but actually the true import of Wiles' formidable accomplishment in proving the Taniyama-Shimura conjecture cannot be conveyed to the reader. Singh does mention that "while science journalists had eulogized over Wiles' proof of Fermat's Last Theorem, few of them commented on the proof of the Taniyama-Shimura conjecture" stressing that the latter was "inextricably linked to" FLT. Singh also reported on Ken Ribet's feeling that "proving the Taniyama-Shimura conjecture had transformed mathematics". But all of this amounts to too little and too late in the book, and a clear statement is lacking concerning the correct relationship between Taniyama-Shimura and FLT. Such statements could be easily found around and Singh might have simply quoted what other had clearly stated. Consider, for instance, the following opinion, advanced by Barry Mazur just in 1991, not long before the results of Wiles' quest became public:[*]
Fermat’s Last Theorem has always been the darling of the amateur mathematician and as things have progressed, it seems that they are right to be enamored of it: Despite the fact that it resists solution, it has inspired a prodigious amount of first-rate mathematics. Despite the fact that its truth hasn’t a single direct application (even within number theory!) it has, nevertheless, an interesting oblique contribution to make to number theory; its truth would follow from some of the most vital and central conjectures in the field. Although others are to be found, Fermat's Last Theorem presents an unusually interesting “test” for these conjectures.
Thus, after close to 350 years of history, and just before the big euphoria aroused by Wiles’ impressive achievement, Mazur summarized the historical status of FLT in a very concise formulation, that makes justice to its purely tangential importance and without really overreaching (even though, one might still question how prodigious is indeed the amount of first-rate mathematics inspired by FLT). It is not that Taniyama-Shimura is important because it allows proving FLT, but just the other way round: FLT suddenly appears as more mathematically important that it could have ever been considered by number-theorists throughout history just because what it does to Taniyama-Shimura (i.e., it provides an illuminating consequence and thus enhances the fundamental importance of the conjecture). But then of course most of the dramatic effect of Singh's account is simply lost.
Singh's book provides a recent example of the potential pitfalls involved in over-dramatizing the history of mathematics. It would be too easy to explain its approach by saying that this is a popularization book that successfully fulfils its aim, and that the over-dramatization it incurs more than faithfully serves its purpose, namely, bringing a broader audience closer to the world of mathematics, its people and its ideas. Whether or not one accepts such a claim in relation with Singh's books, it is very important to notice that this over-dramatic image of the history of science has been essentially shared by scientists themselves and that, until relatively recently, it was commonly found in a considerable amount of mainstream academic historiography of science as well. Indeed, as Yehuda Elkana insightfully stressed more than twenty-five years ago, this view was an outgrowth of a long-engrained tradition in Western culture that identifies "fate in Greek tragedy with the order of nature" and thus sees "natural occurrences and events as inevitable". This point of view, Elkana asserted, was later extended so as to cover not only the natural events in the world, but also the unfolding of human knowledge about the world. Thus he wrote:[*]
The conviction emerged and grew, leading up to its positivistic absoluteness in the Victorian frame of mind, that not only there is one reality with it immutable laws, but also that we humans are on a sure course to find out all, or at least cumulatively more and more about the reality: one nature, one truth about nature. Science, the chief glory of Western culture since the scientific revolution, is an inevitable unfolding of knowledge; what we know we had to know -- if not here, then there, if not now, then at another time, if not discovered by one man, then by another.
Elkana draws attention to the interesting example of Alfred North Whitehead, who explicitly identified the spirit of modern science with Greek tragedy, and attributed a central role to fate in the development of our knowledge of nature. "The absorbing interest in the particular heroic incidents as an example and a verification of the workings of fate -- said Whitehead --, reappear in our epoch as concentration of interest on the crucial experiments". And the foremost, recent crucial experiment that Whitehead had in mind, and that he could present as an illustration of his views, concerned the results announced by the Eddington eclipse expedition of 1919 that measured the deflection of light rays by the sun gravitational field and thus reportedly confirmed Einstein new theory of gravitation. Whitehead described the announcement by the Astronomer Royal at the joint meeting of the Royal Society of London and the Royal Astronomical Society, on November 6, 1919, and his description is phrased in terms that are unmistakably chosen so as to enhance the theatrical character of the scene. He thus wrote:[*]
The whole atmosphere of tense interest was exactly that of the Greek drama: we are the chorus commenting on the decree of destiny as disclosed in the development of a supreme incident. There was dramatic quality in the very staging: the traditional ceremonial, and in the background the picture of Newton to remind us that the greatest of scientific generalizations was, now, after more than two centuries, to receive its first modification.
One needs not doubt the momentous historical significance of this event, and yet a dose of sane skepticism is in order here. One is indeed allowed to wonder if the participants of the meeting actually shared the dramatic feelings retrospectively reported by Whitehead six years later. Even if they did, one may think that the subjective reasons behind such feelings were more substantial and evident than the possible, objective ones. For one thing, Eddington's findings were not hidden from them until the onstage appearance of an Oracle-like Astronomer Royal that would officially open a sealed envelope to announce that "the winner is ...". For another thing, the results were far from unambiguous and it turned out that the strong personal influence of Eddington and of the Astronomer Royal, Sir Frank Dyson, became a decisive factor for the quick, widespread acceptance of the results, and of their implication as a confirmation of Einstein's theory. For instance, no less a physicist than J.J. Thomson, who attended the meeting, explicitly conceded his inability to judge on this question without further examination as well as his willingness to rely on the authority of his prominent colleagues. He thus said:[*]
It is difficult for the audience to weigh fully the meaning of the figures that have been put before us, but the Astronomer Royal and Prof. Eddington have studied the material carefully, and they regard the evidence as decisively in favor of the larger value for the displacement.
A complex mixture of social, institutional, political and cultural circumstances (all of them fully legitimate and human) stand at the background of this interesting chapter in the history of twentieth-century science.[*] They must all be taken in consideration, together with the purely scientific issues involved here, if we want to make full sense of the impressively quick and sweeping process of acceptance of Einstein's new theory on the basis of the astronomical observations of 1919.
Being this the case, one may for a moment conjecture about possible scenarios that might have ensued, had the results of the expedition not been as readily accepted as they were (under the active influence and authority of Eddington and Dyson, and for the many reasons that guided their efforts in this direction) or if the results had showed a preference for Newton's theory over Einstein's. These are by no means imaginary scenarios and they could have easily materialized had the circumstances been different.[*] It is important to remember, in this context, that once the measurements reported by the expedition (and by implication, the confirmation of Einstein's theory with the concomitant refutation of Newton's one) were published in British, (and later in German) newspapers, Einstein was immediately catapulted into world fame. He thus turned into a cultural icon that embodied for decades to come the ideal of the scientist as a secular saint working in isolation from the rest of the world. The events of 1919 played an important role in shaping much of the course of physical science in the twentieth century as well as of its public perception. Thus, the idea of the very possibility of an alternative historical scenario materializing in this regard allows for some further thinking, from which I am led to raise a second, related point stressed by Elkana.
Elkana's criticism outlined above, of the perspective provided by Greek drama as a basis for the conception of the history of science, need not be taken as an a attack on the objectivity of science. Rather it promotes an alternative model for the historiography of science, itself based on a different theatrical conception that of the "epic theater". The model that Elkana has in mind draws on a conception developed by Bertolt Brecht and Walter Benjamin,[*] and it is diametrically opposed to that of the Greek drama in a way the can be summarized in the following schematic table:
Science as Drama/
|Science as Epic Theater (Brecht)|
|We know what will happen: drama arises because we know that it will happen||"Things can happen this way, but they can also happen in a quite different way" (Walter Benjamin)|
|Characters||Human emotions, ideas, and behavior as products of, or responses to the unfolding of the human essence||Human emotions, ideas, and behavior as products of, or responses to, specific social situations|
|Theme||Universal elements of the human situation and fate||Behavior people adopted in specific historical situations|
Elkana describes this model for the history of science as "undramatic" and he characterizes it as follows:[*]
Epic theater, in order to make its point, purposefully avoids historical facts that the audience is aware of, lest they lapse into the tragic mood of knowing what is inevitably coming. Life is unpredictable and events can go in any direction, therefore life is unsensational. What is true of historical inevitability also holds for psychological inevitability, and this, too, is avoided. In short, epic theater is a relaxed, nonsensational, reflective attitude to unpredictable events. To put it in another formulation: the historical question is not what were the sufficient and necessary conditions for an event that took place, but rather, what were the necessary conditions for the ways things happened, although they could have happened otherwise.
The case of the eclipse expedition and its aftermath provides an enlightening example of things that happened in a certain way, but could have happened in a very different way. Indeed, I think it is fair to assert, risking too broad a generalization, that a great deal of interesting research in the history of science over the last two decades has become much closer to the "epic theater" perspective that to the "Greek drama" one.[*] It will be interesting to see how fictional narrative on science (and particularly on mathematics) as well as popularization books on the same topics catch up with this important development.
6. Concluding Remarks - Can Mathematics in Fiction Interfere with Mathematical Reality?:
According to Eco, we read fictional texts since they come to the aid of our metaphysical narrowmindedness and offer an illusion of order within a world whose total structure we are unable to grasp and to describe. Since we know that fictional universes are created by an “authorial entity”, we know that there is a “message” behind them. The very confidence on the existence of this message is, in the first place, what allows us to decipher it, or at least to think that we are on the way to deciphering it. This explains why we feel comfortable in fictional worlds. The actual world, on the contrary does not offer this confidence. Rather, “since the dawn of time, humans have been wondering whether there is a message, and, if so, whether this message makes sense.”[*]
We can now ask ourselves: Is this argument valid for mathematics and mathematics in fiction? We surely know that fictional narratives, even if they are about mathematical themes, are created by an “authorial entity”. But what about mathematics itself? What can we say of that “actual world” about which authors of mathematics in fiction build their fictional universes? One may question, in the first place, whether this “actual world” is indeed actual or itself fictional. One may question whether or not there exists an “authorial entity” behind this “actual world” of mathematics. But no one will deny that the kind of comfort that Eco attributes to our experience with fictional worlds is manifest in a very remarkable way in our encounters with mathematics. True, some people experience difficulties in technically mastering the world of mathematics. But once mastered, it provides perhaps the utmost example of a fictional (or fictional-like) world where the certainty of an underlying message is strongly felt and where, indeed, progress is continually and consistently made on the way to elucidating that message.
Eco also calls attention to that very remarkable phenomenon of intertextuality whereby fictional characters start to migrate from one fictional work into another. When this happens, Eco says, the characters “have acquired citizenship in the real world and have freed themselves from the story that created them.”[*] When one thinks about mathematics in these terms, a rather original explanation seems to arise about the fundamental Platonistic attitude engrained in the mind of the typical working mathematician. Whatever her professed philosophical confession, the typical mathematician will relate to the objects of her investigation as part of an external reality that can be objectively known.[*] Following along Eco’s line, mathematical entities (such as groups, functions, topological spaces, algorithms or whatever) can be seen as fictions that arise within a certain text and then start to migrate to ever new ones until they become ubiquitous, and eventually acquire their status of autonomous , “actual” entities. A mechanism similar to the one that applies to characters of fictional narrative that at some point liberate themselves from the texts in which they first appeared (Sherlock Holmes is a favorite example of Eco) seems to be at play in this case.
Finally, if fiction so strongly fascinates us, asks Eco, may it not be “that we interpret life as fiction, and that in interpreting reality we introduce fictional elements?”[*] Little needs to be said here about how, ever since the seventeenth century, science has been interpreting reality with the help of mathematical ideas. The latter can, in this context at least, be considered like fictions that help us interpret reality. The specific example that Eco refers to, however, seems to point into a different direction that we might also consider here. Eco shows in detail how the text of The Protocols of the Learned Elders of Zion arose from various, purely fictional sources, and how its very existence was effectively taken by its readers to be a confirmation of the message it conveyed. This is a most salient example of fiction intruding into real life with tremendous historical consequences. Can we imagine a similar situation in the case of mathematics? I can think of very few examples of this kind, but there is at least a recent one that cannot be overlooked: Andrew Wiles and FLT.
Wiles’ fascination with FLT reportedly started in his childhood when he read Eric Temple Bell’s The Last Problem (1962). This book together with Bell’s better-known, and indeed legendary, Men of Mathematics (1937) are among the most salient examples of history of mathematics written in the over-dramatized style I discussed above, as a series of essentially undocumented legends about mathematical heroes.[*] This approach, that serious historians like to dislike, caught, and continues to catch to this day, the imagination of many young readers. Some of these young readers became research mathematicians and this was also the case with Andrew Wiles. Had the young, mathematically sensible child, read a moderate, undramatic account of the kind I praised above —for all the historiographical and scholarly qualities I do think it has —it is rather unlikely that FLT would have ignited his imagination as it did. Wiles started his professional career without at first devoting any research to FLT, and became prominent in the fields he investigated. But in 1986, when certain recent developments indicated that FLT had become a mathematical task that might be solved by proving a well-defined, though obviously highly challenging, conjecture, he decided to return to his old love and to give it its final stroke. Thus, it was only his direct, emotionally-laden motivation to prove FLT that provided Wiles with enough fuel to undertake the long and difficult quest that was eventually crowned with sensational success more than eight years later. Bell’s account, then, which was essentially fictional, even if related to real historical events, did intrude the actual world of mathematics and led to its transformation with the help of Wiles.
And yet, the truly ultimate way in which fiction could intrude the actual world of mathematics, I would like to suggest, would be that of a novel on mathematical issues in which some kind of mathematical idea would be suggested (for example a certain way to solve a famous open problem) and that eventually this suggestion would lead some mathematician to develop the actual solution for the problem. I know of no example of this kind in history and I really doubt that it may materialize. Probably also in this respect the mechanisms that control the relation between “reality” and “narrative fiction” are of a different kind when it comes to mathematics.