TY - JOUR
AU - Fillion,, Nicolas
AB - ABSTRACT This paper examines consequences of the computer revolution in mathematics. By comparing its repercussions with those of conceptual developments that unfolded in the nineteenth century, I argue that the key epistemological lesson to draw from the two transformative periods is that effective and successful mathematical practices in science result from integrating the computational and conceptual styles of mathematics, and not that one of the two styles of mathematical reasoning is superior. Finally, I show that the methodology deployed by applied mathematicians in modern scientific computing is a paradigmatic instance of this key lesson. Some fundamental debates in philosophy of mathematics keep reappearing in the literature. In each epicycle, the opposing views — call them thesis and antithesis, to use an old cliché — are rearticulated in light of the mathematics du jour. A plausible lesson to draw from this repeating story would be that both sides of the debate get part of the story fundamentally right, which would suggest that a synthesis is called for. And yet, all too often, the result is an increasing polarization of the debate, where each side’s position becomes more entrenched. The debate I wish to address here concerns conceptual and computational mathematics and a few related themes, including the constructivist and non-constructivist stances in philosophy of mathematics. I will argue for the simple view that better mathematics gets done when we draw from the best of both worlds, and that the cavalier attitudes of the advocates of one exclusive view only impede the progress of mathematics. Finally, I will illustrate the sort of gains that can result from a better integration of both styles of mathematics with the methodology deployed by applied mathematicians in modern scientific computing. 1. THE OTHER REVOLUTION IN MATHEMATICS Mathematics has undergone two profound transformations in the last two centuries. Whether or not the transformations in question should be characterized as revolutionary is subject to controversy, since some historians claim that there are no revolutions in mathematics, e.g., Crowe [1975]. Yet, most scholars seem to grant that there are revolutions in mathematics, in some sense of the term ‘revolution’; see, e.g., the collection by Gillies [1992b]. Historians, philosophers, and mathematicians themselves often emphasize one such transformation, namely, the considerable increase in abstractness in the methods and objects of modern mathematics, e.g., [Marquis, 2016], which has resulted from the resolution of foundational problems that emerged in geometry, real and complex analysis, number theory, and eventually set theory. In fact, scholars maintaining that there are revolutions in mathematics have located many different, but thematically related, revolutions in that period, e.g., [Dauben, 1992; Zheng, 1992; Gillies, 1992a]. In addition, the structuralist turn initiated by mathematicians such as Dedekind in the nineteenth century and pursued in the twentieth century by various schools has led mathematics to become an increasingly qualitative and conceptual discipline. It would perhaps not be an exaggeration to say that an overwhelming majority of the recent work in philosophy of mathematics has focused on this transformation.1 This being said, it is only one side of the coin, the flip side being a re-emphasis on the computational dimension by applied mathematics. Indeed, there has more recently been a revolution of no lesser importance in mathematics, stimulated by the increasingly wide-ranging use of digital computers. The ‘computer revolution’, as sociologists call it, has profoundly transformed our daily lives, finances, industries, etc., in short all sectors of human activity including science and mathematics, just as the industrial revolution did in the nineteenth century. The many repercussions of the computer revolution on mathematics were already felt and discussed by leading mathematicians decades ago (e.g., [Knuth, 1974; Atiyah, 1986]). And today the impact of computers on mathematical practice and applications is stronger than ever before. But perhaps more surprisingly, many scientists have seen this revolutionary transformation of the way in which mathematics is employed in science as having rendered traditional branches of mathematics obsolete. Fraser and Nakane [2003] have made a nice survey of this sort of claim in the field of celestial mechanics; for instance, in his once influential lectures, Sternberg [1969, p. xvi] claimed that [i]n a certain sense, the work of the classical astronomers in perturbation theory is no longer relevant for predicting the motion of the planets. The advent of radar astronomy (and to a certain extent the space probes) has given observational accuracy far surpassing anything known heretofore. This information is incorporated into the equations of motion which are numerically integrated on high speed computers and the various numerical constants are continually readjusted to fit the data. This work […] allows computer prediction of planetary positions far more accurate (by brute computation) than anything provided by classical perturbation theory. In a very real sense, one of the most exalted of human endeavors, going back to the priests of Babylon and before, has been taken over by the machine. A more careful mathematician, scientist, or philosopher might plausibly push back against this claim, remarking that the numerical methods implemented in computer programs are in fact grounded in the very perturbation methods deemed to have become useless. For instance, Kadanoff [2004] describes such claims as an ‘optimization of enthusiasm and misjudgement’. In fact, I will do just that later in the paper and explain precisely what is wrong with this point of view. But a basic fact remains: there certainly has been an important change to practice, as it relies on the use of computers to run numerical simulations on an unprecedented scale. Given this situation, one might ask: what impact, if any, did the computer revolution have on the philosophy of mathematics? Many scientists certainly claimed that science has profoundly changed, but is the computer revolution a scientific revolution in the sense of the tradition of Kuhn [1962], Laudan [1977], etc.? In the literature on the epistemology of computer simulation, a number of philosophers (e.g., [Winsberg, 2010; Humphreys, 2004; 2009]) have described important respects in which this might be the case, such as the increased range of models that are mathematically tractable, the development of new experimental techniques, the opening of new theoretical possibilities, and even the creation of new areas and sub-areas. The claim is that the transformation is so profound that it calls for a new philosophy of science suited to this new science — including its methodological, epistemological, logical, semantic, and ontological dimensions. Others, such as Frigg and Reiss [2009], have argued that this is not the case:2 … the philosophical problems raised by simulations have analogies in the context of modelling, experimenting, or thought experimenting and are therefore not entirely new and far from demanding a novel philosophy of science. (p. 611) The point of their argument is that, since one can show that issues about the context of justification of the use of computers are in continuity with standard issues previously found in philosophy of science, we should work on improving existing philosophy of science to account for new methodological challenges that may otherwise have been neglected instead of attempting to build something entirely new. If the justificatory problems and methods are continuous with the traditional ones rather than revolutionary, then the computer revolution does not warrant a philosophical revolution. There is no need for a new philosophy of science and mathematics because of that. This paper is written from this perspective. So, the philosophical novelty associated with the computer revolution is perhaps not to be found in a new theory of justification of scientific and mathematical knowledge. But emphasizing the continuity helps to improve existing epistemological accounts of scientific practice. And paying close attention to new practices might suggest new interesting questions that demand that one explain why working mathematicians have changed the focus and methods of their studies. In this way it might still be the case that examining the consequences of the computer revolution from a philosophical angle would shed light on the nature of modern mathematics by stressing the epistemological importance of practices that have to some extent been neglected by philosophers. I will argue that one such interesting element is that, more than ever before, computers have enabled a profound integration of the conceptual and computational aspects of mathematics, thereby contributing to the synthesis between styles of reasoning in mathematics alluded to earlier. 2. CONCEPTUAL VERSUS COMPUTATIONAL MATHEMATICS I will begin this discussion by articulating the distinction between conceptual and computational mathematics. The distinction is common, not only in philosophy of mathematics but also in the education literature. In the latter, the distinction is more often articulated in terms of conceptual and procedural mathematics (see, e.g., [Hiebert, 2013]), but there is significant overlap between the two fields of literature. For the sake of this section, I will mostly draw from [Avigad, 2006; Tappenden, 2006; Pincock, 2015]. Avigad elaborates on the distinction to explain the methodological contributions of Dedekind’s theory of ideals, Tappenden develops it to emphasize the Riemannian background of Frege’s work, and Pincock introduces it in relation to Galois theory (specifically, the unsolvability of the quintic in radicals) to clarify the concept of mathematical explanation. The latter two clearly suggest that, in some ways, conceptual mathematics is superior to computational mathematics. On the other hand, Avigad argues for a point closer to the one made here by pointing out their complementarity. Before discussing those three accounts, a simple example3 from [Halmos, 1991] will set the mood: Problem. Suppose that 1025 tennis players want to play an elimination tournament. That means: they pair up, at random, for each round; if the number of players before the round begins is odd, one of them, chosen at random, sits out that round. The winners of each round, and the odd one who sat it out (if there was an odd one), play in the next round, till finally there is only one winner, the champion. What is the total number of matches to be played altogether, in all the rounds of the tournament? Comment: In this question, you are expected not only to find ‘a’ solution, but also to find the best, or the most beautiful way to solve it. A hint is also given: Hint: The ‘patient’ way to solve the problem is to design an algorithm, based on summing a geometric progression. The ‘right’ way is to note that each match has a loser. The patient way alluded to here belongs to computational mathematics. The ‘right way’ consists in simply noticing that, since every player will lose exactly one match except the winner and each match has exactly one loser, there will be |$1025-1=1024$| matches. This second way solves the problem using conceptual mathematics. Of course, the problem is so simple that most of the subtleties associated with those mathematical styles of reasoning are lost, but it is a good starting point. Now, let us turn to more historically important mathematics. A mainstay problem in mathematics consists in finding roots of equations, i.e., for a function |$f$| and a value |$y$|⁠, find the set of arguments |$x$| such that |$f(x)=y$|⁠. We see the importance of the problem by noting that it is the inverse process of evaluation of a function, in which |$f$| and |$x$| are given, and |$y$| is sought. The desire to solve this inversion problem for all univariate polynomial functions, \[ f(x) = a_0 + a_1x+a_2x^2+\cdots+a_nx^n =0\>, \] has been one of the early driving factors in the development of modern mathematics. In the eighteenth century, it was well-known that polynomials of degree four or less could be inverted in a straightforward manner and that the zeros could be expressed algebraically, using radicals. The case of the fifth-degree polynomial proved to be resisting the best attempts. On the basis of new methods introduced by Gauss and Lagrange, Abel showed in 1824 that it is impossible to express the zeros of the quintic in terms of radicals. The strategy behind his proof was as follows: for |$n>4$|⁠, suppose that a formula using radicals expresses the unknown roots, and derive a contradiction from that supposition by means of explicit calculations. His approach was thus computational in nature. However, the story does not end there, as Galois proposed a different approach in 1831, an approach that has been developed into what we now know as Galois theory. The Galois-theory strategy is based on much more abstract concepts and methods articulated in terms of fields, field automorphisms, and field extensions. The condition for the solvability for polynomial equations is given in terms of equivalence to a Galois group, as opposed to being based on explicit computation. The lack of computation and the emphasis on abstract constructions places this strategy firmly in the camp of conceptual mathematics. As Pincock [2015] put it, ‘Galois theory is a central instance of the more abstract, conceptual approach to mathematics that is sometimes known as modern mathematics.’ Note that his statement suggests that modern mathematics is at least predominantly conceptual, a claim that is difficult to accept in light of the computer revolution. The motivation behind Pincock’s claim is that he is interested in the distinction between explanatory and non-explanatory proofs, where the latter only show that a statement is true, rather than also showing why it is true. His view is that proofs are judged more explanatory because they use conceptual rather than computational mathematics; presumably computational mathematics would only be showing that a mathematical statement is true, not why it is. The Galois-theory proof of the unsolvability of the quintic in radicals would thus be a paradigmatic case of an explanatory proof. His analysis builds on that of Tappenden [2006], who analyzes another key episode of the conceptual transformation in modern mathematics, namely, the articulation of Riemann’s paradigm in complex analysis (especially in relation to controversies related to elliptic and Abelian functions, the Dirichlet principle, the concept of function, and conceptions of mathematical rigour and purity). This episode features two opposed schools of thought concerning the foundations of complex analysis: the Riemann school based in Göttingen and the Weierstrass school based in Berlin. Tappenden’s narrative is fascinating and extremely rich in details. But it is told with the firm intent of showing the superiority of Riemann’s approach (as well as explaining that this was the main motivation behind Frege’s work in philosophy of mathematics and logic). According to him, Riemann’s mathematics was revolutionary, exhibiting for the first time a variety of the styles of reasoning that we associate with contemporary mathematics, while Weierstrass’s was a continuation of a broadly computational mathematics that continued what had gone on before. [2006, p. 99] Both clauses in this quotation are correct. But Tappenden also uses various dismissive phrases, e.g., the ‘procrustean restrictions that were bound up with the Weierstrass techniques’ (p. 99), that are more questionable. Weierstrass’s methodology in complex analysis was very much in line with older works by Euler and others. The idea is to proceed to the study of a function in a fully analytical way, first by identifying elementary algebraic operations on complex numbers, then by developing an apparatus to construct infinite power series |$\sum_k a_k(z-z_0)^k$| (or other forms of asymptotic series) that are typically valid in the neighborhood of a point |$z_0$| for some radius |$r$|⁠, and then to find the explicit coefficients of the infinite power series associated with the function of interest. Weierstrass’s view was as follows: At first the purpose of these lectures was to properly determine the concept of analytic dependence; to this there attached itself the problem of obtaining the analytic forms in which functions with definite properties can be represented … for the representation of a function is most intimately linked with the investigation of its properties, even though it may be interesting and useful to find properties of the function without paying attention to its representation. The ultimate aim is always the representation of a function.4 Thus, Weierstass and his school promoted the constructive manipulations of representations of functions instead of abstract, non-constructive existence proofs of functions having the desired properties. The limitations of this approach are relatively easy to see. First and foremost, there might very well be functions that are not representable using infinite series of certain kinds (and indeed Fourier analysis suggested that this was the case, e.g., something as simple as a piecewise step function). Also, most of the work consists in explicitly computing expressions for the coefficients of the terms in infinite series, a task that can be most taxing. Pincock also considers that this approach is not explanatory as it is primarily based on ‘brute force’ [2015, p. 3]. In contrast, Riemann’s approach treated functions in themselves, without any reference being made to their particular representations as infinite series. Riemann always sought to start from an implicit characterization of a function in terms of its defining property and to reason on that basis alone (to the extent that it was possible). Using those defining properties, his most insightful results were based on non-constructive proofs of existence resulting from arguments by contradiction. In addition, he kept his distance from the analytic style of Weierstrass and made extensive use of visualization and geometrical considerations, most famously with his use of what we now call ‘Riemann surfaces’ to represent complex multifunctions. All philosophical and historical authors who have written on Riemann seem to agree that his style is clearly conceptual, and that he actively opposed computational, or algorithmic, procedures. The essence of Tappenden’s message seems right: Riemann was a revolutionary mathematician who contributed to the transformation that led to modern conceptual mathematics. However, the part that is wrong is the characterization of Weierstrass’s approach as a ‘powerful conservative thrust’ that proves to be inferior to Riemann’s from the standpoint of history. Today a significant part of the mathematical work involving special functions and other elements of complex analysis is done with the help of computers. Powerful computer algebra tools such as Maple and Mathematica give us ways of manipulating complex functions by means of their infinite series representations in a painless way, and numerical platforms such as Matlab and NumPy give us tremendously fast ways of using truncated series to get relevant approximate evaluations of functions. That is, we are still using Weierstrass’s computational approach, but the computation is no longer painful. This is why in a recent book review, Higham, a leading applied mathematician, stated that the numerical analysis book under review was aptly eschewing Riemann surfaces, which have ‘almost no traction in the computer world’ [Higham, 2015]. This is not meant to demean conceptual mathematics. Rather, the point is to emphasize that some problems in mathematics require the explicit computational style, and this is particularly true of the modern mathematics that is performed with the help of computers. In a paper discussing Dedekind’s mathematical innovations, Avigad [2006] suggests that a more balanced appraisal of the relative importance of the two styles of mathematics is appropriate. Dedekind contributed significantly to the development of more abstract methods that are typically associated with conceptual mathematics. Avigad describes Dedekind’s contributions to the development of a conceptual way of doing mathematics as follows: His work has certainly had a tangible effect on mathematics, inaugurating practices that were to proliferate in the decades that followed. These include the use of infinitary, set-theoretic language; the use of non-constructive reasoning; axiomatic and algebraic characterization of structures, and a focus on properties that can be expressed in terms of mappings between them; the use of particular algebraic structures, like modules, fields, ideals, and lattices; and uses of algebraic mainstays like equivalence relations and quotient structures. These innovations are so far-reaching that it is hard not to attribute them to a fundamentally different conception of what it means to do mathematics. [2006, p. 160] In articulating, adopting, and promoting this mathematical methodology, Dedekind was attempting to demonstrate the superiority of conceptual mathematics over computational, algorithm-based mathematics. In many passages, Dedekind expresses the view that a theory should not be based on particular representations of mathematical objects, on algorithmic reasoning, and on computation. This was not entirely new, since he is borrowing many methodological insights from Riemann, who was arguing a similar point against Weierstrass’s constructive methods in complex analysis. This is seen in the following forceful argument: My efforts in number theory have been directed towards basing the work not on arbitrary representations or expressions but on simple foundational concepts and thereby — although the comparison may sound a bit grandiose — to achieve in number theory something analogous to what Riemann achieved in function theory, in which connection I cannot suppress the passing remark that Riemann’s principles are not being adhered to in a significant way by most writers — for example, even in the newest works on elliptic functions. Almost always they mar the purity of the theory by unnecessarily bringing in forms of representation which should be results, not tools, of the theory. [Dedekind, 1876, pp. 468–469] The remark on elliptic functions is a reference to Weierstrass’s work. Indeed, Weierstrass does not focus on abstract mappings, but instead he systematically works on the basis of series representations of the functions he analyzes. For Dedekind, it is a methodological mistake to define mathematical objects by means of symbolic expressions and algorithmic procedures to act upon them (for example, series and formal series algebra), since there are typically multiple possible equivalent representations and the choice among them is arbitrary. That is, a function should be characterized in terms of its own properties, not in terms of the properties of its representations, despite the fact that such representations are typically needed to evaluate the function computationally. This is why Dedekind adopts the abstract, non-constructive, axiomatic characterization of structures as his main methodological principle, as do many modern mathematicians. To borrow Avigad’s words once more: ‘[…] in localizing and minimizing the role of representations and calculations, and making them secondary to structural systematization, Dedekind is exhibiting tendencies that have become the hallmark of modern mathematics’ [2006, p. 181]. Yet, Avigad does not fully embrace this transformation, claiming that ‘something important has been lost in the transition to modern mathematics’ (p. 162). Similarly, I will argue below that one of the philosophical lessons from the computer revolution in mathematics is that much is lost if we do not give both computational and constructive methods a central place. 3. EXPANDING THE ANALOGY BETWEEN CHESS AND MATHEMATICS We have seen evidence that one of two profound transformations has changed the conception of what it means to do mathematics. But is it really the case? The analogy with chess is often used to shed light on what mathematics is. This analogy is useful as it helps one to recognize essential elements of the nature, objects, and methods of mathematics that may be difficult to define exactly. For instance, some formalists are fond of using it to flesh out the view that mathematics is similar to a game in which we learn to manipulate some symbols according to a fixed set of rules, much as chess is. On the other hand, opponents of formalism are eager to point out the disanalogies between the two in order to undermine the formalist point of view. An early occurrence of this argument style was used by Frege [1885]. The analogy is most often used to identify what mathematics is; in what follows I will use the analogy to shed light on what it means to do mathematics well. The interesting element is that chess also has dimensions that are analogous to conceptual and computational mathematics, and moreover that there also were a conceptual and a computer revolution in the world of chess, both of which transformed the game in much the same way that, in my opinion, they transformed mathematics. The counterpart of computational mathematics is a style of play known as ‘tactical chess’. A tactical play in chess is one based on a careful and systematic survey of all the combinations of potentially forthcoming moves that a player has the ability to process. On that basis, the player will evaluate the alternatives and go for the move that seems to lead to the best continuation. Often, tactical players are spectacular attacking players who indulge in sacrificial play that leads to long, complicated, and typically forced variations. Many past world champions are known for their superior tactical play, such as Tal, Kasparov, Topalov, and J. Polgár. The counterpart of conceptual mathematics is a style of play known as ‘positional chess.’ There are also many past world champions known for their superior positional play, e.g. Capablanca, Botvinnik, and Karpov. A positional play in chess is typically not based on very detailed, extensive, and precise survey of possible variations. Indeed, strong positional players often only analyze the structural features of a chess position in order to determine what is the most favorable course of action. Perhaps an example will make clear what tactical and positional chess involves, and how they complement each other. Consider the famous position that arose in a game between the British grandmaster Nigel Short (with white) and Dutch grandmaster Jan Timman in 1991, Tilburg (see Figure 1). White stands better in this position for a number of positional reasons: white completely dominates the open d-file and the 7th rank, black has weaknesses on the dark squares in the king’s vicinity, white has more space, so that black has difficulty in maneuvering. In the eye of a strong player, these considerations suffice to hint that white has a decisive advantage. But how does white win? Bringing the knight to g5 to generate a direct attack must be ruled out, for black would deliver mate in one move by taking on g2. Moving the rook from d4 to f4 to add pressure on f7 is also out of the question, since that would leave the d7 rook unprotected. And pushing the pawn from g2 to g4 to destroy the black king’s pawn protection leaves the f3 knight hanging. Playing the queen to f4 (with the idea of bringing it to h6 and have a crushing attack) is rebuffed by playing the king to g7, in which case white still has much work to do to win the game. So, the direct tactical strikes do not appear to deliver a forced win. At the same time, black cannot do much, being tied to the defense of f7. One potential tactical resource is to play the bishop to c8 to attack the d7 rook. However, by playing this move, black relinquishes the pressure on the a8-h1 diagonal; so white can force a win by first playing the knight to g5 and then advancing the pawn to g4. We omit the details of this variation; the important point is that a tactical analysis reveals a forced win for white. Therefore, as black’s only tactical resource fails, he is reduced to a passive defense. Given this background analysis, the positional player indulges in a bit of wishful thinking: the only thing missing to win the game is to have something controlling g7, in order to play the queen to g7 and give checkmate. Then, the idea that no piece except for the white king can fulfil this role arises. From there, once you see the idea of moving the king to h2, you’ve already seen a clear path to victory, and you know that none of the details of the precise variations matter. The idea is simply to walk the king to h6 via g3-f4-g5 and to play the queen to g7, which is checkmate. Because it is focused on a general apprehension of the state of the game and because it lacks reference to particular variations, this would be a positional move akin to the ‘right’ solution in Halmos’s tennis example.5 Fig. 1. View largeDownload slide Short-Timman 1991. White to play and win. Fig. 1. View largeDownload slide Short-Timman 1991. White to play and win. The great players in each camp do not only play different styles, but importantly they also explain the justification for their moves very differently in books, articles, press conferences, etc. At the same time, sometimes a player does not have the choice: some positions demand a tactical analysis, while others demand a positional analysis. And most positions demand a very well crafted analysis that perfectly balances the two aspects. This is why all great players, whatever their preferred style is, are not great in virtue of mastering this one dimension only, but in virtue of mastering the interplay between the two styles of reasoning, by a sense of what sort of consideration dominates the situation at any given moment. Moreover, just as the computer transformed mathematics, it has transformed chess. Of course, the rules of chess have not been changed by the use of computers, just as the ‘rules of mathematics’ have not been changed by computers. In this important sense, the computer revolution has not changed what it means to play chess. What has been changed is the conception of what it involves to play chess well (just as the computer has changed the conception of what it involves to do mathematics well), as witnessed by transformed high-level tournament play. In their analysis, computers use heuristics rather than strict algorithms, heuristics that necessarily have blind spots, i.e., identifiable sets of situations in which they fail and in which the user of the computer must learn to discard the computer evaluation (as I will argue in the next section, the same is true in mathematics).6 The transformation can thus not just be put in terms of players emulating computers. The best way to characterize the contribution of computers to chess is not in having refuted styles, openings, or settled deep questions. Perhaps the most important contribution is that it has helped players to understand better how to integrate the positional and tactical components of the game in sound practical play. In a short article, former world champion Garry Kasparov said this much: There have been many unintended consequences, both positive and negative, of the rapid proliferation of powerful chess software. […] The heavy use of computer analysis has pushed the game itself in new directions. The machine doesn’t care about style or patterns or hundreds of years of established theory. […] It is entirely free of prejudice and doctrine and this has contributed to the development of players who are almost as free of dogma as the machines with which they train. Increasingly, a move isn’t good or bad because it looks that way or because it hasn’t been done that way before. It’s simply good if it works and bad if it doesn’t. [Kasparov, 2010] This is a clear sense in which computers have forced high-level tournament players to reinvent what it means to play chess well. In analogy, and to the extent that a similar computer revolution has occurred in mathematics, I want to argue that a similar transformation has brought mathematicians to recognize that, much of the time, doing mathematics well consists in integrating conceptual and computational mathematics to get good results in an expedient way. 4. COMPUTATION IN DISCRETE AND CONTINUOUS MATHEMATICS One limitation of the chess analogy stems from the fact that chess is in its nature a combinatorial game that is best understood with the tools of discrete mathematics. Computational mathematics differs very much depending on whether one focuses on discrete or continuous mathematics.7 To be sure, in all three cases (i.e., chess, discrete mathematics, and continuous mathematics), results are typically not calculable in a way that leads to an exact demonstration of the solution, partly due to complexity; (see, e.g., [Fillion and Bangu, 2015]). This is not the sense in which computer calculations bring knowledge and understanding. To see the sense in which they bring knowledge and understanding, it is important to take into account the difference between two traditions in computational mathematics. The computation required by various mathematical problems is normally determined by an algorithm, i.e., by a procedure performing a sequence of primitive operations leading to a solution in a finite number of steps, if possible at all. Computation theory is a mathematical reflection on properties of algorithms such as complexity, accuracy, convergence rate, etc. In the context of the computational models used in science, an additional layer of methodological difficulties comes from the fact that computation involves data error and computational error. As a result, the reflection involves a number of subtle concepts of computation, rather than a single one. With a precise model of computation at hand, we can refine our views on what is computationally achievable, and if it turns out to be, what resources are required. The classical model of computation used in most textbooks on logic, computability, and algorithm analysis stems from metamathematical problems addressed in the 1930s; specifically, while trying to solve Hilbert’s Entscheidungsproblem, Turing developed a model of primitive mathematical operations that could be performed by some type of machine affording finite but unlimited time and memory. This model, which turned out to be equivalent to other models developed independently by Gödel, Church, Kleene, Post, and others (see, e.g., [Soare, 1999]), resulted in a notion of computation based on effective computability. From there, we can form an idea of what is ‘truly feasible’ by further adding constraints on time and memory for given implementations on digital computers. Nonetheless, computational practice in continuous mathematics requires an alternative complementary notion of computation, because its objectives and methods are quite different from those of metamathematics. A first important difference is that the Turing model of computation ‘is fundamentally inadequate for providing such a foundation for modern scientific computation, in which most algorithms — with origins in Newton, Euler, Gauss, et al. — are real number algorithms’ [Blum, 2004, p. 1024]. Given this limitation, Blum et al. [1998] generalize the ideas found in the classical model to include operations on elements of arbitrary rings and fields. But the difference goes even deeper: [R]ounding errors and instability are important, and numerical analysts will always be experts in the subjects and at pains to ensure that the unwary are not tripped up by them. But our central mission is to compute quantities that are typically uncomputable, from an analytic point of view, and to do it with lightning speed. [Trefethen, 1992] Even with an improved picture of effective computability, it remains that the concept that matters for a large part of applied mathematics (including engineering) is the different idea of mathematical tractability, understood in a context where there is error in the data, error in computation, and where approximate answers can be entirely satisfactory [Fillion and Corless, 2014; Fillion, 2017; Fillion and Moir, 2018]. This second notion of computability addresses the proper computational difficulties posed by the application of mathematics to the solution of practical problems from the outset. Certainly, both pure and applied mathematics heavily use the concepts of real and complex analysis. From real analysis, we know that every real number can be represented by a nonterminating fraction: \begin{align*} x = \lfloor x\rfloor.d_1d_2d_3d_4d_5d_6d_7\cdots \end{align*} However, in contexts involving applications, only a finite number of digits is ever dealt with. For instance, in order to compute |$\sqrt{2}$|⁠, one could use an iterative method (e.g., Newton’s method) in which the number of accurate digits in the expansion will depend upon the number of iterations. But, since within each iteration (or each term) only finite-precision numbers and arithmetic operations are being used, one must also consider this additional source of error. We will find the same situation in numerical linear algebra, interpolation, integration, differentiation, etc. A similar situation would also obtain if we used the first few terms of a perturbation series expansion for the evaluation of a function or for the solution of a differential equation. Those error components must be analyzed, bounded, and measured in order for one to be in a position to claim that the computer has extracted valuable information from the equations we input. Consider a simple example that illustrates how this assessment is based on the interplay between computational and conceptual aspects of the solution process. Suppose a model is characterized by the following equation: \begin{align*} \dot{x}=x^2-t,\qquad x_0=x(0)=-{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\label{stiffex1}. \end{align*} This is a differential equation of the state |$x$| with respect to time, and it also contains an initial condition. Together, they determine what will happen in the system, but no elementary expression can be given for the solution. Fortunately, we can still examine the solution using numerical methods. What would happen if the initial condition were instead |$0$|⁠, |$-1$|⁠, or |$-3$|? What if the same initial conditions were not given at |$t=0$| but rather at |$t=3$|⁠, |$t=5$|⁠, or |$t=7$|? As it turns out, it would essentially change nothing, except for a very short initial time interval: the trajectories all rapidly converge to the same curve (see Figure 2a). As a result, we can claim that the solution is insensitive to perturbations of the parameter |$x_0$|⁠. Moreover, we are in a position to claim that ‘solutions’ obtained by brute force — i.e., by approximate numerical computation — will be informative in virtue of this local structural property. In technical terms, we say that in such a case small computational error is damped away. In this important sense, we can then consider all systems (with |$x_0<\sqrt[3]{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}$|⁠) equivalent.8 Fig. 2. View largeDownload slide Solutions |$x(t)$| of |$\dot{x}=x^2-t$| for various initial conditions |$x_0$|⁠. (a) Solutions |$x(t)$| for |$x_0\leq0$|⁠; the case |$x_0=-{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$| is in bold. (b) Qualitative change in the solution at |$x_0=\sqrt[3]{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}$| (dotted line). Fig. 2. View largeDownload slide Solutions |$x(t)$| of |$\dot{x}=x^2-t$| for various initial conditions |$x_0$|⁠. (a) Solutions |$x(t)$| for |$x_0\leq0$|⁠; the case |$x_0=-{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$| is in bold. (b) Qualitative change in the solution at |$x_0=\sqrt[3]{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}$| (dotted line). However, for |$x_0\geq\sqrt[3]{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}$|⁠, this local structural justification fails as the systems diverge away from each other, as shown in Figure 2b. At this critical point, there is a qualitative change taking place in the behaviour of the system, technically known as a ‘bifurcation’. Moreover, note that, if we set a value |$x_0$| less than, but close to the critical value, a perturbation of the system could easily push the system on the other side of the bifurcation line. Hence, great care would need to be taken in this region and we would expect brute-force computation to be unreliable in this circumstance. Except for equations found in very idealized or simplified models, it will typically be the case that no easy access to the solutions and the bifurcation regions is known, in which case explicit (approximate) computation will be carried out to investigate both the solution and the general qualitative features needed to warrant a judgement to the effect that the information thus obtained can be relied upon. The merging of computational and conceptual methods from continuous mathematics has also produced impressive results in the field of computer-assisted proofs. As Hilbert [1902] had done at the turn of the twentieth century, Smale [1998] presented a list of eighteen problems for the twenty-first century. The first problem to be solved — and only two of them have been solved so far — was the fourteenth problem. The context of the problem originates in 1963, when Lorenz observed peculiar behaviour in the solutions of a dynamical system — solutions that he obtained with standard numerical methods. As he observed, the system contains an unstable equilibrium point above the origin and, after an initial transient period, the simulation shows solutions to be chaotically dispersed within a bounded region surrounding this unstable equilibrium point, namely, the two visually identifiable ‘wings’ in Figure 3. Such a region is known as a strange attractor. Over the next four decades, many important developments in dynamical systems theory were stimulated by a desire to understand this behaviour mathematically. But a key problem remained: is there actually a strange attractor in the Lorenz system? Computer simulations robustly suggested this to be the case; but simulations use inexact computation, and thus the possibility of its being a deceiving numerical effect could not be excluded. Proving this claim was Smale’s fourteenth problem; as Viana [2000] put it, ‘many of us felt that answering this question […] was a great challenge and a matter of honor for mathematicians’. Tucker [1999] cracked the problem by ingeniously combining computational and conceptual methods. On the one hand, he used numerical methods to calculate an approximate solution. Instead of the usual floating-point calculations, however, he implemented a form of interval arithmetic that guarantees the solution to be within certain bounds of what was approximately calculated. This form of computation, which would be overly inefficient for day-to-day applications, made generating computer-assisted proofs possible. But even with validated numerics, a serious obtacle remained. Near the origin, calculating trajectories using this method would lead to very large error intervals, so much so that the desired conclusions could not be reached. This is where the second part of the method comes in. Near the origin, Tucker instructed the computer to suspend the numerical integration, and instead to use a specific formula. This formula is a simple expression whose sufficient accuracy is asymptotically guaranteed near the origin, in virtue of normal form theory (for the details see [Viana, 2000]). This part of the method, essential to obtaining the desired proof, belongs to conceptual mathematics. As Stewart [2000] put it, ‘numerics can be used with precise error estimates to establish significant features of the flow of a nonlinear differential equation. When these features are combined with appropriate conceptual insights, the existence of chaotic attractors becomes irrefutable.’ Fig. 3. View largeDownload slide Numerical solution of the Lorenz system. Fig. 3. View largeDownload slide Numerical solution of the Lorenz system. As we see, in continuous mathematics, the view of the computer as a number-cruncher is clearly seen to be wrong — a thing that would not be as striking in discrete mathematics. This supports the view that the development of methods of analysis that integrate both brute-force computation and general, higher-level conceptual considerations has been one of the key factors that has made it possible for the computer to transform the way in which we now do mathematics. 5. CONCLUSION As mathematics undergoes transformations, it adds new objects, methods, subdisciplines, and practices to its repertoire. Of course, as a result, philosophy of mathematics is transformed too, but this fact does not necessarily mean that a new philosophy of mathematics is needed to understand the change, at least if this is interpreted as requiring that we replace our views on ontology, methodology, semantics, logic, etc. Instead, it can require only that we refine already developed problems and ideas. There is no question that the developments initiated in the nineteenth century bolstered the conceptual aspect of mathematics — and dramatically improved it. Yet, claims to the effect that this style of mathematics is unqualifiedly superior or that modern mathematics is essentially a conceptual affair are unwarranted. This is clear from the fact that there was another revolution of no lesser importance, the computer revolution, which reasserted the computational aspect of mathematics. Yet, one could argue that the transformation of mathematical practice in the computer age is not, strictly speaking, a revolution. Following Kuhn, we would expect revolutions in science and mathematics to follow the pattern anomaly-crisis-revolution, with the revolution culminating with the overthrow of a paradigm. Nothing suggests that the way in which the computer transformed mathematical practice instantiates this pattern. Dauben [1992] claims that many other transformations fail to instantiate this pattern neatly, but he nonetheless thinks them worthy of the name ‘revolution’, e.g., the discovery of incommensurable magnitudes, the emergence of set theory, and the introduction of non-standard analysis, to name a few. Reflecting upon the broadening of the concept of revolution in history of mathematics, Mehrtens [1992, p. 43] comes to the conclusion that ‘revolutions are not what they used to be’. Based on the analogy with political revolutions, Gillies argues that whereas some revolutions irrevocably overthrow a regime (the Russian model), often the previous regime persists but experiences a considerable loss of importance (the Franco-British model). He then argues that ‘revolutions do occur [in mathematics] but they are always Franco-British’ [Gillies, 1992b, p. 6]. Perhaps this approach could account for features of the computer revolution. Another possibility would be instead to draw an analogy with Quebec’s Quiet Revolution of the 1960s. No official political regime was overthrown or experienced lost importance in the Quiet Revolution. It was a renegotiation of how things were to work within given institutions, in order to make society work better for people and get more from those institutions. This is roughly the sense in which one could say that the computer revolutionized mathematics. As we saw, the analogy with chess, often used in philosophy of mathematics, is again helpful in understanding what changed in mathematics. The development of the principles of positional chess in the nineteenth century has not supplanted the importance of tactical chess, and the computer revolution has not improved play as a result of brute-force tactical analysis. In an important sense, this transformation of chess theory has not changed what it means to play chess simpliciter, but rather what it means to play chess well. I suggested that the same point of view should be adopted with respect to the two transformational periods that have shaped modern mathematics. The most significant gain that resulted from their respective revolutions is that, to an unprecedented degree, we now rely on versatile practices that integrate aspects from both complementary styles of reasoning. I further suggested that the integration of computational and conceptual pieces of mathematics is best seen in the domain of scientific computing directed at problems we find in continuous mathematics, be they pure or applied in character. This is because such problems are typically complex enough that we have no choice but to rely on the assistance of computers; yet our assessment of the quality of the information generated by computers requires that we appreciate fundamentally conceptual aspects of the problems. As such, it is a paradigmatic case of what it now means to do mathematics well, in a way that leads to results that are both sound and useful. Footnotes †I would like to thank a number of colleagues for their valuable input, including Vincent Ardourel, Sorin Bangu, Anouk Barberousse, Jim Brown, Johannes Lenhard, Robert Moir, Derek Postnikoff, Adrian Yee, and two anonymous reviewers. This research was supported by SSHRC Insight Grant #435-2018-0242. 1However, some seem to disagree. In reference to these developments, Tappenden [2006] claims the following: ‘There was a revolution in mathematics in the nineteenth century, and philosophers have, for the most part, failed to notice.’ 2They do acknowledge some exceptions, such as the Duhem problem in computer simulations. See [Lenhard and Winsberg, 2010] and [Jebeile and Barberousse, 2016]. 3I would like to thank Jim Brown for suggesting it. 4K. Weierstrass. Ausgewählte Kapitel aus der Funktionenlehre. Vorlesung, gehalten in Berlin. Mit der akademischen Antrittsrede, Berlin 1857 und drei weiteren Originalarbeiten von K. Weierstrass aus den Jahren 1870 bis 1880/6, edited with commentary by R. Siegmund-Schultze, p. 156. Teubner–Archiv zur Mathematik; 9. Liepzig: Teubner, 1988. Cited in [Tappenden, 2006, p. 111]. 5I want to thank an anonymous reviewer for helping me to make this example more compelling. 6I have checked how two strong chess engines (Komodo 9.2 and Stockfish 7) perform in the Short-Timman position mentioned above. Not so surprisingly, they fail miserably to find the correct continuation. They get lost in all the permutations of moves that make no difference and fail to appreciate the strength of the second move g3. My computer with a 3.5Ghz Xeon processor was looking at an average of about 2 million positions (technically, nodes) per second (with 4Gb hashtable size), and still failed to find the solution after many hours (after looking at more than 16.7 billion positions), with a depth above 24 moves. However, if we just enter the moves Kh2 and Kg3 for the computer, it finds the correct evaluations within milliseconds., and with only Kh2, it still takes a while. A reviewer has pointed that, interestingly, Komodo 12.1 uses extra context from ‘killer moves’ to obtain a better evaluation faster. 7To the extent that the impact of the computer on mathematics has been discussed philosophically, focus tends toward discrete mathematics. The observations made by Knuth [1974], for example, are all about discrete mathematics. Another well-known case is the computer-assisted proof of the four-colour theorem [Tymoczko, 1979], which also falls in this category. In contrast, I will emphasize the case of computation in continuous mathematics, since it clearly reveals the limitations of the view that the contribution of computers is merely ‘brute force’. 8Technically, I am referring to the notion of topological equivalence of curves. 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TI - Conceptual and Computational Mathematics
JF - Philosophia Mathematica
DO - 10.1093/philmat/nkz005
DA - 2019-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/conceptual-and-computational-mathematics-uIiDun2EVy
SP - 199
VL - 27
IS - 2
DP - DeepDyve
ER -