Is There Beauty in Mathematical Theories

Mathematics and Beauty

Iwas pleased to receive an invitation to speak at a gathering largely of philosophers and accepted with alacrity but once having accepted, once having seen the program, I was beset by doubts. As I proceeded, it became ever clearer that the doubts were entirely appropriate. As an elderly mathematician, I am fixed more and more on those problems that have preoccupied me the most over the decades. I cannot, even with the best of intentions, escape them. That will be clear to every reader of this essay and of a second related, but yet unwritten, essay with which I hope to accompany it later.

I left the title suggested to me by the organizer of the symposium, Professor Hösle, pretty much as it was, Is there beauty in purely intellectual entities such as mathematical theories? I abbreviated it, for as a title it was too long, but did not otherwise change it. I do not, however, address directly the question asked. The word beauty is too difficult for me, as is the word aesthetics. I hope, however, that the attentive reader of this essay will, having read it, be able, on his own, to assess soberly the uses and abuses of the first of these words in connection with mathematics.

I appreciate, as do many, that there is bad architecture, good architecture and great architecture just as there is bad, good, and great music or bad, good and great literature but neither my education, nor my experience nor, above all, my innate abilities allow me to distinguish with any certainty one from the other. Besides the boundaries are fluid and uncertain. With mathematics, my topic in this lecture, the world at large is less aware of these distinctions and, even among mathematicians, there are widely different perceptions of the merits of this or that achievement, this or that contribution. On the other hand, I myself have a confidence in my own views with regard to mathematics that does not spring, at least not in the first instance, from external encouragement or from any community of perception, but from decades of experience and a spontaneous, unmediated response to the material itself. My views as a whole do not seem to be universally shared by my fellow mathematicians, but they are, I believe, shared to a greater or lesser extent by many. They are at worst only a little eccentric and certainly not those of a renegade. I leapt at the unexpected possibility to reflect on them, in what I hope is an adequately critical manner, and to attempt to present them in a way persuasive to mathematicians and to others.

I confess, however, that although aware that I agreed—in good faith—to speak not on mathematics as such, but on beauty and mathematics, I soon came to appreciate that it is the possibility of greatness, or perhaps better majesty and endurance, in mathematics more than the possibility of beauty that needs to be stressed, although greatness is unthinkable without that quality that practitioners refer to as beauty, both among themselves and when advertising the subject, although not infrequently with a sincerity that, if not suspect, is often unreflected. This beauty, in so far as it is not just the delight in simple, elegant solutions, whether elementary or not, although this is a genuine pleasure certainly not to be scorned, is, like precious metals, often surrounded by a gangue that is, if not worthless, neither elegant nor intrinsically appealing. Initially perhaps there is no gangue, not even problems, perhaps just a natural, evolutionary conditioned delight in elementary arithmetic—the manipulation of numbers, even of small numbers—or in basic geometric shapes—triangles, rectangles, regular polygons. To what extent it is possible to become a serious mathematician without, at least initially, an appreciation of these simple joys is not clear to me. They can of course be forgotten and replaced by higher concerns. That is often necessary but not always wise. Simple joys will not be the issue in this essay, although I myself am still attached to them. As an aside, I add that, as with other simple human capacities, such as locomotion, the replacement of innate or acquired mathematical skills by machines is not an unmixed blessing.

There is also a question to what extent it is possible to appreciate serious mathematics without understanding much of mathematics itself. Although there are no prerequisites for reading this note, it is not my intention to avoid genuine mathematics, but I try to offer it only in homeopathic doses.

Two striking qualities of mathematical concepts regarded as central are that they are simultaneously pregnant with possibilities for their own development and, so far as we can judge from a history of two and a half millenia, of permanent validity. In comparison with biology, above all with the theory of evolution, a fusion of biology and history, or with physics and its two enigmas, quantum theory and relativity theory, mathematics contributes only modestly to the intellectual architecture of mankind, but its central contributions have been lasting, one does not supersede another, it enlarges it. What I want to do is to examine, with this in mind, the history of mathematics—or at least of two domains, taken in a broad sense, with which I have struggled—as a chapter in the history of ideas, observing the rise of specific concepts in the classical period, taking what I need from standard texts, Plato, Euclid, Archimedes, and Apollonius, then passing to the early modern period, Descartes and Fermat, Newton and Leibniz, and continuing into the nineteenth, twentieth, and even twenty-first centuries.¹ It is appropriate, even necessary, to recount the history of the two domains separately. One can conveniently be referred to as algebra and number theory; the other as analysis and probability. Their histories are not disjoint, but their current situations are quite different, as are my relations to the two. The first is, I fear, of more limited general interest although central to modern pure mathematics. Whether it will remain so or whether a good deal of modern pure mathematics, perhaps the deepest part, has become too abstruse and difficult of access to endure is a question that will be difficult to ignore as we continue these reflections. Even if mankind survives, there may be limits to our ability or, more likely, to our desire to pursue the mathematical reflections of the past two or three thousand years.

As with other simple human capacities, the replacement of innate or acquired mathematical skills by machines is not an unmixed blessing

I have spent more time, more successfully, with the first domain but am nevertheless even there if not a renegade an interloper and regarded as such. In the second, I am not regarded as an interloper, simply as inconsequential. Nevertheless, if for no other reason than for the sake of my credibility as an impartial commentator on mathematics and its nature, I shall try to express clearly and concretely my conviction that the problems of both are equally difficult and my hope that their solutions equally rich in consequences. Unfortunately, for reasons of time and space the description of my concerns in the second domain must be postponed to another occasion.

Theorems and Theories

The difference between theorems and theories is more apparent in the first domain than in the second. Any mathematical theory will contain valid general statements that can be established rigorously accordingly to the commonly accepted logical principles, thus proved, and without which the theory would lack its structural coherence. They are referred to as theorems, but there is another sense to this word that refers to those statements that, independently of their necessity to the theory are provable within it, and, above all, of an importance that transcends their structural value within the theory. Fermat’s theorem, to which we shall return later, is a striking instance and belongs to the first domain and to pure mathematics. Without theorems of this sort, a theory in pure mathematics would be an insipid intellectual exercise. On the other hand, theorems of this sort are seldom, perhaps never, established without the concepts that are only available within a theory. It is the latter, as should appear from the following observations, that give pure mathematics its honorable place in intellectual history. In analysis, familiar in the form of differential and integral calculus, the test of a theory is less this type of theorem, than its pertinence outside of mathematics itself.

The more I reflected on this lecture, the more I was aware that in presuming to speak, above all to a lay audience, on the nature of mathematics and its appeal, whether to mathematicians themselves or to others, I was overreaching myself. There were too many questions to which I had no answer. Although I hope, ultimately, to offer appreciations of two quite different mathematical goals, I do not share the assurance of the great nineteenth-century mathematician C.G.J. Jacobi of their equal claims. In a letter to A.-M. Legendre of 1830, which I came across while preparing this lecture, Jacobi famously wrote,

Il est vrai que M. Fourier avait l’opinion que le but principal des mathématiques était l’utilité publique et l’explication des phénomènes naturels; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c’est l’honneur de l’esprit humain, et que sous ce titre, une question des nombres vaut autant qu’une question du système du monde.¹

I am not sure it is so easy. I have given a great deal of my life to matters closely related to the theory of numbers, but the honor of the human spirit is, perhaps, too doubtful and too suspect a notion to serve as vindication. The mathematics that Jacobi undoubtedly had in mind when writing the letter, the division of elliptic integrals, remains, nevertheless, to this day unsurpassed in its intrinsic beauty and in its intellectual influence, although in its details unfamiliar to a large proportion of mathematicians. Moreover, the appeal to the common welfare as a goal of mathematics is, if not then at least now, often abusive. So it is not easy to find an apology for a life in mathematics.

In Jules Romains’s extremely long novel of some 4750 pages Les hommes de bonne volonté [The Men of Goodwill] a cardinal speaks of the Church as “une œuvre divine faite par des hommes” [“a divine work done by men”] and continues “La continuit’e vient de ce qu’elle est une œuvre divine. Les vicissitudes viennent de ce qu’elle est faite par les hommes.” [“Continuity comes from it being divine work. The vicissitudes come from it being done by men.”] There are excellent mathematicians who are persuaded that mathematics too is divine, in the sense that its beauties are the work of God, although they can only be discovered by men. That also seems to me too easy, but I do not have an alternative view to offer. Certainly it is the work of men, so that it is has many flaws and many deficiencies.²

Humans are of course only animals, with an animal’s failings, but more dangerous to themselves and the world. Nevertheless they have also created—and destroyed—a great deal of beauty, some small, some large, some immediate, some of enormous complexity and fully accessible to no-one. It, even in the form of pure mathematics, partakes a little of our very essence, namely its existence is, like ours, like that of the universe, in the end inexplicable. It is also very difficult even to understand what, in its higher reaches, mathematics is, and even more difficult to communicate this understanding, in part because it often comes in the form of intimations, a word that suggests that mathematics, and not only its basic concepts, exists independently of us. This is a notion that is hard to credit, but hard for a professional mathematician to do without. Divine or human, mathematics is not complete and, whether for lack of time or because it is, by its very nature, inexhaustible, may never be.

I, myself, came to mathematics, neither very late nor very early. I am not an autodidact, nor did I benefit from a particularly good education in mathematics. I learned, as I became a mathematician, too many of the wrong things and too few of the right things. Only slowly and inadequately, over the years, have I understood, in any meaningful sense, what the penetrating insights of the past were. Even less frequently have I discovered anything serious on my own. Although I certainly have reflected often, and with all the resources at my disposal, on the possibilities for the future, I am still full of uncertainties. These confessions made to the innocent reader, I am almost ready to begin. Before doing so, I would like to describe a theorem with a sketch of its proof, because it may be the most striking illustration available of the relation between theorems and theories that I have in mind.

Fermat’s Theorem

It is easy to solve the equation \(a^2+b^2=c^2\) in integers, none of which is \(0\). One writes it as \(a^2=c^2-b^2=(c+b)(c-b)\) and takes \(c+b=r^2\), \(c-b=s^2\), \(a=rs\). On the other hand, for \(n>2\) there is no solution of \(a^n+b^n=c^n\) with none of \(a\), \(b\), \(c\) equal to \(0\). This is the famous theorem of Fermat. It is, of course, not my intention to provide a proof here, nor even to provide a history of the proof, of which there are a number of accounts, many reliable. I just want to point out certain elements. If there is a solution for \(n\) equal to a prime \(p\), and this is the decisive case, then one considers, after Gerhard Frey and Yves Hellegouarch, the curve in the \((x, y)\) plane defined by \(y^2 = x(x − a^p )(x + b^p)\). The form of this curve is special: quadratic in \(y\) and cubic in \(x\), what is called an elliptic curve. Moreover the coefficients are rational numbers. There are many conjectures of greater or less precision—rather greater than less and undoubtedly correct, although we are a long way from constructing the theory they entail and that entails them—establishing a correspondence between equations with rational coefficients and what are called automorphic forms, a notion to which we shall return. For elliptic curves the correspondence takes a particularly simple form whose meaning is clear to a sizeable fraction of mathematicians, and was formulated somewhat earlier than the general form, although its proof demands, among other things, an appeal to consequences of the little of the general theory that is available. In general and in this case as well, they affirm that associated to any equation with rational coefficients are one or several automorphic forms. It is surprising because these are apparently much more sophisticated objects. By and large automorphic forms are none the less easier to enumerate than curves. In particular, those automorphic forms with small ramification, a basic technical concept, that might correspond to elliptic curves can be counted and there are not enough of them to account for the Frey–Hellegouarch curve. So it cannot exist and there is no solution of Fermat’s equation for \(n > 2\). Fermat’s theorem is, and probably will remain, the quintessential example of an apparently elementary theorem that is a consequence of a very sophisticated, very demanding theory.

I have included in this article, as an aid in the explanation of the historical evolution of the context in which this proof is effected, two diagrams (Diagrams A and B). In the first (Diagram A), a large number of mathematical theories and concepts are given, within rectangular frames connected by lines in an attempt to express the relations between them. The historical development proceeds from top to bottom. In the hope that it will provide some temporal orientation to the reader unfamiliar with the concepts themselves, there is a second diagram (Diagram B) with the names of some of the better-known creators of the concepts. It is not otherwise to be taken seriously. Not all names will be equally familiar; they are not equally important.

The labels in the elliptical frames stand apart. There are four of them, in two columns and two rows. The columns refer to the theories at their head: contemporary algebraic geometry and diophantine equations on the left and automorphic forms on the right. The first row corresponds to central concepts in these two theories, motives on the left, functoriality on the right. Neither of these words will evoke anything mathematical at first. Both are, in the sense in which they are used in the diagram, problematical. It is clear, at least to a few people, what is sought with these two concepts. It is also clear that at the moment they are largely conjectural notions. Both express something essential about the structure of the theories to which they are attached. This structure is expressed not so much in terms of what mathematicians call groups but in terms of their representations, which form what mathematicians call a Tannakian structure. That is a somewhat elusive concept and I will make no attempt to formulate it here. Its mystery is heightened as much as explained by an appeal to a laconic observation of Hermann Weyl in the preface to his celebrated exposition Gruppentheorie und Quantenmechanik published in 1928: “Alle Quantenzahlen sind Kennzeichen von Gruppendarstellungen” [“All quantum numbers are characters of group representations”].

The variety of wavelengths emitted by various atoms and molecules was in the early years of the last century completely inexplicable and a central puzzle for the physicists of the time. It was explained only in the context of the new quantum theory and, ultimately, with the notion of a group representations and of the product of two representations, thus pretty much in what mathematicians refer to as a Tannakian context. This was a magnificent achievement and Weyl is referring to it. Although automorphic forms or motives are mathematical notions, not physical phenomena, and thus, especially automorphic forms, much closer, even in their definition, to groups than the spectra of atoms and molecules are to the rotation group and its representations, the clarifying connection between the ellipses on the left and those on the right is every bit as difficult to discover: between diophantine geometry or motives and automorphic forms or functoriality; and between \(\ell\)-adic represenations and Hecke operators. We are hoping not merely to elucidate the two concepts of motives and functoriality in the two theories to which they are here attached, diophantine equations and automorphic forms, but to create them. Although there is little doubt, at least for me, that the proposed conjectures are valid, it is certain that a great deal of effort, imagination, and courage will be needed to establish them.

Nevertheless some of them, the very easiest ones, have been proved. These were adequate to the proof of Fermat’s theorem. Rather, some few examples of functoriality, which had implications not only for the column on the right but also for the correspondence between the two columns, were available to Andrew Wiles, who recognized that with them another much more difficult conjecture, the Shimura–Taniyama conjecture, again a part of the correspondence but more difficult, was within his reach. This great achievement represents a second stage, not simply the creation of the two theories in the two columns, but the proof, or at least a partial proof, that the one on the left is reflected at the level of algebraic curves in the one on the right. This is the essence of the proof of Fermat’s theorem, contemporary in its spirit. If the theorem were false, it would imply the existence of an elliptic curve, a motive, thus a diophantine object attached to the left-hand column. Since the diophantine theory is reflected in the theory of automorphic forms, there would be in the right-hand column an object attached to it, a corresponding automorphic form that, thanks to the nature of the correspondence would have some very specific properties, properties that permit us to conclude that it could not possibly exist. So Wiles’s glory is perhaps not so much to have established Fermat’s theorem as to have established the Shimura–Taniyama conjecture.

Indeed, if it had never been conjectured that Fermat’s equations had no nontrivial integral solutions and if mathematicians had arrived by some other route at the Shimura–Taniyama conjecture and its proof—not in my view entirely out of the question although some essential steps in Diagram A were clearly inspired by the search for a proof of Fermat’s theorem—and if then some modest mathematician had noticed that, as a consequence, the equation \(a^n +b^n = c^n\), \(n > 2\) had no integral solutions except the obvious ones with \(abc = 0\), little attention would have been paid to him or to the equation. It might be inferred from this observation that the relation between the intrinsic value of a mathematical notion and the depth of its historical roots is subtler than I suggested when beginning this essay. I find none the less that is of great value as a mathematician to keep the historical development in mind.

The last row offers for completeness the parameters that are presently most commonly used for comparing the two columns, or rather for deciding whether any proposed correspondence of the objects in the two columns is correct. The existence of the \(\ell\)-adic representations is an acquisition of the last few decades and, like the fundamental theorem of algebra, which will be formulated below, expresses something fundamental about the relation between geometry and irrationality.

Two striking qualities of mathematical concepts are that they are simultaneously pregnant with possibilities for their own development, and are of permanent validity

When struggling with the difficulties, and they are many, that must be overcome in any attempt to establish the correspondence here adumbrated, I am troubled by a question of a quite different nature. Certainly, for historical reasons if for no other, any mathematical theory that yields Fermat’s theorem has to be taken seriously. Fermat’s theorem now established, what is the value of developing the theory further? What will the correspondence give? This is by no means clear. My guess is that it will be difficult to create the theory entailing the correspondence of the two columns without establishing a good part of what is known as the Hodge conjecture, perhaps the outstanding unsolved problem in higher-dimensional coordinate geometry. It has remained inacessible since its first—slightly incorrect, but modified later—formulation in 1950. I see, however, no reason that the theory could not be established without coming any closer to the Riemann hypothesis, on which a great deal of effort has been expended over the past 150 years, but the theory does suggest a much more general form of the hypothesis and puts it in a much larger context. A different and very hard question is what serious, concrete number-theoretical results does the theory suggest or, if not suggest, at least entail. A few mathematicians have begun, implicitly or explicitly, to reflect on this. I have not.

Pure Mathematics

From Then to Now

The proof of Fermat’s theorem is a cogent illustration of the concrete consequences of the correspondence symbolized by the first pair of ellipses in Diagram A. The diagram itself and the explanations, partial and unsystematic, in the following paragraphs are an attempt, more for my own sake than for that of the reader, to integrate this correspondence—the central theme of this essay, devoted to pure mathematics and, God willing, the first of two essays—with various insights or contributions over the centuries that would be regarded by all mathematicians as decisive and to demonstrate that it is the legitimate issue of these contributions. It would take more than a lifetime to appreciate them all, but each and every one offers, each time that it is taken up and its content and implications just a little better understood, a great deal of immediate pleasure and satisfaction. In my life as a mathematician I have spent very little, too little time reflecting on the past of mathematics, and this lecture is an occasion to remedy the neglect and redeem myself.

There are two major themes arising in antiquity and accumulating mass and structure over the centuries that contribute to the correspondence symbolized by the two ellipses: geometry, of which Apollonius is a major classical representative, for our purposes more telling than Euclid; and arithmetic—a word often used among mathematicians as a substitute for the phrase “theory of numbers”, which is more embracing—for which I have chosen in the restricted space of the diagram Theaetetus and Euclid as representatives, rather than Pythagoras. Although Book X of the Elements is readily available in well-edited English translations and contains much of which, so far as I know, Theaetetus was unaware, his name is likely to have more sentimental appeal to philosophers, because of his modest but poignant appearance in the Platonic dialogue that bears his name, first briefly and off-stage, as a soldier wounded and ill with dysentery, and then, in the recollected dialogue, as a very young man and a foil for the questions of Socrates, although he is allowed some mathematical interjections. The work of Apollonius on conic sections, in which the equations of ellipses, hyperbolas, and parabolas familiar to most of us appear, although without symbols and without Cartesian coordinates, are, with the editorial help needed for penetrating the unfamiliar style of exposition, a pleasure to read even in translation. They have been published in several languages. In particular, they were made available in Latin in 1566. Descartes had, apparently read and understood the first parts, much to his profit and much to ours.

A question about numbers is as valuable as a question about the system of the world

The first column in both diagrams refers to the development of Cartesian geometry, based on Descartes interpretation of Apollonius, from its introduction in Sur la géeométrie to the present. The central feature of Cartesian geometry is the use of coordinates and it is preferable, for many reasons, to refer to it as coordinate geometry, but there are also good reasons for recalling the name of Descartes, whose contribution to mathematics was of enduring value, although as a mathematician he was brash and not always frank in acknowledgement of his debt to Apollonius. The use of coordinates ultimately allowed the introduction of what are called the projective line, the projective plane, or higher dimensional projective spaces, whose purpose is, one might say, largely to avoid the constant introduction of special cases, when some of the solutions of algebraic equations might otherwise slip off to infinity. Even these solutions are captured in projective geometry.

This enables us to exploit systematically the consequences of the fundamental theorem of algebra. This theorem asserts in its contemporary form that every equation of the form

\[\begin{equation}
X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0=0.\tag{1.1}
\end{equation}\]
with complex numbers \(a_{n-1},…, a_{1}, a_{0}\) as coefficients has exactly \(n\) roots, or, better expressed, because some of the roots may be multiple, that the left side factors as
\[\begin{equation}
X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0 \ =(X-\alpha_1)(X-\alpha_2)… (X-\alpha_n).\tag{1.2}
\end{equation}\]

The most familiar example is

\[\begin{align*}
X^2+bX+a=0,
\end{align*}\]

for which \(\alpha_{1}=(-b+\sqrt{b^{2}-4a}/2), \alpha_{2}=(-b-\sqrt{b^{2}-4a}/2)\). The adjective “complex”, whose meaning and consequences may perhaps not be familiar to every reader, allows \(b^2 −4a\) to be negative. The modern facility with the notion of a complex number was only slowly acquired, over the course of the eighteenth and early nineteenth centuries.

The origins of the theorem lie, however, not in algebra as such or in geometry, but in the integral calculus and in formulas that will be familiar to a good many readers, those who have encountered the formulas,

\[
\begin{align*}
%\begin{split}
\int\frac{1}{x-a}dx&=\ln(x-a),\\
\int\frac{1}{(x-a)^n}dx&=-\frac{1}{n-1}\frac{1}{(x-a)^{n-1}}, \quad n>1,\\
\int\frac{1}{x^2-a}dx&= \frac{1}{2\sqrt{a}}\ln\left(\frac{x-\sqrt{a}}{x+\sqrt{a}}\right),\\
\int\frac{1}{\sqrt{x^2-a}}dx&=\ln (x+\sqrt{x^2-a}) \tag{1.3} \\&=\,\text{arccosh}\,\left(\frac{x}{\sqrt{a}}\right) +\text{const.},\\
\int\frac{1}{\sqrt{x^2+a}}dx&=\ln (x+\sqrt{x^2+a})\\&=\,\text{arccos}\,\left({\sqrt{-a}}\right) +\text{const.},
%\end{split}
\end{align*}
\]

or similar formulas. The third is a formal consequence of the first. The third is an immediate consequence of the first two. The first equality of the fourth line or of the fifth can be deduced from the parametrizations \(x + y = z\), \(x − y = a/z\) or, rather,

\[\begin{align*}
%\begin{split}
x&=x(z)=\frac12\left(z+\frac{a}{z}\right), \tag{1.4}\\
y&=y(z)=\frac12\left(z-\frac{a}{z}\right)
%\end{split}
\end{align*}\]

of the plane curve \(x^2 − y^2 = a\). The second equality of these two lines is a consequence of the relations between the logarithm and the trigonometric functions or the hyperbolic functions, for example,

\[\begin{align*}
\theta&=\frac{1}{2i}\ln\left(\frac{1+i\tan \theta}{1-i\tan\theta}\right)\\
\theta&=\frac{1}{2}\ln\left(\frac{1+\tanh a}{1-\tanh a}\right)
\end{align*}\]

These equalities, a reflection of the intimate relation between the trigonometric functions and the hyperbolic functions and between them both and the exponential functions were not immediately evident in the eighteenth century when the integrals first began to be investigated.

In contrast to the arts, mathematics is a joint effort

The passage from the third to the fourth and fifth lines requires understanding that the line with the parameter \(z\) is for many purposes almost the same as the curve \(x^2 − y^2 = a\), for as \(z\) runs over the line, the point \((x(z), y(z))\), runs over this curve. So the calculation of integrals brings with it a freer use of geometric abstractions that carries us forward in the left-hand columns of the two diagrams.

It is, however, more important to understand that the fundamental theorem of algebra arose in large part as an essential element of the urge to calculate integrals of the form

\[\begin{equation}\tag{1.5}
{\displaystyle\int}\frac{X^m+b_{m-1}X^{m-1}+\cdots +b_1X+b_0}{X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0}dx,
\end{equation}\]

but these integrals are seldom if ever discussed. First of all, the general formulas are less important than the special formulas; secondly they require a calculation of the roots of the denominator and that is not an easy matter. It can in most cases only be effected approximately. These are the integrals that can, in principle, be treated in introductory courses. Even though a good deal of experience is implicit in the treatment, they are definitely at a different level than many other integrals that, at first glance, appear to be of the same level of difficulty. A typical example is

\[\begin{align}\tag{1.6}
\int\frac{1}{\sqrt{x^3+1}}.
\end{align}\]

These are the integrals with which Jacobi was familiar and that inspired his eloquence. They are astonishingly rich in geometric implications.

In order to treat (1.5) even on a formal level, the denominator has to be factored as in (1.2) or, if one is uneasy about complex numbers, as a product in which quadratic factors also appear

\[\begin{equation*}
X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0 \ =(X-\alpha_1)… (X^2+\beta_1 X+\beta_2)\dots.
\end{equation*}\]

For example, the factorization

\[\begin{align*}
X^4-1=(X-1)(X+1)(X^2+1),
\end{align*}\]

is then preferable to

\[\begin{align*}
X^4-1=(X-1)(X+1)(X+\sqrt{-1})(X-\sqrt{-1}).
\end{align*}\]

Although such uneasiness was overcome two hundred years ago, traces of the quadratic factors will certainly remain in introductory courses in the integral calculus. For the purposes of integration, the factorization (1.2) has to be found explicitly and that entails supplementary skills.

So the difficulties raised by the formal calculation of the integrals (1.5) raised two difficulties, one formal and logical, the other mathematical, both formidable although in different senses. The first requires the acceptance of complex numbers as genuine mathematical objects; the second a proof of the possibility of the factorization (1.2) in the domain of complex numbers, no matter what real or complex coefficients are taken on the left-hand side.

Once mathematicians were easy with the factorization (1.2) and certain that it was always possible, there were two directions to pursue, or rather many, but two that are especially pertinent to Diagram A and Diagram B. One was the study of the collection of all solutions in complex numbers of one or several algebraic equations. This is the first column, most of which has to be reckoned geometry. The other is the study of all solutions of one algebraic equation in one unknown, thus the equation (1.1), but with coefficients from a restricted domain, above all with ordinary fractions or ordinary integers as coefficients. Restrictions on the coefficients entail restrictions on the roots and the nature of these restrictions is to be revealed.

The presence of the two different long columns on the left of Diagram A, in which the one on the far left refers largely to the first direction and the one in the middle refers to the second, is to a considerable extent arbitrary, to a considerable extent a consequence of my own much more extended experience with the topics leading to automorphic forms than with those leading to diophantine equations, but not entirely artificial. The development of algebra in the late middle ages and early renaissance was every bit as much a prerequisite for Descartes as for the analysis of the roots of equations of degree three and four and the beginnings in the writings of Lagrange and others of a general analysis of the roots of the equation (1.1). Nevertheless the two columns, one carried by the study of integrals into the study of plane curves in complex coordinate geometry, really surfaces, namely Riemann surfaces, because to fix a complex number \(a + b\sqrt{-1}\) requires two real numbers \(a\) and \(b\), the other carried by the explicit solutions of equations of degree three and four discovered by Tartaglia, Ferrari and others, through the reflections of Lagrange into the study of groups, above all Galois groups, are useful as a symbol of the accumulation and the fusion of concepts that had to occur for modern pure mathematics to arise.

The configuration on the right of Diagram A is more unfamiliar. The history and the role of the theory of quadratic forms is not well understood even by mathematicians, but would be technical and too lengthy to undertake here. The triangular patch with Galois theory, groups, and automorphic forms at its vertices refers to some extent to little known, or at least little understood, overlapping of mathematics and physics on which I will not be able to resist dwelling. The triangle, class field theory, Galois theory, automorphic forms, is more arcane, but it is perhaps here where the pivotal difficulties lie. I shall attempt to adumbrate the pertinent issues in the final section.

Notes

1. I am not concerned, however, with history in any systematic way. Just as the present is richer, perhaps even more tolerable and acceptable, when viewed with a knowledge and understanding of the past, even though the knowledge and the understanding are for most, maybe all, of us necessarily individual, provisional, accidental, and partial, so is the judgement of a mathematician likely to be juster and more valuable the more insight he has into the waxing and waning importance over time of various mathematical notions. I cannot pretend to have much insight and certainly no systematic knowledge.

2. It may be excessive to introduce the Devil as well, but there is an appealing fable that I learned from the mathematician Harish-Chandra, and that he claimed to have heard from the French mathematician Claude Chevalley. When God created the world, and therefore mathematics, he called upon the Devil for help. He instructed the Devil that there were certain principles, presumably simple, by which the Devil must abide in carrying out his task but that apart from them, he had free rein. Both Chevalley and Harish-Chandra were, I believe, persuaded that their vocation as mathematicians was to reveal those principles that God had declared inviolable, at least those of mathematics for they were the source of its beauty and its truths. They certainly strived to achieve this. If I had the courage to broach in this paper genuine aesthetic questions, I would try to address the implications of their standpoint. It is implicit in their conviction that the Devil, being both mischievous and extremely clever, was able, in spite of the constraining principles, to create a very great deal that was meant only to obscure God’s truths, but that was frequently taken for the truths themselves. Certainly the work of Harish-Chandra, whom I knew well, was informed almost to the end by the effort to seize divine truths.

   Like the Church, but in contrast to the arts, mathematics is a joint effort. The joint effort may be, as with the influence of one mathematician on those who follow, realized over time and between different generations—and it is this that seems to me the more edifying—but it may also be simultaneous, a result, for better or worse, of competition or cooperation. Both are instinctive and not always pernicious but they are also given at present too much encouragement: cooperation by the nature of the current financial support; competition by prizes and other attempts of mathematicians to draw attention to themselves and to mathematics. Works of art or of literature, however they may have been created in the past or in other cultures, are presently largely individual efforts and are judged accordingly, although movements in styles and in materials are perhaps more a consequence of the human impulse to imitate than of spontaneous individual inspiration.

   Harish-Chandra and Chevalley were certainly not alone in perceiving their goal as the revelation of God’s truths, which we might interpret as beauty, but mathematicians largely use a different criterion when evaluating the efforts of their colleages. The degree of the difficulties to be overcome, thus of the effort and imagination necessary to the solution of a problem, is much more likely than aesthetic criteria to determine their esteem for the solution, and any theory that permits it. This is probably wise, since aesthetic criteria are seldom uniform and often difficult to apply. The search for beauty quickly lapses in less than stern hands into satisfaction with the meretricious. So the answer to Vittorio Hösle’s question whether there is beauty in mathematical theories might be: there can be, but it is often ignored. This is safer and appears to demand less from the judges. On the other hand, when the theorem in which the solution is formulated is a result of cumulative efforts by several mathematicians over decades, even over centuries, and of the theories they created, and when there may have been considerable effort—the more famous the problem the more intense—in the last stages, it is not easy, even for those with considerable understanding of the topic, to determine whose imagination and whose mathematical power were critical, thus where the novelty and insight of the solution lie, and whose contributions were presented with more aplomb and at a more auspicious moment.

Footnotes

1. Translation: It is true that Mr. Fourier had the opinion that the principal object of mathematics was public utility and the explanation of natural phenomena; but a philosopher like him ought to have known that the sole object of science is the honor of the human spirit, and thus a question about numbers is as valuable as a question about the system of the world. ↩