7
Excerpts from Riemann, Topology, and Physics Michael Monastyrsky Editor's Note: The following are excerpts from a "dual" book recently published by 'Birkhfiuser Boston. The first part of the book is a modified translation of a paper that originally appeared in the Soviet journal Priroda (Nature) on the oc- casion of the 150th anniversary of Riemann's birth. The second part of the book deals with various applications of to- pology to contemporary physics. The author graduated from Moscow University in 1967 and received his Ph.D. in the Institute of Theoretical and Experimental Physics, Academy of Sciences, USSR. He is currently a member of the Theoretical Department of that Institute. His mathematical interests include applications of modern mathematics (topology and group theory) to Field Theory and Condensed Matter. On Riemann's Early Life Mathematicians are born, not made. Henri Poincar6 Georg Friedrich Bernhard Riemann was born on Sep- tember 17, 1826, in the village of Breselenz, located near the city of Dannenberg in the kingdom of Han- nover. His father, Friedrich Bernhard Riemann, was a Lutheran pastor who participated in the anti-Napole- onic campaign of 1812-1814, the "wars of liberation," serving as a lieutenant. He was in the army of the Austrian general Count Ludwig Wallmoden (1769- 1862), which gained distinction in the siege of Ham- burg. The army, which combined Russian, Prussian, and other allied troops, smashed the units of Marshall Davout which were in the Mecklenburg province. He was already middle-aged when he married Charlotte Ebell, the daughter of a court councilor. Bernhard was the second of their six children, two boys and four daughters. As a boy, his health was poor and, in general, illnesses and premature deaths haunted all the members of his family: his mother died when he was twenty, and his brother and three of his sisters also died quite young. Riemann was always very attached to his family and maintained the closest contact with family members throughout his life. Timid and reserved by nature, he felt at ease and free in the company of his relatives. At the age of five, history, especially the history of Poland, interested him most. An interest in history and in general in humanitarian subjects is character- istic of many great mathematicians. One has only to think of Karl Friedrich Gauss who, as a student, wa- vered between philology and mathematics as his spe- cialty, and Carl Gustav Jacob Jacobi who participated in a seminar on ancient languages. Soon the family began to notice the striking ability of the young Riemann to make calculations. At the age of six, under the tutelage of his father who was quite an educated man, he solved arithmetic problems. When he was ten, a teacher named Schulz began to work with him, but the pupil soon outstripped his master. At the age of fourteen, Riemann entered di- rectly into the third (senior) class of the gymnasium in Hannover; after two years he transferred to the gym- nasium of the city of Lfineburg where he continued to study until he was nineteen. Riemann was not a bril- liant student, although he earnestly studied such sub- jects of the classical gymnasium curriculum as Hebrew and theology. Schmalfuss, the director of the gymnasium, noticed the mathematical talents of Riemann and permitted him the use of his personal library. On one occasion, he gave Riemann Legendre's course on the theory of numbers to read. Riemann studied this book, which has almost 900 pages, for six days. Various aspects of what he learned there were used some years later in his own work on the theory of numbers. In 1846, in accordance with his father's wishes, Rie- mann matriculated at G6tfingen University in the fac- ulty of theology. However, his interest in mathematics was so strong that he asked his father to allow him to transfer to the faculty of philosophy. At this time, such well-known scholars were teaching at the univer- sity as the astronomer Carl Goldschmidt, who lectured on terrestrial magnetism, the mathematician Moritz Stern (1807-1894), who lectured on numerical 46 THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 2 9 1987 Springer-Verlag New York

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Page 1: Excerpts from  riemann, topology, and physics

Excerpts from Riemann, Topology, and Physics

Michael Monastyrsky

Editor's Note: The following are excerpts from a "dual" book recently published by 'Birkhfiuser Boston. The first part of the book is a modified translation of a paper that originally appeared in the Soviet journal Priroda (Nature) on the oc- casion of the 150th anniversary of Riemann's birth. The second part of the book deals with various applications of to- pology to contemporary physics.

The author graduated from Moscow University in 1967 and received his Ph.D. in the Institute of Theoretical and Experimental Physics, Academy of Sciences, USSR. He is currently a member of the Theoretical Department of that Institute. His mathematical interests include applications of modern mathematics (topology and group theory) to Field Theory and Condensed Matter.

On Riemann's Early Life

Mathematicians are born, not made. H e n r i P o i n c a r 6

Georg Friedrich Bernhard Riemann was born on Sep- tember 17, 1826, in the village of Breselenz, located near the city of Dannenberg in the kingdom of Han- nover. His father, Friedrich Bernhard Riemann, was a Lutheran pastor who participated in the anti-Napole- onic campaign of 1812-1814, the "wars of liberation," serving as a lieutenant. He was in the army of the Austrian general Count Ludwig Wallmoden (1769- 1862), which gained distinction in the siege of Ham- burg. The army, which combined Russian, Prussian, and other allied troops, smashed the units of Marshall Davout which were in the Mecklenburg province.

He was a l ready midd le -aged when he marr ied Charlotte Ebell, the daughter of a court councilor. Bernhard was the second of their six children, two boys and four daughters. As a boy, his health was poor and, in general, illnesses and premature deaths haunted all the members of his family: his mother died when he was twenty, and his brother and three of his sisters also died quite young.

Riemann was always very attached to his family and maintained the closest contact with family members

throughout his life. Timid and reserved by nature, he felt at ease and free in the company of his relatives.

At the age of five, history, especially the history of Poland, interested him most. An interest in history and in general in humanitarian subjects is character- istic of many great mathematicians. One has only to think of Karl Friedrich Gauss who, as a student, wa- vered between philology and mathematics as his spe- cialty, and Carl Gustav Jacob Jacobi who participated in a seminar on ancient languages.

Soon the family began to notice the striking ability of the young Riemann to make calculations. At the age of six, under the tutelage of his father who was quite an educated man, he solved arithmetic problems. When he was ten, a teacher named Schulz began to work with him, but the pupil soon outstr ipped his master. At the age of fourteen, Riemann entered di- rectly into the third (senior) class of the gymnasium in Hannover; after two years he transferred to the gym- nasium of the city of Lfineburg where he continued to study until he was nineteen. Riemann was not a bril- liant student, although he earnestly studied such sub- jects of the classical gymnasium curriculum as Hebrew and theology.

Schmalfuss, the director of the gymnasium, noticed the mathematical talents of Riemann and permitted him the use of his personal library. On one occasion, he gave Riemann Legendre's course on the theory of numbers to read. Riemann studied this book, which has almost 900 pages, for six days. Various aspects of what he learned there were used some years later in his own work on the theory of numbers.

In 1846, in accordance with his father's wishes, Rie- mann matriculated at G6tfingen University in the fac- ulty of theology. However, his interest in mathematics was so strong that he asked his father to allow him to transfer to the faculty of philosophy. At this time, such well-known scholars were teaching at the univer- sity as the astronomer Carl Goldschmidt, who lectured on terrestrial magnetism, the mathematician Moritz S te rn (1807-1894) , w h o l ec tu red on numer ica l

46 THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 2 �9 1987 Springer-Verlag New York

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methods and definite integrals, and the "king of mathematicians," Carl Friedrich Gauss (1777-1855). At the time, Gauss, who was at the height of his powers, gave a small course on the method of least squares. However, given Gauss's extremely unsoci- able character and his sec luded way of life, it is doubtful that Riemann had any personal contacts with him at this time. Professor Stern did take note of Rie- mann 's ability: he subsequently recalled about Rie- mann that "He already sang like a canary. ''I

O n Topology

In G6ttingen in 1850, Riemann's purely mathematical interests were concentrated on the problems of func- tions of a complex variable. Riemann was unusually fortunate that after his two-year sojourn in Berlin, where he received a brilliant background in analysis, he returned to G6ttingen. It is quite likely that no- where else was the scholarly atmosphere so full of geometric, or more precisely, of topological ideas as at G6ttingen University. It was in GOttingen, in 1848, that the first book on topology was published: Vorstu- dien zur Topologie (Preliminary Studies on Topology) in "G6ttinger Studien" by Professor Listing. Listing had begun his studies of topology under the influence of Gauss. Gauss himself had worked on this subject a great deal, as can be seen from his scholarly legacy. Riemann was well acquainted with Listing and his work but, in point of fact, aside from the basic defini- tion and several properties of knotted curves, could get nothing from it. This is not said in reproach to Listing, but only reflects the real state of the branch of mathematics which Gottfried Wilhelm Leibniz (1646- 1716) called "analysis situs" (analysis of position). In his book Characteristica Geometrica (1679) he tried, as we would say today, to study the properties of figures associated with their topological rather than their metric parameters. He wrote that, in addition to the coordinate representation of figures: "We need a dif- ferent analysis, purely geometric or linear, which also

Georg Friedrich Bernhard Riemann

defines the position (situs) as algebra defines quan- tity." It is interesting to note that Leibniz tried to in- terest Christian Huygens (1629-1695) in these ideas, but the latter did not evince much interest in them. In the subsequent 150 years, except for Euler's famous formula concerning polyhedra (V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces) and the resolution of the problem Of the seven bridges of Koenigsberg, nothing more appeared in the theory of "Analysis situs."

The term " topology," introduced by Listing, be- came affixed to this branch of mathematics only at the beginning of our century; Riemann used exclusively the term "Analysis situs."

Riemann's first scientific successes are associated with the introduction of topological methods into the theory of functions of a complex variable.

On Riemann Surfaces

Riemann began his work where Gauss left off. He based his study of analytic functions on the property of conformality, and this allowed him to avoid the use of explicit analytic formulas. The property of confor- mality was known already to Gauss. (See his work cited in the "'Astronomische Abhandlungen.") This is the only reference to the literature in his dissertation, and it provided the initial premises from which Riemann began his own investigations. The basic task of the in- vestigation was to consider the behavior of an analytic function, not on the plane, but on a certain surface, as

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Riemann himself put it, "on a surface spread over the plane." In his general examination of this question, Riemann turned to "analysis situs." The basic idea is that the behavior of an analytic function on any sur- face can be reduced to a s tudy of a function on a simple-connected domain and to the determination of the jumps of the function at the cuts. For this, he con- ducted a thorough, in essence, purely topological in- vestigation. Here already appear the first results on Riemann surfaces. We will examine them in greater detail.

The connectivity of a surface is defined by using a system of cuts. This is, in fact, a combinatorial defini- t i o n - t h e embryo of the future homology theory. The surface is called simple-connected if any cut divides it into separate parts. If this is not the case, the surface is said to be multi-connected. In this work, surfaces with a b o u n d a r y are pr imar i ly considered. R iemann showed that the connectivity of a surface ought not to depend on the system of cuts made. Specifically, if the number of cuts of a certain system equals n, and the number of simple-connected pieces is m, then the dif- ference n - m does not depend on the system of cuts and will be one and the same for the given figure. This number he called the order of connectivity. For ex- ample, in the case of a disk, it equals - 1; in the case of a ring, 0. His definition of the degree of connectivity differs from the modern definition by 1. Riemann ap- plied the concept of connectivity also to a two-dimen- sional surface and established a relation of the order of connectivity of the surface with the order of connec- t ivity of the boundary . The number of separate boundary curves of an n-connected surface is either equal to n or less than it by an even number.

After all these preparations, Riemann then ad- vanced to the basic problem: to define the behavior of an analytic function having given singularities on a mul t i -connected surface. He solved it first for a simple-connected domain and then, using a system of cuts and computing the jumps of the function on the cuts, reduced the general case to this special one. In the case of simple-connected domain, he proved two basic theorems.

The first theorem states that if a function u(x,y) sat- isfies the Laplace equation in a domain f~ (such a function is called harmonic),

0 2 u 0 2 u

AU = ~X2 + - - = 0 ' O y 2

then the function u has derivatives of all orders and, moreover, is the real part of an analytic function f(z). Given this, the function f is defined by u uniquely up

to the addition of a purely imaginary constant term. The second theorem-- the theorem of existence--

says that inside a simple-connected domain (it is suffi- cient to limit oneself to the case of the disk), there exists one and only one function u continuous up to the boundary with a given boundary value f and which satisfies the Laplace equation inside the do- main.

For proof of this second theorem, Riemann used a variational principle. Let's look at the integral in the disk

+ | - - | dxdy, (3) ~ \ O x l \Oyl

where the following conditions are placed on the function u: u is continuous up to the boundary where it takes on the boundary value v, and inside the do- main the integral (3) is finite. Let's consider the class of all functions u, satisfying these conditions. Since the integral (3) is everywhere positive, it follows that there is a lower bound on its values. Let's suppose that this bound is obtained for a certain specific func- tion u = t/, having the given boundary value v. This means that the integral (3) attains its minimum u = t/. Having the variation of the integral (3) equal to zero is a necessary condition for a minimum:

(oula [Ou~ 2

which, as is well known, is equivalent to the equation &u = 0. Thus, having proposed the existence of the solution to the variational problem, we immediately find a corresponding harmonic and, consequently, an analytic function.

Reimann actively used this means of resolution which he later called the "Dirichlet Principle." He first became acquainted with it in Dirichlet's lectures, al- though the method was known already to Gauss, George Green (1793-1841), and William Thomson (Lord Kelvin, 1824-1907). In the "Dirichlet Principle," there exist dangerous pitfalls which are not immedi- ately obvious. The crux of the matter is that it is im- possible to assert a priori that: if a variational integral has a minimum, then there exists an actual function where the minimum is attained. One can explain the situation which arises by an example taken directly from the calculus of variations. Let's consider the fol- lowing problem: to find a curve of the shortest length among all smooth curves (having continuous curva- ture) connecting two points A and B and passing through a third point C which is assumed to be non- collinear with A and B. It is easy to see that the length of the broken line ACB will be the lower bound for the lengths of the curves under consideration. On the

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other hand, it is obvious that ABC has a break at the point C and therefore does not belong to the given class of curves.

The debate associated with the "Dirichlet Principle" played an important role in the history of mathematics and also to some extent in Riemann's life as well.

On the Dirichlet Principle

The fact is that already in his doctoral dissertation, and especially in "The Theory of Abelian Functions," in order to assert the existence of a required function with a given type of singularities, Riemann used the variational principle we have already discussed-- the "Dirichlet Principle." Weierstrass showed that it is im- possible, in the general case, to deduce the existence of the desired function from the variational principle. This required a special proof which Riemann did not give.

Klein describes the reaction to Weierstrass's criti- cism as follows:

The major i ty of ma thema t i c i ans t u rned away from Riemann . �9 . . R i e m a n n had a quite different opinion. He fully recog- n ized the justice and correctness of Weiers t rass ' s critique; bu t he said, as Weiers t rass once told me: tha t he appea led to the Dirichlet Principle only as a conven ien t tool that was r ight at h a n d , a n d tha t his existence t h e o r e m s are still correct.

Weierstrass himself was also convinced of that. He en- couraged his student Hermann Schwarz (1843-1921) to make a thorough study of the Riemann theorem of existence and to seek other proofs, which he suc- ceeded in doing.

In 1869, immediately after Weierstrass's critical re- marks appeared in print, Schwarz proved the exis- tence of the solution of the Dirichlet problem without using the variational method. His alternative method consisted of the following: he first solved the Dirichlet problem for the disk with the help of the Poisson inte- gral, to construct a harmonic function u(x,y) in the disk, taking the value f on the boundary. Then he showed how to pass to an arbitrary domain obtained as the union of a finite number of disks. Another re- markably interesting proof which Poincar6 proposed was based on the theory of the potent ial-- the method of balayage (sweeping out).

Riemann's "mistake" had yet one more remarkably useful consequence. It stimulated specialists in alge- braic geometry to find a purely algebraic proof of Rie- mann ' s theorem, specifically the Riemann-Roch theorem. The outstanding work of Rudolph Friedrich Alfred Clebsch (1833-1872), who invented the term " g e n u s of a surface ," lies in this direction. Paul Gordon (1837-1912), Max Noether (1844-1921), and

finally David Hilbert (1862-1943) in 1899 succeeded in giving a proof of the variational principle. It is difficult to recall another example in the history of nineteenth- century mathematics when a struggle for a rigorous proof led to such productive results.

It is characteristic that for physicists Riemann's work was completely convincing. We will cite an excerpt from the report of A. Sommerfeld, "Klein, Riemann and Mathematical Physics":

R i e m a n n ' s d isser ta t ion at first was foreign to his con tempo- rary m a t h e m a t i c i a n s w h o r ev iewed it as if it were a book pub l i shed for the family. The fact that it was closer in its way of r eason ing to phys i c s t han to ma thema t i c s is a t tes ted to by a story of one of m y colleagues�9 Once he s p e n t his vacation toge ther wi th He lmho lz and Weiers t rass . Weiers t rass took R i e m a n n ' s d isser ta t ion a long on hol iday in order to deal with w h a t he felt w a s a complex work in qu ie t c i r cums tances . He lmholz did not u n d e r s t a n d wha t complicat ions m a t h e m a t - ical specialists could find in R i e m a n n ' s work; for h i m Rie- m a n n ' s exposi t ion was exceptionally clear�9

Why was Riemann's explanation so clear to physi- cists but presented difficulties to mathematicians? This, of course, is not explained by the obtuseness of mathematicians and the genius of physicists but rather by the requirements of proof which the one and the other demand.

On the Zeta Function

The first duty a newly elected correspondent of the Berlin A c a d e m y had to fulfill, accord ing to the Academy's charter, was to send a report about his most recent works. Riemann chose his work on the distribution of prime numbers , "A topic perhaps, which will not be bereft of interest if one recalls that for a prolonged period of time it attracted the attention of Gauss and Dirichlet." The work which its author spoke of so humbly is entitled On the Number of Primes Less Than a Given Magnitude. It placed before mathe- maticians problems which determined the develop- ment of several branches of mathematics for a whole century.

In order to investigate the distr ibution of prime numbers, Euler had already studied the zeta function

(the notation ~ is due to Riemann). Euler had ob- tained the relation

1 ~(s) = ~=1~- ~ = 1-[(1 - p-S)-1. (15)

(The product is taken over all prime numbers, while the sum is over all positive integers. Euler considered this relation for real s.) From this formula follows im- mediately the existence of an infinite number of prime numbers (the series for 4(1) diverges). But all efforts of mathematicians were directed toward obtaining more

THE MATHEMATICAL 1NTELLIGENCER VOL. 9, NO. 2, 1987 49

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precise information about the distribution of primes. Let us denote the number of primes less than a

given number x by ~(x). There was already a conjec- ture, apparently enunciated by Euler himself, that as x --* %

*r(X) 1. (16)

x/log x

Despite attempts by such mathematicians as Euler, Legendre, and Gauss, this conjecture had not been proven. Gauss, with his characteristic energy and love of computation, even constructed a table of primes of magnitude less than three million.

The strongest result which had been obtained before the time of Riemann was due to the great Russian mathematician P. L. Chebyshev (1821-1894) and was published in 1854 in the M~moires de l'Acaddmie des Sciences de Saint Petersbourg. Chebyshev proved that

~(x) A 1 ~ - - ~ A 2

x/log x

(where 0.992 < A 1 < 1; 1 < a 2 K 1.105), but he did not prove that the limit exists. To prove the inequality, he used relation (15). There is no information on whether Riemann knew of the work of Chebyshev.

Riemann also began his work with the Euler identity (15), but he considered the series (15) for complex values of s. This was a completely new step in the study of the t-function (we will follow the accepted modern terminology and call it the Riemann X-func- tion). Riemann studied the convergence of the series (15) for complex values and deduced the functional equation for the t-function.

How many important conclusions can be derived from analytic properties of this function can be seen if one looks only at the example of the problem about the distribution of primes. For example, the law of the distribution of primes (formula 16) is equivalent to the assertion that the X-function of Riemann does not have complex zeros, of which the real part is equal to one: a ( a + it) ~ 0 when t # 0. However, most remarkable is the assertion that all zeros of the functions X(s), with the exception of trivial ones ( - 2, - 4 . . . . . -2n) , lie on the straight line ReX(s) = 1/2. This is the famous, still unproven, Riemann hypothesis.

The majority of results in this work were rigorously substantiated by subsequent generations of mathema- ticians. In particular, Jacques Hadamard (1865-1963) and Charles de la Vall6e-Poussin (1866-1962) proved the validity of the formula for ~(x). But as Hadamard himself wrote, "As for the properties for which he gave only a formula, it took me almost three decades before I could prove them, all except one." The latter property is, in fact, the Riemann hypothesis. Addi-

tional material connected with t(s) was found in Rie- mann's manuscripts preserved in the archives of G6t- tingen University. Unfortunately, one cannot obtain any sort of proof on the basis of these papers; they only give an idea about the reflections which led Rie- mann to his hypothesis. Hadamard recalled an asser- tion contained in Riemann's papers: "These properties are deduced from one of its representations which I was unable to simplify enough to publish." This sen- tence calls to mind the note made by Pierre Fermat in connection with his no less famous theorem about the impossibility of solving the equation x n + y" = z", when n > 2 in whole numbers. It is interesting that the analogy of this hypothesis for other fields, the con- gruence of L-functions, was proved by Andr6 Weil in 1941. Thus, one can say that, in a certain sense, the field of complex and real numbers we are accustomed to is more complicated than others.

On the Fate of Riemann's Work

While he lived, Riemann's work did not afford its cre- ator that influence which he might have expected. This relates in the first place to those works which today are considered perhaps his main contribution to science: the theory of Abelian integrals, Riemann sur- faces, and Riemannian geometry. There are many ob- jective and subjective reasons which explain this cir- cumstance. His views on geometry were, of course, completely novel for a wide circle of mathematicians. One ought not to forget that the very idea of non-Eu- clidian geometry had difficulty being accepted, even eliciting wild fury from a majority of philosophers. For example, here is what E. K. Duehring wrote in his composition, "Kritische Geschichte der allgemeinen Prin- zipien der Mechanik" ("A Critical History of the General Principles of Mechanics"), which earned the Benecke prize in 1872 from the philosophical faculty of G6t- tingen University.

Thus the late Goettingen mathematics professor, Riemann, w h o - - w i t h his lack of independence except for Gaussian self-mystification--was led astray even by Herbart 's philo- sophistry, wrote (in his paper, "On the Hypotheses which Lie at the Foundations of Geometry," Goettinger Abhand- lungen, Vol. 13, 1868): "But it seems that the empirical con- cepts on which are based the spatial definitions of the phys- ical universe, the concept of a rigid body and of a light ray, no longer are valid in the infinitely small. Thus, it is permissible to think that physical relations in space in the infinitely small do not correspond to the axioms of geometry; and, in fact, this should be allowed if this would lead to a simpler explana- tion of phenomena ."

Years passed before the ideas of Riemann and Helmholtz found their mathematical realization in the works of Poincar6 and Einstein. The works on the

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theory of Abelian funct ions-- the most brilliant for the ideas they con ta ined- -were written in Riemann's characteristically intuitive manner; in places they were based on unproven statements, and they did not sat- isfy those standards of rigor which had been estab- lished at the time in mathematics. As an example of the extreme incomprehension of the value of Rie- mann's work, we again refer to the work of Duehring:

Evidence of the completely dependent character of Riemann's work on Abelian functions is that in it the same method of intuitive presentation, in the same completely arbitrary form, is taken and extended simply by faith in the authority of the teacher.

The "teacher" referred to here is Gauss. Today this article is considered a curiosity, but one must re- member that in his time, Duehring was a well-known and influential philosopher. 1 However, if one does not take into account the extremely incompetent criticism by Duehring, well-known for his pathological anti- Semitism and fierce struggle with Helmholtz, it is also true that among qualified mathematicians, there was no clear understanding of Riemann's work.

Weierstrass's criticism of the "Dirichlet Principle," on which Riemann's work was based, played a role here. However, gradually Riemann's ideas received full recognition. Hilbert proved the "Dirichlet Prin- ciple," and all Riemann's reasonings acquired a firm basis. Riemann's greatness as a mathematician lies in the fact that almost all of his works proved to be not an ending, but rather the beginning of new, productive research.

One can cite the theory of automorphic functions, the Atiyah-Singer theorem on the index of differential operators on manifolds of arbitrary d imens ions- -a generalization of the Riemann-Roch theorem, the problem of moduli in the theory of complex mani- folds, and much more.

The nature of Riemann's successes in so many areas of physics and mathematics consists of his universal approach to natural phenomena and his unusual flair for comprehending connections between apparently heterogeneous phenomena. In his philosophical re- flections he wrote, "My main work relates to a new understanding of well-known laws of Nature."

Here it is perhaps appropriate to note the differ- ences between two great mathematicians, Riemann and Weierstrass, whose names often stand side by side in contemporary mathematics.

Weierstrass's logical approach, in contrast to Rie- mann's intuitionist approach, consisted of a precise and sequential process of reasoning. He held to this

1 A critique of many of Duehring's ideas is found in the work of Friedrich Engels, "Anti-Duehring."

approach in his articles and lectures. His concepts of rigor became subsequently the standard in mathemat- ical works. Weierstrass is considered a classic example of a pure mathematician, and his pronouncement about the connection of mathematics with its applica- tions is, therefore, all the more interesting:

Between mathematics and the natural sciences, deeper mu- tual relations ought to be established than those which would take place if, for example, physics would see in mathematics only an auxiliary, even if a necessary, discipline, while math- ematicians would consider questions proposed by physicists only as a bountiful collection of examples for their methods. �9 . . To the question, can one really obtain anything directly applicable from those abstract theories with which today's contemporary mathematicians occupy themselves, I can an- swer that Greek mathematicians studied the properties of conic sections in a purely theoretical way long before the time when anyone could foresee that these curves represent paths along which the planets move. I believe that there will be found many more functions with such properties; for ex- ample, I believe that the well-known O-functions of Jacobi allow one, on the one hand, to find out into how many squares any given number decomposes, which allows one to rectify the arc of an ellipse and, on the other, make it possible to find the true law of the oscillations of a pendulum.

The clear difference between Weierstrass and Rie- mann can be seen not only in the difficult concrete physics problems they solved but in the effort to build theories explaining natural phenomena.

These two contrasting views on the aims of mathe- matics can be traced at all stages of the development of science.

Jacobi presented them in clear form:

Monsieur Fourier held to opinions as if the main aim of math- ematics is its social utility and the explanation of phenomena of Nature; but as a philosopher, he ought to have known that the single goal of science is to promote the courage of human reason and, thus, any sort of quest ion in the theory of numbers has no less value than a question about the system of the world.

In the real development of science, by far the most fruitful is the coexistence of both points of view which, following Niels Bohr, one can call great truths. (A great truth is a truth whose negation is also a great truth.) We see a confirmation of this in the works of Jacobi on mechanics and of Riemann on the theory of numbers. Nonetheless, in mathematical science, there exist periods when one of these tendencies is more prevalent than the other.

A typical "intuitionist," Klein, who did much for developing Riemann's ideas, gave an appreciation, in humorous form, of the mathematics of the end of the nineteenth century:

Mathematics in our day reminds me of major small-arms pro- duction in peacetime. The shop window is filled with models which delight the expert by their cleverness and their artful and captivating execution. Properly speaking, the origin and

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significance of these things--that is, their ability to shoot and hit the enemy, recedes in one's consciousness and is even completely forgotten.

Just such a situation arose in the 1940s and 1950s. It was very aptly characterized by the well-known Amer- ican theoretical physicist Freeman Dyson: "The mar- riage between mathematics and physics, which was so fruitful in past centuries, recently ended in divorce." Different causes promoted this. Now, of course, it is completely impossible to imagine a mathematician working in an experimental physics laboratory. The times of Riemann will not come again. But in recent years, we are again seeing a renaissance of interest in phys ics among " p u r e " mathemat ic ians . This is bearing its first fruits, and precisely in those areas of mathematics in which Riemann worked.

At the same time, completely new connections among various subjects which occupied Riemann have arisen in remarkable fashion. For example, in the theory of nonlinear waves (also for other classes of equations, it is true), the O-function of Riemann ap- pears in the discovery of periodic solitons. The theory of gravitation, relying on Riemannian geometry, has been united with the theory of strong interactions (by the t heo ry of gauge fields). There has ar isen a quantum theory of gravitation where methods of alge- braic geometry and topology are actively used; these methods originate in the work on Abelian functions. It

is too early now to talk about a new "renaissance" in the interaction of physics and mathematics, but defi- nite successes are at hand.

More than a hundred years have passed since Rie- mann's death. Mathematics has been enriched in a major way by new ideas and results: Cantor's theory of sets has transformed its face, and the degree of ab- straction has achieved exceptional levels- -one has only to recall the theory of categories and functors, the theory of formal schemes of Grothendieck, and others. And it is remarkable that from the height of these theories, Riemannian concepts, which seemed com- pletely murky and unrigorous to his contemporaries, have received a very adequate description in the lan- guage of contemporary algebraic geometry and to- pology.

The check on the fruitfulness of any mathematical idea is the possibility of making progress on complex concrete problems left by preceding generations. Now, as before, the Riemann hypothesis poses a chal- lenge to contemporary mathematicians.

One cannot doubt that Riemann's works will in- terest not only historians of science but also research mathematicians for many years.

Inst. Theor. & Exper. Physics Bolshova Chremusschikinskaya 25 117259 Moscow USSR

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