Riemann Quotes

If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann Hypothesis been proven?

I tell you, with complex numbers you can do anything.

No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.

Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were, sitting in the dining car, having a nice conversation, and that train is being pulled along at a terrific clip by certain locomotives named The Bertrand Russell and Riemann and Euler and others. And our friend Lawrence is running alongside the train, trying to keep up with us—it’s not that we’re smarter than he is, necessarily, but that he’s a farmer who didn’t get a ticket. And I, Rudy, am simply reaching out through the open window here, trying to pull him onto the fucking train with us so that the three of us can have a nice little chat about mathematics without having to listen to him panting and gasping for breath the whole way.

The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe its properties.

Riemann Hypothesis,

Sólo Riemann, incomprendido y solitario, se preocupó por establecer una nueva concepción del espacio en la que se segregaba al espacio de su inmovilidad y se posibilitaba su participación en los sucesos físicos.

Even today, I am in total awe of the following wondrous chain of ideas and interconnections. Guided throughout by principles of symmetry, Einstein first showed that acceleration and gravity are really two sides of the same coin. He then expanded the concept to demonstrate that gravity merely reflects the geometry of spacetime. The instruments he used to develop the theory were Riemann's non-Euclidean geometries-precisely the same geometries used by Felix Klein to show that geometry is in fact a manifestation of group theory (because every geometry is defined by its symmetries-the objects it leaves unchanged). Isn't this amazing?

The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.… The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It

Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a "force" has no independent life of its own; it is only the apparent effect caused by the distortion of geometry. By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed.

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

Near Shepherd’s Bush two thousand Beta-Minus mixed doubles were playing Riemann-surface tennis.

[Sobre la teoría de la relatividad] Debido a que el campo gravitatorio queda determinado por la configuración de masas y varía al variar dicha configuración, la estructura geométrica de este espacio depende también de los factores físicos. El espacio ya no es, pues, según esta teoría - exactamente como lo había presentido Riemann - absoluto, sino que su estructura depende de influencias físicas

Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study.

Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical conceptsãthe four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.

(The secret of unification, we will see, lies in expanding Riemann's metric to N-dimensional space and then chopping it up into rectangular pieces. Each rectangular piece corresponds to a different force. In this way, we can describe the various forces of nature by slotting them into the metric tensor like pieces of a puzzle. This is the mathematical expression of the principle that higher-dimensional space unifies the laws of nature, that there is "enough room" to unite them in N-dimensional space. More precisely, there is "enough room" in Riemann's metric to unite the forces of nature.)

One can set up a completely self-contained mathematical theory, which proceeds from the elementary laws that are valid for individual points to processes in the actually given continuously filled space, without distinguishing whether it is gravity, electricity, magnetism, or the equilibrium of heat that is being treated.

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

It was in 1742 that Christian Goldbach put forward his famous conjecture that every even number greater than 2 can be expressed as the sum of two primes.

(which has inspired at least one novel, Apostolos Doxiadis's Uncle Petros and Goldbach's Conjecture29).

Man wird "verrückt", man "ver-rückt

Is Euclidian geometry true or is Riemann geometry true? He answered, The question has no meaning. As well ask whether the metric system is true and the avoirdupois system is false; whether Cartesian coordinates are true and polar coordinates are false. One geometry can not be more true than another; it can only be more convenient. Geometry is not true, it is advantageous.

Wenn man von der Liebe oder der Ehe illusionäre Erwartungen hat und mehr fordert, als man selbst zu investieren bereit ist, muß man immer wieder enttäuscht werden; gewöhnlich erkennt man diesen Zusammenhang indessen nicht und bleibt auf der Suche nach der «großen Liebe». Man findet daher in den Partnerbeziehungen hysterischer Persönlichkeiten die häufigsten Trennungen und Neuanfänge; weil die letzteren jeweils für die vergangenen Enttäuschungen entschädigen sollen, werden neue Beziehungen von Beginn an überfordert, worin bereits wieder der Keim zum Scheitern liegt.

Furious, the beast writhed and wriggled its iterated integrals beneath the King’s polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann’s Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier-—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, “Hurrah! Victory!!

Mathematicians call it “the arithmetic of congruences.” You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, … up to 11. If the time is eight o’clock, and you add 9 hours, what do you get? Well, you get five o’clock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 ≡ 5 (mod 12), pronounced “eight plus nine is congruent to five, modulo twelve.

Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t merely that his mind worked faster, that his memory was more retentive, or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists — Georg Friedrich Bernhard Riemann, Jules Henri Poincaré, Srinivasa Ramanujan — Nash saw the vision first, constructing the laborious proofs long afterward.

The Riemann zeta function was a simple enough looking infinite series expressed in terms of a complex variable. Here, “complex” means not difficult or complicated, but refers to a variable of two distinct components, “real” and “imaginary,” which together could be thought to range over a two-dimensional plane. In 1860, Georg Friedrich Bernhard Riemann made six conjectures concerning the zeta function. By Ramanujan’s time, five had been proven. One, enshrined today as the Riemann hypothesis, had not

Grossmann went home to think about the question. After consulting the literature, he came back to Einstein and recommended the non-Euclidean geometry that had been devised by Bernhard Riemann.11 Riemann (1826–1866) was a child prodigy who invented a perpetual calendar at age 14 as a gift for his parents and went on to study in the great math center of Göttingen, Germany, under Carl Friedrich Gauss, who had been pioneering the geometry of curved surfaces. This was the topic Gauss assigned to Riemann for a thesis, and the result would transform not only geometry but physics.

On a flat surface with just the normal x and y coordinates, any high school algebra student, with the help of old Pythagoras, can calculate the distance between points. But imagine a flat map (of the world, for example) that represents locations on what is actually a curved globe. Things get stretched out near the poles, and measurement gets more complex. Calculating the actual distance between two points on the map in Greenland is different from doing so for points near the equator. Riemann worked out ways to determine mathematically the distance between points in space no matter how arbitrarily it curved and contorted.

Boston and Chicago are two great seats of mathematical research located in major American cities. Until they won in 2004, if you asked a baseball fan in Boston what they most hoped to see in their lifetime, they would have answered a World Series win for the Boston Red Sox. Chicago Cubs fans are still waiting. Ask a mathematician in either of those cities or anywhere else in the world what they would most hope to see in their lifetime, and they would most likely answer: "A proof o the Riemann hypothesis!" Perhaps mathematicians, like Red Sox fans, will have their prayers answered in our lifetimes, or at least before the Cubs win the World Series.

Published mathematical papers often have irritating assertions of the type: “It now follows that…,” or: “It is now obvious that…,” when it doesn't follow, and isn't obvious at all, unless you put in the six hours the author did to supply the missing steps and checking them. There is a story about the English mathematician G.H. Hardy, whom we shall meet later. In the middle of delivering a lecture, Hardy arrived at a point in his argument where he said, “It is now obvious that….” Here he stopped, fell silent, and stood motionless with furrowed brow for a few seconds. Then he walked out of the lecture hall. Twenty minutes later he returned, smiling, and began, “Yes, it is obvious that….” If he

The realization that symmetry is the key to the understanding of the properties of subatomic particles led to an inevitable question: Is there an efficient way to characterize all of these symmetries of the laws of nature? Or, more specifically, what is the basic theory of transformations that can continuously change one mixture of particles into another and produce the observed families? By now you have probably guessed the answer. The profound truth in the phrase I have cited earlier in this book revealed itself once again: "Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." The physicists of the 1960s were thrilled to discover that mathematicians had already paved the way. Just as fifty years earlier Einstein learned about the geometry tool-kit prepared by Riemann, Gell-Mann and Ne'eman stumbled upon the impressive group-theoretical work of Sophus Lie.

One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a two-dimensional universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has no meaning to them. Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them) gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our planet heading straight off over the seas would eventually return from the other horizon. The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it.10 By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well. In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,” the physicist Max Born has declared.

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).

Spațiile lui Riemann sunt lipsite de orice fel de omogenitate. Fiecare dintre ele se caracterizează prin forma expresiei care definește pătratul distanței dintre două puncte infinit învecinate. ... Rezultă de aici că doi observatori aflați învecinați pot să repereze într-un spațiu riemannian punctele care se află în imediata lor vecinătate, dar nu pot, fără stabilirea unei noi convenții, să se repereze unul față de celălalt. Fiecare vecinătate este deci ca o mică bucată de spațiu euclidian, dar racordarea dintre o vecinătate și următoarea nu e definită și poate fi făcută într-o infinitate de moduri. Spațiul riemannian cel mai general se prezintă, astfel, ca o colecție amorfă de bucăți juxtapuse fără a fi legate’ (Albert Lautmann, Les schèmas de structure, Hermann, 1938, pp.23, 34-35); această mulțime poate fi definită independent de orice referire la o metrică, prin condiții de frecvență sau, mai curând, de acumulare valabile pentru un ansamblu de vecinătăți, condiții total diferite de cele care determină spațiile metrice și tăieturile lor (chiar dacă un raport între cele două feluri de spațiu trebuie să decurgă de aici). Pe scurt, dacă urmăm frumoasa descriere a lui Lautmann, spațiul riemannian este un pur patchwork. Are conexiuni sau raporturi tactile. Are valori ritmice care nu se regăsesc în altă parte, chiar dacă pot fi traduse într-un spațiu metric. Eterogen, în variație continuă, este un spațiu neted, în măsura în care este amorf, nu omogen. (Gilles Deleuze et Félix Guattari)

Vielleicht wird unser Vorurteil auch dadurch bestärkt, daß es – von außen gesehen – oft vom Leben Begünstigte zu sein scheinen, […] denen wir daher sozusagen das Recht nicht zusprechen wollen, daß sie erkrankten; kennt man ihre Lebensgeschichte, wird man seine Meinung revidieren müssen; letztlich leiden wir alle an nicht genügend verarbeiteter Vergangenheit; bei wem sie so beschaffen war, daß er sein Leben dennoch fruchtbar gestalten konnte, weil er aus ihr mehr Hilfen als Schädigungen mitbekam, der sollte aus der Dankbarkeit dafür Verständnis und Toleranz gegenüber den weniger Glücklichen aufbringen.

Wir alle machen unsere ersten Erfahrungen am anderen Geschlecht an unseren Eltern und Geschwistern. Die Beziehung der Eltern zueinander, die an ihnen erlebte Ehe oder sonstige Gemeinschaft, die Erfahrungen mit unseren Geschwistern formen unsere Erwartungen von Partnerschaft, Liebe und Sexualität. Hatten wir das Glück, unsere Eltern auch als Paar lieben zu können, ohne sie idealisieren zu müssen, ohne sie andererseits bedauern oder verachten, ja vielleicht hassen zu müssen; konnten wir ihre Begrenztheit, ihre Sorgen und Probleme, ihr Bemühen miterleben, aber auch ihre Freuden, ihr Zueinander-Stehen, ihr Verständnis für und ihr Vertrauen zueinander, haben wir mehr Aussichten, einen Partner zu finden, der solchen Erwartungen entspricht, und haben zugleich für unser eigenes Partner-Sein ein realisierbares Bild vorschweben.

11 — I have explained where Wagner belongs—not in the history of music. What does he signify nevertheless in that history? The emergence of the actor in music: a capital event that invites thought, perhaps also fear. In a formula: "Wagner and Liszt."— Never yet has the integrity of musicians, their "authenticity," been put to the test so dangerously. One can grasp it with one's very hands: great success, success with the masses no longer sides with those who are authentic,—one has to be an actor to achieve that!— Victor Hugo and Richard Wagner—they both prove one and the same thing: that in declining civilizations, wherever the mob is allowed to decide, genuineness becomes superfluous, prejudicial, unfavorable. The actor, alone, can still kindle great enthusiasm.— And thus it is his golden age which is now dawning—his and that of all those who are in any way related to him. With drums and fifes, Wagner marches at the head of all artists in declamation, in display and virtuosity. He began by convincing the conductors of orchestras, the scene-shifters and stage-singers, not to forget the orchestra:—he "redeemed" them from monotony .... The movement that Wagner created has spread even to the land of knowledge: whole sciences pertaining to music are rising slowly, out of centuries of scholasticism. As an example of what I mean, let me point more particularly to Riemann's [Hugo Riemann (1849-1919): music theoretician] services to rhythmic; he was the first who called attention to the leading idea in punctuation—even for music (unfortunately he did so with a bad word; he called it "phrasing"). All these people, and I say it with gratitude, are the best, the most respectable among Wagner's admirers—they have a perfect right to honor Wagner. The same instinct unites them with one another; in him they recognize their highest type, and since he has inflamed them with his own ardor they feel themselves transformed into power, even into great power. In this quarter, if anywhere, Wagner's influence has really been beneficial. Never before has there been so much thinking, willing, and industry in this sphere. Wagner endowed all these artists with a new conscience: what they now exact and obtain from themselves, they had never extracted before Wagner's time—before then they had been too modest. Another spirit prevails on the stage since Wagner rules there: the most difficult things are expected, blame is severe, praise very scarce—the good and the excellent have become the rule. Taste is no longer necessary, nor even is a good voice. Wagner is sung only with ruined voices: this has a more "dramatic" effect. Even talent is out of the question. Expressiveness at all costs, which is what the Wagnerian ideal—the ideal of décadence—demands, is hardly compatible with talent. All that is required for this is virtue—that is to say, training, automatism, "self-denial." Neither taste, voices, nor gifts: Wagner's stage requires one thing only—Teutons! ... Definition of the Teuton: obedience and long legs ... It is full of profound significance that the arrival of Wagner coincides in time with the arrival of the "Reich": both actualities prove the very same thing: obedience and long legs.— Never has obedience been better, never has commanding. Wagnerian conductors in particular are worthy of an age that posterity will call one day, with awed respect, the classical age of war. Wagner understood how to command; in this, too, he was the great teacher. He commanded as the inexorable will to himself, as lifelong self-discipline: Wagner who furnishes perhaps the greatest example of self-violation in the history of art (—even Alfieri, who in other respects is his next-of-kin, is outdone by him. The note of a Torinese). 12 The insight that our actors are more deserving of admiration than ever does not imply that they are any less dangerous ... But who could still doubt what I want,—what are the three demands for which my my love of art has compelled me?

South American Riemann-Surface Tennis Championship, which were being played in silent and diminished reproduction on the screen of the television box at the foot of the bed.

Perhaps Cardano's curious combination of a mystical and a scientifically rational personality allowed him to catch these first glimmerings of what developed to be one of the most powerful of mathematical conceptions. In later years, through the work of Bombelli, Coates, Euler, Wessel, Argand, Gauss, Cauchy, Weierstrass, Riemann, Levi, Lewy, and many others, the theory of complex numbers has flowered into one of the most elegant and universally applicable of mathematical structures. But not until the advent of the quantum theory, in the first quarter of this century, was a strange and all-pervasive role for complex numbers revealed at the very foundational structure of the actual physical world in which we live-nor had their profound link with probabilities been perceived before this. Even Cardano could have had no inkling of a mysterious underlying connection between his two greatest contributions to mathematics-a link that forms the very basis of the material universe at its smallest scales.

With a great deal of effort, seeking help from friends better versed in mathematics than himself, Einstein learns Riemann’s math—and writes an equation where R is proportional to the energy of matter. In words: spacetime curves more where there is matter. That is it.

Russell

The prime advantage of defining the geometry of the pebble's surface in terms of general coordinates is that we can then proceed to define tensors that are true in any coordinate system, tensors that describe both the pebble's curvature (the Riemann curvature tensor) and physics (the field equations, for example).

The Riemann hypothesis states that the roots of the zeta function (complex numbers z, at which the zeta function equals zero) lie on the line parallel to the imaginary axis and half a unit to the right of it. This is perhaps the most outstanding unproved conjecture in mathematics with numerous implications. The analyst Levinson undertook a determined calculation on his deathbed that increased the credibility of the Riemann-hypothesis. This is another example of creative work that falls within Gruber and Wallace's (2000) model.

Riemann is an event server that allows for fairly advanced aggregation and routing of events and can form part of such a solution

Moreover, there is no known reason why the geometry of space and time should be described by the particular types of curved geometry defined by Riemann. There exist other more complicated varieties that could in principle have been employed by Nature. Only observation can at present tell us which mathematics is chosen by Nature for employment in particular situations. This may of course merely be a transient manifestation of our relative ignorance of the bigger picture in which everything that is not excluded is demanded.

Finding a method to categorize transcendental numbers was one of David Hilbert's great unsolved maths problems (along with the Riemann Hypothesis), and it remains justbas unsolved today.

The Riemann Hypothesis states that all the non-trivial zeroes of the zeta function are on this line. If we can prove the Riemann Hypothesis is true, then we'll also have proved the method for counting the prime numbers. In some weird twisted act of mathematical logic, at a fundamental level the alignment of these zeroes stems from the same logic as the density of the primes. It doesn't seem to make sense. But if we can understand this mysterious alignment, we understand where the numbers are hiding their primes.

There has been some progress on proving the Riemann Hypothesis, but it remains unsolved. In 1914, Hardy managed to prove that there are infinitely many zeroes on that line, but he couldn't prove that there aren't any extra zeroes off the line. We currently know that 40 percent of the non trivial zeroes are definitely on that line, but we need to know it's true for 100 per cent. So much as a single zero somewhere else, and the Riemann hypothesis would be disproved, causing our apparent understanding to come crashing down. But it hasn't. Everything has uncannily indicated that we are on the right track, but we can't yet prove it for sure.

In the chapter on prime numbers, I mentioned Bernhard Riemann's 1859 paper 'On the Number of Primes Less than a Given Magnitude'. In it he found a method of calculating how many primes there are below any given number. This would give mathematicians an amazing insight into the distribution and nature of prime numbers. The only problem was that he couldn't prove that this method definitely worked. He did, however, prove that if an apparent alignment in the zeta function was real, then the prime counting method was real. Then he failed to prove that too.

The problem is that many mathematicians have done the same thing Riemann himself did: surged ahead using the prime counting method , assuming that someone would prove it later on. It looks like a safe bet: as we know, using computers, the first 10 trillion zeroes have been checked, and all of them are on that line. That said, mathematical theories have been disproved with numbers bigger than that, so there could be a zero off the line that we've simply not reached yet.

Bernhard Riemann was a very pure case of the intuitive mathematician. This needs some explaining. The mathematical personality has two large components, the logical and the intuitive. Both are present in any good mathematician, but often one or the other is strongly dominant.

If Weierstrass is a rock climber, inching his way methodically up the cliff face, Riemann is a trapeze artist, launching himself boldly into space in the confidence—which to the observer often seems dangerously misplaced—that when he arrives at his destination in the middle of the sky, there will be something there for him to grab. It is plain that Riemann had a strongly visual imagination, and also that his mind leaped to results so powerful, elegant, and fruitful that he could not always force himself to pause to prove them. He was keenly interested in philosophy and physics, and notions gathered from long, deep contemplation of those two disciplines—the flow of sensations through our senses, the organizing of those sensations into forms and concepts, the flow of electricity through a conductor, the movements of liquids and gases— can be glimpsed beneath the surface of his mathematics.

He (Riemann) showed that there are three main ways space can be curved- positive curvature, negative curvature, and zero curvature. Thes correspond respectively to elliptic, hyperbolic, and Euclidean flat geometries.

For Space, when the position of points is expressed by rectilinear co-ordinates, ds = \sqrt{ \sum (dx)^2 }; Space is therefore included in this simplest case. The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression…. I restrict myself… to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression…. Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum (dx)^2 }, are… only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses… I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle.

On our unnamed alien hero’s home world, Vonnadoria, mathematics has transformed his people, giving them the ability to create a utopian society where knowledge is limitless and immortality attainable. But when Cambridge professor Andrew Martin cracks the Riemann hypothesis, opening a door to the same technology that the alien’s planet possesses, the narrator is sent to Earth to erase all evidence of the solution and kill anyone who had seen the proof. He struggles to pass undetected long enough to gain access to Martin’s research. But as he takes up the role of Professor Martin in order to blend in with the humans, he begins to see a kind of hope and redemption in the humans’ imperfections, and he questions his marching orders. Mathematics or not, he becomes increasingly convinced that Martin’s family deserves to live, forcing him to confront the possibility of forgoing everything he has ever known and become a human. TOPICS AND QUESTIONS FOR DISCUSSION 1. In the preface, our narrator explains his purpose and asks his people back home to set aside prejudice in the name of understanding. How is this plea to his fictional reader also directed at us, the actual readers? What prejudices must we set aside to understand our alien hero? 2. Our hero’s entrance into human life is . . . rocky, at best. How did his initial impressions of human life—noses, clothes, rain, and Cosmopolitan—shape the rest of his journey? Which of his first disconcerting realizations did you find the most surprising? 3. Starting with the possibility that the purpose of humanity is to “pursue the enlightenment of orgasm,” our hero is constantly seeking the solution to the meaning of human life. What does he

use four-dimensional Riemannian geometry for his famed theory of relativity. Within seven decades, Theodr Kaluza at the University of Königsberg, Germany, would use five-dimensional Riemannian geometry to integrate both gravity and light. Light is now viewed as a vibration in the fifth dimension. Oskar Klein made several improvements, including the calculation of the size of the fifth dimension—the Planck length, which is 10-33 centimeters, much too small to detect experimentally. One hundred thirty years after Riemann’s famous lecture, physicists would extend the Kaluza-Klein constructs to develop ten-

The rules of evidence can deliver very persuasive results, sometimes contrary to the strictly argued certainties of mathematics. […] Hypothesis: No human can possibly be more than nine feet tall. Confirming instance: A human being who is 8'11¾" tall. The discovery of that person confirms the hypothesis … but at the same time casts a long shadow of doubt across it!

The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper about the distribution of prime numbers,

This is the remarkable paradox of mathematics," observed commentator John Tierney. "No matter how determinedly its practitioners ignore the world, they consistently produce the best tools for understanding it. The Greeks decide to study, for no good reason, a curve called an ellipse, and 2,000 years later astronomers discover that it describes the way the planets move around the sun. Again, for no good reason, in 1854 a German mathematician, Bernhard Riemann, wonders what would happen if he discards one of the hallowed postulates of Euclid's plane geometry. He builds a seemingly ridiculous assumption that it's not possible to draw two lines parallel to each other. His non-Euclidean geometry replaces Euclid's plane with a bizarre abstraction called curved space, and then, 60 years later, Einstein announces that this is the shape of the universe.

Amikor visszajöttem, az asztalon két sör állt, és az apám éppen a naptáramba irkált valamit. Bikst: lambda/3x (Gik Gst–Gis Gkt) – olvastam. – Hát ez meg mi? – kérdeztem. – Ez a Riemann–Christoffel-féle tenzor, a világegyetem mérésére szolgál – mondta. – És én mihez kezdjek vele? – Te? Semmit. Ihatsz még egy vodkát.

Riemann conjecture,