How did they stand it? But they knew nothing else and simple affection is sufficient to sustain the human spirit amid shared hardships.

before the post-Napoleonic shift of attitudes, and in some countries for longer, there was a clear distinction between universities, whose purpose was to teach and train whatever of a cognitive elite the nation was thought to require, and academies or societies, which existed for the purpose of research—this

it? Well, yes, strictly speaking, it does, and if you want

(In strict fairness to Napoleon, I should add that on a later razzia through western Germany, when Gauss was installed at Göttingen, the Emperor spared the city because “the greatest mathematician of all time is living there.”)

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….”

Legendre to throw another conniption. Fortunately

Euler wrote mainly in Latin, but this is not much of an obstacle to appreciating him,

The complainers were saying: “If you measure the time from the starting instant of the common era to the very end of the year 1999, you only have 1,999 complete years. You should wait until 2,000 complete years have elapsed.” They were imposing measuring logic on a system created according to counting logic. The revelers, on the other hand, were saying: “Here comes year number 2,000! Whoopee!”—pure

The first thing to be said about Pafnuty Lvovich Chebyshev is that his last name is a data-retrieval nightmare.

“The other morning, Dirichlet spent about two hours with me. He gave me notes I need for my Habilitation thesis—they are so comprehensive that my work has been substantially lessened.

The non-Euclidean geometry described by Lobachevsky in the 1830s was, seen from this point of view, a philosophical heresy. Riemann's paper was an enlargement of that heresy; and this might be why he presented his ideas at such a very general level that their connection with non-Euclidean geometry would have escaped all but the most mathematically adept in his audience. (But not, of course, Gauss. Gauss had in fact invented non-Euclidean geometry for himself, but had not published his findings, “for fear,” as he wrote in a letter to a friend, “of the hue and cry of the blockheads.” Nineteenth-century Germans took their philosophy seriously.)

Riemann and Dedekind seem to have attended each other's lectures, though whether they paid each other the requisite fees, I have not been able to discover.

Gauss's brain, meanwhile, was pickled and stored in the university's physiology department, where it remains to this day.

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.

His many books included The Psychology of Invention in the Mathematical Field (1945), still well worth reading for its insights into the thought processes of mathematicians; I have used some of its ideas for this book.

The number of rational numbers is infinite, and the number of irrational numbers is infinite; but the second infinity is bigger than the first. How on earth do they all fit on the real line? How does such an inconceivably vast number of irrationals squeeze in among the rationals, if the rationals themselves are everywhere dense? I have no space to go into such things here. My advice is not to think about these matters too much. That way lies madness.

Hilbert had a student who one day presented him with a paper purporting to prove the Riemann Hypothesis. Hilbert studied the paper carefully and was really impressed by the depth of the argument; but unfortunately he found an error in it which even he could not eliminate. The following year the student died. Hilbert asked the grieving parents if he might be permitted to make a funeral oration. While the student's relatives and friends were weeping beside the grave in the rain, Hilbert came forward. He began by saying what a tragedy it was that such a gifted young man had died before he had had an opportunity to show what he could accomplish. But, he continued, in spite of the fact that this young man's proof of the Riemann Hypothesis contained an error, it was still possible that some day a proof of the famous problem would be obtained along the lines which the deceased had indicated. “In fact,” he continued with enthusiasm, standing there in the rain by the dead student's grave, “let us consider a function of a complex variable….”

Hilbert was seen day after day in torn trousers, a source of embarrassment to many. The task of tactfully informing Hilbert of the situation was delegated to his assistant, Richard Courant. Knowing the pleasure Hilbert took in strolls in the countryside while talking mathematics, Courant invited him for a walk. Courant managed matters so that the pair walked through some thorny bushes, at which point Courant informed Hilbert that he had evidently torn his pants on one of the bushes. “Oh no,” Hilbert replied, “they've been that way for weeks, but nobody notices.”

One of Hilbert's students stopped showing up to classes. On enquiring the reason, Hilbert was told that the student had left the university to become a poet. Hilbert: “I can't say I'm surprised. I never thought he had enough imagination to be a mathematician.”

Mathematicians did not go home from their New Years' parties in the small hours of January 1, 1900 (or 1901, if you like—see Chapter 6.ii) thinking, “It's the twentieth century! We must move to a higher level of abstraction!” any more than Europeans woke up on the morning of May 30, 1453, thinking, “The Middle Ages are over! We'd better start disseminating printed books, challenging the authority of the Pope, and discovering the New World!” I should hate to have to stand before a jury of my

Mathematicians did not go home from their New Years' parties in the small hours of January 1, 1900 (or 1901, if you like—see Chapter 6.ii) thinking, “It's the twentieth century! We must move to a higher level of abstraction!” any more than Europeans woke up on the morning of May 30, 1453, thinking, “The Middle Ages are over! We'd better start disseminating printed books, challenging the authority of the Pope, and discovering the New World!”

In twentieth-century math the objects that had been invented to encapsulate important facts about number themselves became the objects of inquiry, and the techniques that had been developed for investigating numbers and sets of numbers were turned on those objects themselves. Mathematics broke free, as it were, from its mooring in number and soared up to a new level of abstraction.

The North Sea can be pretty rough, and the probability that such a small boat would sink was not exactly zero. Still, Hardy took the boat, but sent a postcard to Bohr: “I proved the Riemann Hypothesis. G.H. Hardy.” If the boat sinks and Hardy drowns, everybody must believe that he has proved the Riemann Hypothesis. Yet God would not let Hardy have such a great honor and so He will not let the boat sink.

I don't think Landau's Handbuch has ever been translated into English. Number theorist Hugh Montgomery, the star of my Chapter 18, taught himself German by reading his way through the Handbuch, one finger on the dictionary. He tells the following story. The first 50-odd pages of the book are given over to a historical survey, in sections each of which is headed with the name of a great mathematician who made contributions in the field: Euclid, Legendre, Dirichlet, and so on. The last four of these sections are headed “Hadamard,” “von Mangoldt,” “de la Vallée Poussin,” “Verfasser.” Hugh was extremely impressed with the contributions of Verfasser, but was puzzled to know why he had not heard the name of this fine mathematician before. It was some time before he learned that “Verfasser” is a German word meaning “author” (ordinary nouns are capitalized in German).

We ought not believe those who today, with a philosophical air and a tone of superiority, prophesy the decline of culture, and are smug in their acceptance of the Ignorabimus principle. For us there is no Ignorabimus, and in my opinion there is none for the natural sciences either. In place of this foolish Ignorabimus, let our resolution be, to the contrary: “We must know, we shall know.”

It had been arranged that, following the speech, Hilbert would give a shorter version of the address over the radio—at that time, of course, a very new thing. That shorter version was recorded and was actually released as a 78 R.P.M. gramophone record. (“Celebrity mathematician” was apparently not an oxymoron in Weimar Germany.) It can now be found on the Internet.

With singular courage, Landau asked the Council leader, a 20-year-old student named Oswald Teichmüller, to write out as a letter his reasons for organizing the boycott. Teichmüller did so, and the letter somehow survived. Teichmüller was a very intelligent man and in fact became a fine mathematician.88 It is clear from his letter that his motivation for the boycott was ideological. He believed, wholeheartedly and sincerely, in the Nazi doctrines, including the racial ones, and felt it improper that German students should be taught by Jews. We are accustomed to think of Nazi activists as thugs, low-lifes, opportunists, and failed artists of one sort or another, which indeed most of them were. It is salutary to be reminded that they also included in their ranks some people of the highest intelligence.

Carl Ludwig Siegel, the son of a Berlin letter carrier, was a lecturer at the University of Frankfurt. An accomplished number theorist, he understood very well, as any mathematician who reads it must, that Riemann's 1859 paper was only, to employ the terminology of Erving Goffman that I introduced in Chapter 4.ii, a “front” display— a summary for formal presentation of what must have been a far greater amount of “back” work. Siegel spent such time as he could spare at Göttingen, going through Riemann's private mathematical papers from the period, to see if he could gain any insight into the activity of Riemann's mind when he was constructing the paper.

Sir Michael Berry110 likes to quote the Nobel Prize-winning physicist Richard Feynman in this context, “A great deal more is known than has been proved.”

The adele is certainly a very abstruse concept. Nothing is so abstruse that it doesn't find its way into physics eventually, though.