Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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6. The Great Fusion
6. The Great Fusion
Pages 82-98
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From page 82... ...
Since my visit occurred early in July 2001, Taiye's age at the time was, in the modern western reckoning, 95~/2 years and a few days. So why was everyone telling me that he was 97?
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From page 83... ...
The English language takes the world to be mainly a countable place: one cow, two fishes, three mountains, four doors, five stars. Somewhat less frequently, our language takes the worI(1 to be measurable: one blade of grass, two sheets of paper, three head of cattle, four grains of rice, five gallons of gasoline.
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From page 84... ...
for each noun: one head of cow, two sticks of fish, three plinths of mountain, four fans of door, five grains of star. In the entire Chinese language there are only two words that can always be let loose grammatically without a measure word: "day" and "year." Everything else cows, fishes, mountains, doors, stars is a kind of stup~that must be divided up and measured out before we can talk about it.
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From page 85... ...
" pure counting logic. Yet these same revelers might fall back on measuring logic if asked the age of their new baby: "Oh, he's just half a year old." Which is to say, his age is 0.5 years measuring logic, at least by contrast with the traditional Chinese approach.
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From page 86... ...
The Riemann Hypothesis was born out of an encounter, what my chapter heading calls a great fusion, between counting logic and measuring logic. To put it in precise mathematical terms; it arose when some ideas from arithmetic were combined with some from analysis to form a new thing, a new branch of the mathematical tree, analytic number theory.
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From page 87... ...
· Analysis The study of limits. This fourfold scheme was well established in people's minds around 1800, and the great fusion I am going to describe in this chapter was a fusion of ideas which, until 1837, had lived separate lives under two of the above headings, arithmetic and analysis.
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From page 88... ...
In the early eighteenth century, when calculus first became known to the general educated public, the notion of infinitesimals came in for much scorn. The Irish philosopher George Berkeley (1685-1753 the California town is name(1 after him)
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From page 89... ...
The proposition "a whale is big" represents language at its best, giving terse expression to a complicated fact; while the true analysis of"one is a number" leads, in language, to an intolerable prolixity. (They weren't kidding.
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From page 90... ...
When the New York Times asked George Mallory why he wanted to climb Mount Everest, Mallory replie(l: "Because it's there." V The connection between measuring and continuity is this.
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From page 91... ...
DirichIet (1805-1859) was, names notwithstanding, German, from a small town near Cologne, where he got most of his education.3i The fact that he was a German deserves a brief detour by itself; for the fusion of ideas from arithmetic and analysis, carried out by DirichIet and Riemann, happened within a broader social change in mathematics at large, the rise of the Germans.
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From page 92... ...
The rising nationalist passions stimulated by the long wars against Napoleon, and by the Romantic Movement, were an added spur to reform, in spite of having been thwarted (as the nationalists saw it) by the failure of the Congress of Vienna to unify the German-speaking peoples.
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From page 93... ...
He must have been an exceptionally quick study, for by age 16 he had acquired all the qualifications necessary for university entrance. Already hooked on mathematics, he set off for what was still the world capital of mathematical knowledge, Paris, carrying with him the book he treasured above all others, Gauss's Disquisitiones Arithmeticae.
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From page 94... ...
DirichIet married Rebecca Mendelssohn, one of the sisters of the composer Felix Mendelssohn, thereby forming one of the many Mendelssohn-mathematics connections.33 We have some sketches of DirichIet and his teaching style during his Berlin years from Thomas Hirst, an English mathematician and diarist who spent much of the 1850s traveling in Europe, taking in mathematics wherever he could find it. During the fall and winter of 1852-1853 he was in Berlin, where he befriended DirichIet and attended his lectures.
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From page 95... ...
The title of DirichIet's groundbreaking paper was, I am sorry to say, Beweis des Satzes, class jede unbegrenzte arithmetische Progression, deren erstes Gl/ied und Differenz ganze Zahl/en ohne gemeinschaftl/ichen Factor sind, unendl/ich viel/e Primzahl/en enthdl/t "Proof of the theorem that each unlimited arithmetic progression, whose first member and difference are whole numbers without common factor, contains infinitely many prime numbers." Take any two positive whole numbers an(1 repetitively a(l(1 one to the other. If the two numbers have a common factor, every resulting number has that factor, too; repetitively a(l(ling 6 to 15 gives you 15, 21,27,33,39,45, .
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From page 96... ...
, but each of the six sequences contains the same proportion of primes. In other words, if you imagine each sequence stretching out to the neighborhood of some very large number N
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From page 97... ...
, pronounced"eight plus nine is congruent to five, module twelve." The phrase "module twelve" means "I am working from a clock-face with twelve hours marked, O to 11." This may seem trivial, but in fact the arithmetic of congruences goes very deep and is full of strange and difficult results. Gauss was a great grand master of it; not one of the seven sections of Disquisitiones Arithmeticae is free from that "_" sign.
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From page 98... ...
If it was Riemann who turned the key, it was DirichIet who first showed it to him and demonstrated that it was a key to something or other; and it is to DirichIet that the immortal glory of inventing analytic number theory properly belongs. But what, exactly, is this Golden Key?
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