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Inthe tradition of Fermat's Enigma and Pi, Marcus du Sautoy tells the illuminating, authoritative, and engagingstory of Bernhard Reimann and the ongoing quest tocapture the holy grail of mathematics--the formula to predict prime numbers.Oliver Sacks, author of The Man Who Mistook His Wife for a Hat, calls TheMusic of the Primes "an amazing book. . . . I could not put it down once Ihad started." Simon Winchester, author of The Professor and the Madman,writes, "this fascinating account, decoding the inscrutable language of themathematical priesthood, is written like the purest poetry. Marcus du Sautoy's enthusiasm shines through every line of this hymnto the joy of high intelligence, illuminating as it does so even the darkestcorners of his most arcane universe."… (plus d'informations)
I can say, with some modesty, that I am familiar with the subject of mathematics more than an average person. Despite that I hadn’t ever read a non-technical book on mathematics, so this book ended up being a very engrossing and a pleasant surprise. Marcus has done a tremendous job detailing history of search of the proof of Reimann Hypothesis interspersed with personal stories and personalities of mathematicians involved – one person at a time.
What is Reimann Hypothesis? I am not smart enough to explain, but I regurgitate what I gleaned from the book. As mathematicians are wont to do, they wanted to understand what and why certain numbers are prime and what is the distribution of prime numbers across number line. When that proved to be difficult of an endeavor, one of them decided to find pattern in number of prime numbers less than a certain number, say primes less than 100, prime less than 1000, and so on. There was some success in generating a formula for such counting (see later), but it wasn’t exact. Meanwhile when mathematician B. Riemann decided to pass on complex numbers to simple looking function called Zeta function (1+1/2^n+1/3^n+…) in 1850s, he observed that values of “n” for which function takes value of zero relate to counting of primes (I don’t understand how, though!). Moreover, he guessed that all such “n” have the real part of 1/2. This guess is the Riemann Hypothesis. Line passing through this point is called Reimann’s critical line.
Book gave me a fascinating glimpse into how theoretical mathematicians work, and how does mathematics advance. They often work by posing ad-hoc puzzles (“what happens when kick a rotating sphere of liquid”), trying new mathematical operations or statistics on existing formulas (“taking moments of function”, “passing imaginary numbers to known equations”), looking for patterns and trying to find equation and explanation of those patterns, translating numbers and equations from one branch of math to another (“from numbers to graph”), trying to prove intuitions, and aiming for elegant, closed form solutions and formulas to problems. To an application oriented engineer, this was innocently ridiculous – trying anything to anything and spending decades trying to make sense of it toward no immediately discernible end!
In doing so, mathematical properties sprout with educated guesses on observed patterns called ‘conjectures,’ which when supported by available data become ‘hypotheses’ which make a novel yet untested prediction. Hypotheses graduate to ‘theory’ non-trivial novel prediction comes true and proven mathematically, and eventually to ‘law’ when validated in multiple domains across time to multiple level of rigour with no scope for any further enhancement or exception.
Growth of mathematics is also, surprisingly, driven by friendly bets between prominent mathematicians. Bets, often of little material value, but taken seriously enough, is reason for many important advancements in the math. For instance, 100 million Reiman’s zeros were calculated by costing CPU processing in 1970s just to dis(prove) a bet! I wonder if this is true of other disciplines of science.
Till very recently before advent of internet exchange of ideas through letters was usual form of brainstorming and communication among mathematicians. Yet, many also worked in isolation through their rough sketches never sharing ideas till they were comfortable of elegance of proof, or of not being challenged or ridiculed, or not wanting to share credit of their discovery too early! Consequently, the amount of knowledge, which is lost to humanity, because their family couldn’t comprehend the importance of scribbles in their papers when they died and just threw away is heart wrenchingly disappointing. Ramanujan’s and Reimann’s notes on having figured out proofs but couldn’t be bothered to write them out because they thought them trivial have costed mathematics efforts of centuries.
Book underscores a unique manner of working of theoretical mathematicians – their rigorous reliance on proof. Unlike other spheres of science, where theoretical predictions are useless unless observed under experimentation, mathematicians are content with solid proof despite lack of observations, and as corollary, they still do want solid proof despite what would be overwhelming experimental observations in other sciences. And why is it so? Because math being math, incongruence with theory may occur even after trillions of observations – 10^10^10^34 is theoretical number when Gauss’s Prime Conjecture is proven false. This reliance of solid theoretical proofs also means that proofs once established don’t change forever. While theory of gravity has been refined every century, mathematical proofs established from 1600s have stood test of time.
Role of computers is hinted on later parts of books, but it is aptly concluded that computer can only help in disproving a hypothesis by providing counter examples but cannot prove a hypothesis simply because any number of matching examples (evidence) don’t count and cannot constitute a proof.
Now let’s come to the personalities of mathematicians narrated the book. If you imagine mathematicians to be hard working geniuses & nerds, you will not be wrong, on average. Like all geniuses, most of them show their signs before they turn ten. For instance, Gauss (of the Gaussian distribution) was correcting his father’s math at age of three where normal children do not know beyond counting. Most mathematicians are pacifists reveling in their own cocoon with little care or awareness of rest of the world. Some were eccentric – for instance, G H Hardy, lived his life fighting with, tempting, and challenging God, with often comical outcome where he ended up believing in God to prove that he didn’t believe in Him.
Ramanujan’s level of genius is, of course, unfathomable. If partition number is defined as number of ways to partition N things (partition number of 3 is 3, of 4 is 5) then he literally dreamt up formula to calculate partition number of any N just in flashing insight without intermediate steps which included square root, pi, sum, mod, natural logarithm, integration, differentiation, imaginary number, and trigonometry! If not Namagiri Goddess, then how?!
Of course, there are sprinkling of revolutionaries, womanizers, and party stars. While the book does reference tumults in the rest of world – world wars, famines, Nazi politics – and their impact on mathematicians, and hence on mathematics, they are not dwelled into more than mere mentions.
Incentives for theoretical mathematics are few and far – most of them do this simply for the joy and, occasionally, for the fame within mathematician community. There is no Nobel Prize in math. Field Medal is given only once in four years for living mathematics under forty years of age and accompanies just CAD 15,000. Opportunities for commercialization are rare – though of course, cryptography and encryptions are important exceptions. Makes you wonder how cheap and easy to sponsor a prize in math and yet why hasn’t anyone done so?
While reading the book, I also realized that my level of thinking as commoner, despite my admissions in the opening line of this review, is closer to what mathematicians were doing 400 years ago. Hence, what mathematicians are doing today, reflected in the later chapters of the book, went completely over my head! Marcus has done excellent job in talking about mathematics without using mathematics, but I do feel that book can be slightly better by exposing readers to real math – say equations or screenshots of seminal papers – even with clear expectations that reader won’t understand. Else there is risk of getting too lost in analogies!
Search for Reimann Hypothesis led to multiple discoveries and developments on the way, and in recent years even found connection to Quantum Physics, where distance between consecutive Reimann Zeros is found to mimic distance between energy levels in atoms! Was this coincidence or some underlying pattern? Being able to predict next number in moments of Reimann Zeta function by observed energy level of atoms confirmed this as a tangible connection. Not only is the universe stranger than we imagine, but it is also stranger than we can imagine!
Longevity of Reimann Hypothesis has given it an otherworldly status. Some mathematicians take this as challenge of ultimate proportion. Others shy away knowing that it may as well be end of their career. Many proofs in other areas of mathematics depend on Reimann Hypothesis being true. Will we ever prove this almost 200 hundred year old hypothesis? There were 300 million zeros discovered which follow Reimann Hypothesis – as of book’s writing in 2003 – and 10 trillion as of writing of this review, giving overwhelming evidence about its truth, but alas no solid conceptual proof exists yet. It’s even proved that there are many more zeros within an arbitrarily small band around critical line than outside, but that doesn’t prove that all zeros are on that line. It’s even proved that there are infinite zeros on the Reimann’s critical line, but that still doesn’t prove that all zeros are on that line.
Godel’s incompleteness theorem says that we cannot prove that same set of math axioms won’t lead to two different answers (contradictions), and there are going to be some true statements which cannot be proven. It’s amazing how mathematical logic can even prove such abstract generalized statements! A possible consequence of this is that Reimann’s Hypothesis may be one of those statements which cannot be proven. But mathematicians believe in elegance of solution and haven’t yet surrendered that such important hypothesis won’t have a proof.
* * * Below are some noteworthy facts about Prime numbers and Reimann hypotheses which have nothing to do with the book review but elicited aha for me and documented just for fun. - Number of primes up to number N is approximately 1/ln(2) + 1/ln(3) + … + 1/ln(N) - Harmonic series, 1/n, is called so because pot filled with water up to 1, 1/2, 1/3, … levels produce harmonious sound but not at other levels! - Mathematicians have created a 26 variable equation to generate all primes. This works when results are positive numbers, but most often, they are not. - AT&T played key role in mathematical research, even as telephone company they had little applications for mathematics, but it was not purely altruistic behaviour towards science & humanity, but a commercial and public relation consideration since they had restrictions on how much profit to absorb because they were a monopoly. - How to test if a number is Prime, without necessarily trying out all divisors? X^p=X(modulo p) only if p is Prime for any X. This is important to find keys for Public Key Cryptography. ( )
Some interesting biographical info about several of the key players. I didn't find the discussions of their work on primes that illuminating, however. It needed something akin to a "timeline of ideas" to show the various threads and how they were connected. ( )
Very nice history about prime numbers. Riemann hypothesis is not easy to explain it and more difficult to understand it. It's a pity that the demonstration was lost... ( )
We use prime numbers everyday, every internet transaction uses these for security, and de Sautoy here gives us a history of the prime number, from the very early Greek mathematicians, to the modern day.
The primary focus of the book is the Reimann Hypothesis. Du Sautoy gives a very readable history of all the mathematicians who have looked at proving and trying to disprove this. Some of the maths was a bit beyond me, but I did enjoy having my brain stretched. ( )
Wow, I am not mathematically inclined at all but this was a thrill to read. what a talent to bring complex mathematics and the prime numbers to more people. Thanks to Du Sautoy. This book enriched my life. ( )
Informations provenant du Partage des connaissances anglais.Modifiez pour passer à votre langue.
For the memory of Yonathan du Sautoy October 21, 2000
Premiers mots
Informations provenant du Partage des connaissances anglais.Modifiez pour passer à votre langue.
One hot and humid morning in August 1900, David Hilbert of the University of Göttingen addressed the International Congress of Mathematicians in a packed lecture hall at the Sorbonne, Paris.
Citations
Informations provenant du Partage des connaissances espagnol.Modifiez pour passer à votre langue.
La comprensión del significado de la creatividad matemática es, en el mejor de los casos, difícil, pero la forma de proceder de Ramanujan siempre tuvo algo de misterioso: afirmaba que la diosa Namagiri, protectora de su familia y consorte de Narashima, el dios león, cuarta encarnación de Vishnu, le aportaba sus ideas en sueños. En la aldea de Ramanujan algunos creían que la diosa tenía el poder de exorcizar los demonios; para Ramanujan, Namagiri era la explicación de los relámpagos de iluminación que desencadenaban su flujo ininterrumpido de descubrimientos matemáticos. Ramanujan no es el único ejemplo de matemático para quien el mundo de los sueños resulta ser un territorio fértil para la exploración matemática. Dirichlet tenía las Disquisitiones arithmeticae bajo la almohada, esperando recibir la inspiración para comprender las afirmaciones a menudo crípticas que contenía el libro. En los sueños es como si la mente se liberara de las barreras del mundo real y tuviera la libertad de abrir caminos que se excluyen en estado consciente. Ramanujan parecía capaz de inducir este estado onírico en sus horas de vigilia: un trance así está muy cerca del estado mental que la mayoría de los matemáticos intenta conseguir.
Derniers mots
Informations provenant du Partage des connaissances anglais.Modifiez pour passer à votre langue.
We still await the person whose name will live for ever as the mathematician who made the primes sing.
Inthe tradition of Fermat's Enigma and Pi, Marcus du Sautoy tells the illuminating, authoritative, and engagingstory of Bernhard Reimann and the ongoing quest tocapture the holy grail of mathematics--the formula to predict prime numbers.Oliver Sacks, author of The Man Who Mistook His Wife for a Hat, calls TheMusic of the Primes "an amazing book. . . . I could not put it down once Ihad started." Simon Winchester, author of The Professor and the Madman,writes, "this fascinating account, decoding the inscrutable language of themathematical priesthood, is written like the purest poetry. Marcus du Sautoy's enthusiasm shines through every line of this hymnto the joy of high intelligence, illuminating as it does so even the darkestcorners of his most arcane universe."
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What is Reimann Hypothesis? I am not smart enough to explain, but I regurgitate what I gleaned from the book. As mathematicians are wont to do, they wanted to understand what and why certain numbers are prime and what is the distribution of prime numbers across number line. When that proved to be difficult of an endeavor, one of them decided to find pattern in number of prime numbers less than a certain number, say primes less than 100, prime less than 1000, and so on. There was some success in generating a formula for such counting (see later), but it wasn’t exact. Meanwhile when mathematician B. Riemann decided to pass on complex numbers to simple looking function called Zeta function (1+1/2^n+1/3^n+…) in 1850s, he observed that values of “n” for which function takes value of zero relate to counting of primes (I don’t understand how, though!). Moreover, he guessed that all such “n” have the real part of 1/2. This guess is the Riemann Hypothesis. Line passing through this point is called Reimann’s critical line.
Book gave me a fascinating glimpse into how theoretical mathematicians work, and how does mathematics advance. They often work by posing ad-hoc puzzles (“what happens when kick a rotating sphere of liquid”), trying new mathematical operations or statistics on existing formulas (“taking moments of function”, “passing imaginary numbers to known equations”), looking for patterns and trying to find equation and explanation of those patterns, translating numbers and equations from one branch of math to another (“from numbers to graph”), trying to prove intuitions, and aiming for elegant, closed form solutions and formulas to problems. To an application oriented engineer, this was innocently ridiculous – trying anything to anything and spending decades trying to make sense of it toward no immediately discernible end!
In doing so, mathematical properties sprout with educated guesses on observed patterns called ‘conjectures,’ which when supported by available data become ‘hypotheses’ which make a novel yet untested prediction. Hypotheses graduate to ‘theory’ non-trivial novel prediction comes true and proven mathematically, and eventually to ‘law’ when validated in multiple domains across time to multiple level of rigour with no scope for any further enhancement or exception.
Growth of mathematics is also, surprisingly, driven by friendly bets between prominent mathematicians. Bets, often of little material value, but taken seriously enough, is reason for many important advancements in the math. For instance, 100 million Reiman’s zeros were calculated by costing CPU processing in 1970s just to dis(prove) a bet! I wonder if this is true of other disciplines of science.
Till very recently before advent of internet exchange of ideas through letters was usual form of brainstorming and communication among mathematicians. Yet, many also worked in isolation through their rough sketches never sharing ideas till they were comfortable of elegance of proof, or of not being challenged or ridiculed, or not wanting to share credit of their discovery too early! Consequently, the amount of knowledge, which is lost to humanity, because their family couldn’t comprehend the importance of scribbles in their papers when they died and just threw away is heart wrenchingly disappointing. Ramanujan’s and Reimann’s notes on having figured out proofs but couldn’t be bothered to write them out because they thought them trivial have costed mathematics efforts of centuries.
Book underscores a unique manner of working of theoretical mathematicians – their rigorous reliance on proof. Unlike other spheres of science, where theoretical predictions are useless unless observed under experimentation, mathematicians are content with solid proof despite lack of observations, and as corollary, they still do want solid proof despite what would be overwhelming experimental observations in other sciences. And why is it so? Because math being math, incongruence with theory may occur even after trillions of observations – 10^10^10^34 is theoretical number when Gauss’s Prime Conjecture is proven false. This reliance of solid theoretical proofs also means that proofs once established don’t change forever. While theory of gravity has been refined every century, mathematical proofs established from 1600s have stood test of time.
Role of computers is hinted on later parts of books, but it is aptly concluded that computer can only help in disproving a hypothesis by providing counter examples but cannot prove a hypothesis simply because any number of matching examples (evidence) don’t count and cannot constitute a proof.
Now let’s come to the personalities of mathematicians narrated the book. If you imagine mathematicians to be hard working geniuses & nerds, you will not be wrong, on average. Like all geniuses, most of them show their signs before they turn ten. For instance, Gauss (of the Gaussian distribution) was correcting his father’s math at age of three where normal children do not know beyond counting. Most mathematicians are pacifists reveling in their own cocoon with little care or awareness of rest of the world. Some were eccentric – for instance, G H Hardy, lived his life fighting with, tempting, and challenging God, with often comical outcome where he ended up believing in God to prove that he didn’t believe in Him.
Ramanujan’s level of genius is, of course, unfathomable. If partition number is defined as number of ways to partition N things (partition number of 3 is 3, of 4 is 5) then he literally dreamt up formula to calculate partition number of any N just in flashing insight without intermediate steps which included square root, pi, sum, mod, natural logarithm, integration, differentiation, imaginary number, and trigonometry! If not Namagiri Goddess, then how?!
Of course, there are sprinkling of revolutionaries, womanizers, and party stars. While the book does reference tumults in the rest of world – world wars, famines, Nazi politics – and their impact on mathematicians, and hence on mathematics, they are not dwelled into more than mere mentions.
Incentives for theoretical mathematics are few and far – most of them do this simply for the joy and, occasionally, for the fame within mathematician community. There is no Nobel Prize in math. Field Medal is given only once in four years for living mathematics under forty years of age and accompanies just CAD 15,000. Opportunities for commercialization are rare – though of course, cryptography and encryptions are important exceptions. Makes you wonder how cheap and easy to sponsor a prize in math and yet why hasn’t anyone done so?
While reading the book, I also realized that my level of thinking as commoner, despite my admissions in the opening line of this review, is closer to what mathematicians were doing 400 years ago. Hence, what mathematicians are doing today, reflected in the later chapters of the book, went completely over my head! Marcus has done excellent job in talking about mathematics without using mathematics, but I do feel that book can be slightly better by exposing readers to real math – say equations or screenshots of seminal papers – even with clear expectations that reader won’t understand. Else there is risk of getting too lost in analogies!
Search for Reimann Hypothesis led to multiple discoveries and developments on the way, and in recent years even found connection to Quantum Physics, where distance between consecutive Reimann Zeros is found to mimic distance between energy levels in atoms! Was this coincidence or some underlying pattern? Being able to predict next number in moments of Reimann Zeta function by observed energy level of atoms confirmed this as a tangible connection. Not only is the universe stranger than we imagine, but it is also stranger than we can imagine!
Longevity of Reimann Hypothesis has given it an otherworldly status. Some mathematicians take this as challenge of ultimate proportion. Others shy away knowing that it may as well be end of their career. Many proofs in other areas of mathematics depend on Reimann Hypothesis being true. Will we ever prove this almost 200 hundred year old hypothesis? There were 300 million zeros discovered which follow Reimann Hypothesis – as of book’s writing in 2003 – and 10 trillion as of writing of this review, giving overwhelming evidence about its truth, but alas no solid conceptual proof exists yet. It’s even proved that there are many more zeros within an arbitrarily small band around critical line than outside, but that doesn’t prove that all zeros are on that line. It’s even proved that there are infinite zeros on the Reimann’s critical line, but that still doesn’t prove that all zeros are on that line.
Godel’s incompleteness theorem says that we cannot prove that same set of math axioms won’t lead to two different answers (contradictions), and there are going to be some true statements which cannot be proven. It’s amazing how mathematical logic can even prove such abstract generalized statements! A possible consequence of this is that Reimann’s Hypothesis may be one of those statements which cannot be proven. But mathematicians believe in elegance of solution and haven’t yet surrendered that such important hypothesis won’t have a proof.
* * *
Below are some noteworthy facts about Prime numbers and Reimann hypotheses which have nothing to do with the book review but elicited aha for me and documented just for fun.
- Number of primes up to number N is approximately 1/ln(2) + 1/ln(3) + … + 1/ln(N)
- Harmonic series, 1/n, is called so because pot filled with water up to 1, 1/2, 1/3, … levels produce harmonious sound but not at other levels!
- Mathematicians have created a 26 variable equation to generate all primes. This works when results are positive numbers, but most often, they are not.
- AT&T played key role in mathematical research, even as telephone company they had little applications for mathematics, but it was not purely altruistic behaviour towards science & humanity, but a commercial and public relation consideration since they had restrictions on how much profit to absorb because they were a monopoly.
- How to test if a number is Prime, without necessarily trying out all divisors? X^p=X(modulo p) only if p is Prime for any X. This is important to find keys for Public Key Cryptography. (