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Mathematics: The New Golden Age (1988)

de Keith Devlin

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A modern classic by an accomplished mathematician and best-selling author has been updated to encompass and explain the recent headline-making advances in the field in non-technical terms.
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Ben scritto, meno dettagliato e ampio dell'equivalente scritto da Odifreddi (Matematica del '900), è un'ottima introduzione non tecnica ma seria. ( )
  kenshin79 | Jul 25, 2023 |
Devlin explains in the preface to this book that his aim was to highlight the breadth and excitement of current mathematical research by selecting 10 topics from the past 25 years in which significant developments had been made and which were capable of being described to the lay reader. It's an interesting challenge and I think he was at least partly successful. However, it's worth noting that this book was written in the mid-1980s, and published in 1988; "the past 25 years" refers to the period between 1960 and 1985. I'm not sure how this book managed to sit on my shelves unread for nearly 25 years; I'm a keen reader of this sort of popular exposition of mathematics. The time lag has made it an interesting read. Some topics in mathematics retain their interest and relevance over many decades or centuries and the methods used to tackle them may not change much in that time. Others wax and wane in interest or are transformed by theoretical or methodological advances. Given that Devlin intended to pick areas of research which were very active it is not surprising that the same is true of the topics he tackles here. Some of the descriptions remain current; others describe areas which have undergone significant transformation or where questions which were outstanding at the time Devlin was writing are now settled, often in unexpected ways.

This is perhaps best illustrated by considering the 4-colour problem, one of those to which Devlin devotes a chapter. It's one that pops up in many, many popular mathematics books over the years because its description is accessible even to children but its solution eluded many of mathematics' finest minds for 150 years. Before 1976 it was an interesting unsolved problem; a popular exposition might describe both the unsuccessful attempts of the past and speculate about how it might be solved in the future. After 1976, that question was settled. The problem had been solved by an unexpected means of attack - an exhaustive enumeration of 10,000 or so special cases and a mixture of hand and computer analysis which proved both that this set was complete and that each case consisted of an 'irreducible configuration.' It was a proof of a different and unexpected kind and some would argue that it transformed mathematics. Devlin was writing after this proof and so was able to consider its effect on mathematics more widely with nearly 10 years of hindsight. The same is not true of Fermat's Last Theorem, another of the topics he tackles. Unsolved at the time of writing, it was finally dealt with by Andrew Weil some 10 years later. Devlin make one successful prediction regarding the proof - he says that "if one is ever found, it will involve a great deal more than elementary considerations."

Enough about historical perspective. In addition to the two topics mentioned, Devlin also covers prime numbers and factoring, infinite sets and undecidable propositions, the class number problem, chaos theory and fractals, simple groups, Hilbert's tenth problem, knots, algorithmic efficiency and a collection of 'hard problems' in complex numbers including the Riemann hypothesis. You don't need to recognise all of these topics for this book to be interesting and accessible. But I suspect that if you have not heard of any of them you'll find the book very hard going. Devlin tries not to assume much knowledge on the part of the reader - he gives an explanation of complex numbers in chapter 3, for instance - but he does assume a familiarity with some basics of algebra and elements of mathematical notation. In some chapters he also moves rapidly from these basic explanations to some challenging concepts, a number of which defeated me on first reading despite having a degree in the subject.

These moments are rare, however. Overall this book does a good job of explaining the history of the problems discussed and describing many aspects of them which will be new even to those who may have encountered the topics in many similar books. It was new to me, for instance, to discover that the 4-space manifold has unique properties with regard to differentiation, something which has significant impact on much of theoretical physics given that this manifold describes space-time. The existence and accuracy of Heawood's formula (which places an upper limit on the number of colours needed for a map on a surface of a particular genus) was also new to me, as was its accuracy for every surface except the Klein Bottle.

Further reading is provided at the end of every chapter should you wish to investigate any of the topics in more depth. The book also has both an author index and subject index, an unusual but helpful division. Worth reading by the aspiring student of mathematics, those like me who studied it but moved elsewhere and those with interest and ability in the topic but without formal education beyond secondary school. ( )
  kevinashley | Aug 4, 2013 |
Indeholder "Preface", "Acknowledgements", "1. Prime Numbers, Factoring, and Secret Codes", "2. Sets, Infinity, and the Undecidable", "3. Number Systems and the Class Number Problem", "4. Beauty From Chaos", "5. Simple Groups", "6. Hilbert's Tenth Problem", "7. The Four-Colour Problem", "8. Fermat's Last Theorem", "9. Hard Problems About Complex Number", "10. Knots and Other Topological Matters", "11. The Efficiency of Algorithms", "Author Index", "Subject Index".

"Preface" handler om at han har udvalgt emner, der er blevet udviklet mellem 1960 og 1985 med overvægt af de yngre.
"Acknowledgements" handler om hvem der har hjulpet og at alle fejl er hans egne.
"1. Prime Numbers, Factoring, and Secret Codes" handler om primtal, ARCL primtalstest, Mersenne-primtal, Fermat-primtal, perfekte tal og RSA kryptering.
"2. Sets, Infinity, and the Undecidable" handler om uendelige mængder og et sødt bevis for at mængden af alle delmængder af en mængde har flere elementer end den oprindelige mængde, så R har fx større mægtighed end N.
"3. Number Systems and the Class Number Problem" handler om Heegner, klassetal og Gauss. Zagier og Gross fik åbenbart fod på det generelle klassetalsproblem i 1983.
"4. Beauty From Chaos" handler om fraktaler. Mandelbrot-mængder, Julia-mængder osv.
"5. Simple Groups" handler om simple grupper og problemet med at klassifikere dem. Smukt og enkelt forklaret. Galois grupper, undergrupper. Undergruppe defineret udfra a*a=e.
"6. Hilbert's Tenth Problem" handler om diofantiske ligninger og hænger sammen med beregnelighed.
"7. The Four-Colour Problem" handler om firfarveteoremet, som blev knust med en udtømmende søgning i et endeligt antal grafer.
"8. Fermat's Last Theorem" handler om x^n + y^n = z^n for heltal og n større end 2. Uløst da bogen her blev udgivet i 1988.
"9. Hard Problems About Complex Number" handler om Riemann-hypoesen, Farey-tallene og Martens-formodningen.
"10. Knots and Other Topological Matters" handler om knudeteori og hvordan man knytter en gruppe til en knude. Poincaré-formodningen. Manifolde.
"11. The Efficiency of Algorithms" handler om P=NP. Simplex-metoden, Karmarkar.
"Author Index" gør det muligt at finde ud af der står noget om G. Dantzig på side 267. Ret nyttigt, hvis man ikke læser bogen fra ende til anden og tager notater. Men fx står der noget om Cohen på side 40 uden at det er med i index'et, så helt til at stole på er det ikke.
"Subject Index" er et sædvanligt opslagsregister.

En glimrende bog med masser af spændende nye ting fra matematikkens nyere tid. Fx et polynomium, hvis positive værdier alle er primtal. Polynomiet er skruet sammen på en måde, der nogenlunde viser hvad der foregår og det er Devlin god til. Han viser glimt af hvad der foregår og stopper så der, hvor det bliver for langhåret for andre end professionelle matematikere.
Glimrende og inspirerende bog ( )
  bnielsen | Nov 17, 2008 |
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A modern classic by an accomplished mathematician and best-selling author has been updated to encompass and explain the recent headline-making advances in the field in non-technical terms.

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