Keith DevlinReseñas

Interesting, but slow going. For some reason this book was a bit unsatisfying. Something was lacking in the way it was organized.

Indeholder "Preface", "Zero. The Gauntlet is Thrown", "One. The Music of the Primes: The Riemann Hypothesis", " Appendix 1. Euklid's proof that there are infinitely many primes", " Appendix 2. How do mathematicians work out infinite sums?", " Appendix 3. How Euler discovered the Zeta function", "Two. The Fields We Are Made Of: Yang-Mills Theory and the Mass Gap Hypothesis", " Appendix. Group Theory: The Mathematics of Symmetry", "Three. When Computers Fail: The P versus NP Problem", "Four. Making Waves: The Navier-Stokes Equations", "Five. The Mathematics of Smooth Behavior: The Poincaré Conjecture", "Six. Knowing When the Equation can't Be Solved: The Birch and Swinnerton-Dyer Conjecture", " Appendix: Notation for infinite sums and products", "Seven. Geometry Without Pictures: The Hodge Conjecture", "Further Reading", "Index".

"Preface" handler om at Clay Institute i maj 2000 annoncerede syv priser på hver en million dollar for løsningen på syv vanskelige og vigtige matematiske problemer. Det gav efterspørgsel på en bog om problemerne. Faktisk mindst to, for Clay Institute bad Keith Devlin og Ian Stewart om at skrive en generel introduktion til hvert problem i den officielle problemformulering. Denne bog er anderledes, for ideen er blot at give en ide om hvad de syv problemer går ud på. Udover Clay Math Institute bogen kan man også finde en film på www.claymath.org som forsøger det samme.
"Zero. The Gauntlet is Thrown" handler om ???
"One. The Music of the Primes: The Riemann Hypothesis" handler om ???
" Appendix 1. Euklid's proof that there are infinitely many primes" handler om ???
" Appendix 2. How do mathematicians work out infinite sums?" handler om ???
" Appendix 3. How Euler discovered the Zeta function" handler om ???
"Two. The Fields We Are Made Of: Yang-Mills Theory and the Mass Gap Hypothesis" handler om ???
" Appendix. Group Theory: The Mathematics of Symmetry" handler om ???
"Three. When Computers Fail: The P versus NP Problem" handler om ???
"Four. Making Waves: The Navier-Stokes Equations" handler om ???
"Five. The Mathematics of Smooth Behavior: The Poincaré Conjecture" handler om ???
"Six. Knowing When the Equation can't Be Solved: The Birch and Swinnerton-Dyer Conjecture" handler om ???
" Appendix: Notation for infinite sums and products" handler om ???
"Seven. Geometry Without Pictures: The Hodge Conjecture" handler om ???
"Further Reading" handler om forslag til yderligere læsning.
"Index" er et almindeligt opslagsregister.

Clay Mathematical Institute udlovede i 2000 syv priser på hver en million dollar. Til hver pris hørte en lille opgave med at løse et matematisk problem, som ingen tidligere havde løst.½

Like many philosophers, it seems best to read about Descartes rather than trying to read what he wrote himself.

Ben scritto, meno dettagliato e ampio dell'equivalente scritto da Odifreddi (Matematica del '900), è un'ottima introduzione non tecnica ma seria.

Interessante

Devlin posited some interesting ideas here, but the thing that I liked the most was his sheer passion about math and his desire to share that love with others. It worked, too -- I'm picking up some books on math so I can further my education and maybe catch some of that love. Definitely worth reading.

Kept feeling it's much ado about nothing. I understand the stats and the significance, but I think the author is a bit overexcited. However, the descriptions of the historical characters are quite interesting.

Very nice looking book, well-written. As is usual w/ books about math that actually include even a little math, it is slow going.

Detailed Review:

* Prologue: Gives the means of arriving at the definition, a science of patterns, and also emphasizes the importance of abstraction and notation. Remarks that computers have allowed visualization that was not possible before; Julia sets are older than our ability to easily see what they look like.

* Chapter 1: Counting
Covers the history of the development of the concept of number, as far as can be inferred from archaeological evidence. The Rhind papyrus has a description of how to calculate the volume of the frustum of a pyramid. The numbers used are concrete; but it it s clear that the reader is expected to consider them variables as necessary. A modern day person would use integration to derive a formula for the volume.

Moving on to the Greeks, who were really most interested in geometry. Discusses the Pythagorean theorem and the interpretation of numbers as lengths. The formula for (a + b)^2 and (a -b)^2 is presented as the Greeks would have understood it, geometrically. A proof of the irrationality of root 2 is shown; is that how the Greeks discovered it? A discussion of prime numbers; evidently the fundamental theorem of arithmetic is in Euclid's work, as well as the proof that there is an infinite number of primes.

The primes allow us to leap 1000 or so years forward, and discuss the prime density function, the ratio of the number of primes less than or equal to N to N. Wikipedia discusses the prime _counting_ function: https://en.wikipedia.org/wiki/Prime-counting_function. The proof that the number of primes are infinite is well presented; I've seen other place where the proof fails to properly cover the two possibilities, leaving the reader perplexed and uncertain.

Chebyshev discovers that between every number and its double there is at least one prime. And finally, an upper bound, on the function, 1/ln N, was proved.

The next topic, Finite Arithmetic, is a discussion of abstract algebra. It was then I wanted to find my Abstract Algebra textbook, only to discover that it was almost certainly in my currently inaccessible cubicle.

Chapter 4: Shape
About geometry and algebraic geometry. There is a solid discussion of Euclid and a sidebar on the golden ratio with some unexplained relationships. The regular solids and conic sections are discussed. Properties are described w/out much justification but with fine illustrations. Kepler's wierd obsession with the plantonic solids and the planetary orbits gets a mention. Then algebra is introduced, and there is a perplexing discussion of a general formula for a circle, that yields no insight to me (p.118). Old problems of compass and straightedge construction of various figures are discussed. These all rely on the fact that compass and straightedge constructions only yield a subset of mathematical operations; basic arithmetic and the taking of square roots, and that all the solutions require a calculation that uses more than these. This can't be covered in detail in a book like this, of course. The next step is non-Euclidean geometries: hyperbolic, spherical, Riemannian, projective.

Keith Devlin con questo libro aveva deciso di prendersi una pausa dal suo lavoro di matematico, con la scusa che invecchiando ci si inaridisce, e ha scritto questa "biografia" di Fibonacci. Il guaio è che noi non abbiamo praticamente nessuna notizia biografica del nostro, e quindi tutto il testo è costellato di frasi del tipo "mah, dovrebbe essere successo questo ma non è che lo sappiamo" che alla lunga diventano stucchevoli. Quello che può al limite interessare sono le supposizioni su quale sia effettivamente stato il ruolo di Fibonacci nel fiorire dei libri dei maestri d'abaco: un po' poco, diciamocelo. Nulla da eccepire sulla traduzione di Daniele Didero.

"The Man of Numbers: Fibonacci's Arithmetic Revolution" enthusiastically summarizes the little that is known about Leonardo of Pisa, later more famously called Fibonacci. Those who read medieval primary texts have become used to the dearth of direct evidence related to such texts, as well as the admirable, if Herculean, labors medievalists are forced to perform to prove the most basic biographical details. In the case of Leonardo of Pisa, the proof for his role in the "arithmetic revolution" has been fairly well-established, and is nicely summarized here. Well-known for his Fibonacci sequence, his greater contribution may have been the role he played in the transmission of arithmetic and algebra from Moslem North Africa to medieval Italy. Interestingly, this transmission appears to have proceeded along two tracks: First, in a formal, Latin primer on algebra--the famous Liber Abaci (1202)--via the educated elite, and second, through transmission to the Tuscan mercantile community in a format more suitable for the problems that would interest them via a lost primer--Di minor guisa--on commercial arithmetic for the "abbacus schools." Thus, Leonardo of Pisa seems to have played a significant role in both the rebirth of classical arithmetic and science, and the economic revolution that was already beginning to pull much of Italy into its cultural renaissance.

I read this book long ago. This is probably as close as one can get to give a light overiview of the seven problems recognised by the clay institute for a million dollar prize. The author here takes up an impossible task of explaining these problems to a lay audience. Even if he didn't entirely succeed in this, this book can be used to spark someone's interest for deeper study. Worth the read at least for the chapters on Riemann hypothesis and the P vs NP problem.

A slim volume, but well worth reading. Little is known about Leonardo of Pisa's life, but much more is now known of his legacy and the era in which he lived. It also gives a glimpse how mathematical notation changed and became even more symbolic since his time.

As someone who spends no time ever trying to solve these things, I found it to be a really approachable summary of what the problems were and roughly the techniques that one must be fluent in to stand a reasonable chance of understanding them 'really well'. Gave good references for further reading too. I liked Devlin's style, he was conscious not to get too bogged down whilst not being so high level as to be irritating (I thought anyway). If you read this book within a week or two I;d suggest its one fo the few times anyone would ever think about all 7 major problems in quite such quick succession (well - 6 now after Poincare's conjecture was proven - can I get a refund for that one?)...

Indeholder "Preface", "Chapter I. Naive set theory", " 1. What is a set?", " 2. Operations on sets", " 3. Notation for sets", " 4. Sets of sets", " 5. Relations", " 6. Functions", " 7. Well-orderings and ordinals", "Chapter II. The Zermelo-Fraenkel axioms", " 1. The language of set theory", " 2. The cumulative hierarchy of sets", " 3. Zermelo-Fraenkel set theory", " 4. Axioms for set theory", " 5. Summary of the Zermelo-Fraenkel axioms", " 6. Classes", " 7. Set theory as an axiomatic theory", " 8. The recursion principle", " 9. The axiom of choice", "Chapter III. Ordinal and cardinal numbers", " 1. Ordinal numbers", " 2. Addition of ordinals", " 3. Multiplication of ordinals", " 4. Sequences of ordinals", " 5. Ordinal exponentiation", " 6. Cardinality. Cardinal numbers", " 7. Arithmetic of cardinal numbers", " 8. Cofinality. Singular and regular cardinals", " 9. Cardinal exponentiation", " 10. Inaccessible cardinals", "Chapter IV. Some topics in pure set theory", " 1. The Borel hierarchy", " 2. Closed unbounded sets", " 3. Stationary sets and regressive functions", " 4. Trees", " 5. Extensions of Lebesgue measure", " 6. A result about the GCH", "Chapter V. The axiom of constructibility", " 1. Constructible sets", " 2. The constructible hierarchy", " 3. The axiom of constructibility", " 4. The consistency of constructible set theory", " 5. Use of the axiom of constructibility", "Chapter VI. Independence proofs in set theory", " 1. Some examples of undecidable statements", " 2. The idea of a boolean-valued universe", " 3. The boolean-valued universe", " 4. V^B and V", " 5. Boolean-valued sets and independence proofs", " 6. The non-provability of CH", "Bibliography", "Glossary of notation", "Index".

Mængdelære på et rimeligt abstrakt niveau.

Dated and too simple for me, but not a bad book for ppl who are scared of math.  Very attractive, with bright pictures, etc.

If you like math (especially probability), you'll like this book about two of the fields giant thinkers.

Not bad - sort of an introduction to the major themes of mathematics. It has some history but it doesn't get bogged down by trying to stay chronological or include every historical detail. It sort of "whets the appetite" for studying math beyond the textbook.

It's good. It was fun. It just didn't grab me. I think I wanted more detail. I felt like it was little too popularized or "dumbed down". And yet, I didn't feel it was "sparkly" enough to appeal to students.

Entertaining but not practical for classroom use.

Yes, I did feel real sad after reading this book. Reason: Why weren't we taught Mathematics the way the author teaches in this book?

Author's work on making users understand Calculus is simply amazing in this book. Highly recommended for anyone who wants to be in touch with and for those who're 'afraid' of Math.

Nice book introducing the logic of proofs and thinking outside the box. Lots of interesting easy exercises to do, if you are so minded.

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.

A slim volume, but well worth reading. Little is known about Leonardo of Pisa's life, but much more is now known of his legacy and the era in which he lived. It also gives a glimpse how mathematical notation changed and became even more symbolic since his time.

Keith Devlin’s book “Introduction to mathematical thinking.” is the textbook, the unnecessary and ridiculously inexpensive ($10) textbook, for his class on Coursera.com. The class obviously designed to be an introduction to mathematical thinking, a transition from the problem solving math of secondary school to college level mathematics where simply finding an answer is not the final goal.

I wanted to take a look at the free, non-credit classes that Coursera offers and this looked like a good one to try. It has been 30 years since I last took college calculus, and I have not looked at a math book since then. I knew I could do the work, I wanted to see just how a free, non-credit class, with 50,000 students worked.

Both the class and the book are excellent. Devlin begins by showing us that imprecision is often acceptable in spoken English. “One American dies every hour from heart disease” is his favorite example. Literally it says that there is one single American who dies, and apparently recovers, from heart disease every hour. We all understand the true meaning because in English we have background knowledge which allows us to make sense from nonsense based on the context. Mathematics requires precision because with it we will be dealing with concepts with which we do not have the background to guide our understanding.

Dr. Devlin focused on developing logical thinking and managed to arrange the lessons and exercises such that the mathematical logic required quickly evolves from simple “and” “or” statements into doing formal proofs, no small feat for a class only seven weeks long. The book was not really necessary for the class, the video lectures were very close to the text and the problem sets could be printed out and worked on offline but I do feel that having the book helped me. I find it impossible to highlight a point in a video lecture.

I was concerned about how a class this large could be taught, in all the math classes I have taken the learning takes place not during the lecture but answering questions that come up after attempting to work problems related to the lecture. Advanced students were recruited to act as Teaching Assistants to keep an eye on the discussion threads and answer questions when they could. Threads that had heavy traffic were brought to professor Devlin’s attention and he, most often, just confirmed what the TA’s had said. Learning directly from the book is possible but would mean losing what for me was the most helpful part of the class, the forums.

I learned a lot taking the class, principally that you can forget a lot in thirty years. I was over a week behind in the work and struggling not to fall farther behind when it came time for the final exam. Needless to say I will not be getting a certificate of completion. I also learned several other things, Coursera classes really are college level, someone only a few years out of high school would have had to work to finish the class and would have been prepared for higher mathematics classes and would be better able to think logically.

It was expected that out of the over 50,000 students that started the class as few as 5,000 would successfully finish it but as Dr. Devlin pointed out, teaching 5,000 students in traditional classes of 25 students would represent one professor’s entire career. This one class introductory level class freed up one career to teach higher level classes.

The Man of Numbers, by Keith Devlin, is an account of Leonardo of Pisa, better
known as "Fibonacci". Leonardo is best known for the number sequence, the
"Fibonacci Numbers", named after him. (1, 1, 2, 3, 5, 8, 13, 21, 34, ... Can
you guess the pattern?)

Far more important than this sequence, however, was Leonardo's introduction of
the familiar Arabic numerals to Europe. These are the numbers (0, 1, 2, 3, 4,
5, 6,...) that we use now for nearly everything, and they replaced the older
Roman numerals (I, II, III, IV, V, VI,...) that were in use in Europe prior to
the thirteenth century.

The unfortunate fact is that very little is known about Leonardo, apart from
some of his writing. This makes his story rather difficult to tell, so Devlin
makes up for the lack of hard data by describing life during Leonardo's time,
and speculating intelligently about various aspects of his education, travels
and motivations for his work. Most interestingly, he describes the tremendous
impact the introduction of Arabic numerals had on Western culture, and the way
ordinary calculation was so profoundly affected.

Devlin has a well-earned reputation as a master of telling mathematical
stories, and while I would not consider it his best work, this book does not
disappoint on that score.

In 1654, Blaise Pascal, the precocious son of a French tax collector sent Pierre de Fermat a letter with a proposed solution to an old but still lingering problem. The problem goes thusly:

Suppose two people are playing consecutive rounds of a game of chance. There is an agreed upon number of rounds a player has to win in order to take the pot. For each round, they each pay into the pot equally. At some point in the game, they are interrupted and the pot has to be divided. Without knowing the current “score”, how do you divide the pot fairly?

This was known as “the problem of the points” in Pascal’s day. The exchange of letters between Pascal and Fermat started the development of mathematics into the realm of probabilities and applied statistics.

Keith Devlin’s Unfinished Game uses these letters as a springboard to showcase the history of applied statistics. From John Graunt to the Bernoulli family to Gauss and Bayes, he explores the implications of the letters’ solutions. Interestingly enough, no one had thought to use mathematics to make prediction about human population or games of chance before.

The book is succinctly written, and can easily be knocked out in an afternoon. There’s a bit of jumping around in history, and Devlin tries competently to weave together the threads of discovery. There are some digressions, however, that he gets carried away with (most notably, the bit about DNA and prosecutorial fallacies), but overall, the book does a decent job of explaining statistical history. It’s hard to fathom that a single set of letters marked the starting point of both the modern insurance racket and mathematical epidemiology.

http://lifelongdewey.wordpress.com/2012/04/13/519-the-unfinished-game-by-keith-d...½

As a young man Leonardo of Pisa, aka Fibonacci, went into the family trading business, which required knowledge of arithmetic. Between the ages of 10 and 12 his parents sent him to a religious school to learn to read and write, and to learn the Roman system of arithmetic.

In Pisa at the time, and throughout Europe in the late twelfth century, religious schools were the only schools, and only boys were accepted as students. Wax tablets were used for used for writing, and the reading board, a type of abacus, was used in arithmetic using the Roman system, and its Roman numerals.

Many children today struggle to learn to add, subtract, multiply, divide, and take percentages. Think how much harder basic arithmetic was in Leonardo's time, using the Roman system of arithmetic. Leonardo's great contribution to the advancement of knowledge in the West was the introduction of the algorithms for basic arithmetic using the Hindu-Arabic system, with its ten place-valued digits.

Sometime in the 1180's Leonardo's father took a diplomatic post in the Islamic port of Bugia on North Africa's Barbary Coast. Leonardo followed him there a year later, and during his stay learned the Hindu-Arabic system of arithmetic.

In 1202 Leonardo completed the first edition of Liber Abbacci, a book that literally changed the Western World. No copies of this first edition survive, but three copies of the second edition, completed in 1228, still survive. Our current use of the Hindu-Arabic system for arithmetic in the West can be traced back directly to Liber Abbacci, and the multitude of later books more or less based on it.

The mathematical content of this book, as little as there is, is interesting. But the historical content overwhelms the mathematical, and most of the book is about life in the twelfth and thirteenth centuries. I was looking to find more mathematics, but was not disappointed when I did not find it. Highly recommended for anyone interested in mathematics, or history, and especially for those interested in the history of mathematics.