Recently a student of mine – a middle school teacher – asserted that he did not believe that 0.999… = 1. I tried several methods of convincing him that he was wrong; for example calculating 1 = 3(1/3) = 3(0.333…) = 0.999…, and showing that 0.999 … < 1 led to a contradiction. I browbeat him into submission, but I don’t think I convinced him. After reading the book under review, I see I could have handled things better. I learned that in non-standard analysis, 0.999 … < 1 is true, and more importantly I learned that understanding even the simplest infinite processes involves mastering some tricky metaphors.
This is an important and flawed book. It has generated much commentary, pro and con, and if your curiosity is whetted by my review I suggest you check the reviews of the book in Amazon.com. Like The Number Sense by Stanislaus Dehaene (reviewed below), this book applies cognitive science to an analysis of mathematical thought. But unlike that book, Where Mathematics Comes From goes far beyond an analysis of arithmetic skill, and analyzes some very sophisticated concepts of higher mathematics. Also, compared with Dehaene’s book, this book does not depend very much on laboratory science, but depends almost entirely on theoretical cognitive science.
This is a long book, and a somewhat difficult read, unless you happen to be well versed in the jargon of both mathematics and cognitive science. Even though the authors cover a lot of territory in 450 pages, and the quality of the writing is generally good, I had the feeling that I was reading the same thing over and over. If you want an “executive summary” to get the gist of what the authors believe they have accomplished, I suggest looking at the section “A Portrait of Mathematics” on pages 377 – 379.
In the first four chapters the authors describe the brain’s innate arithmetic, which is quite rudimentary and similar to the innate arithmetic of many other species, and then details how humans have learned to extend these basic concepts to an arithmetic which enables efficient calculation and obeys certain “laws”, such as the commutative property for addition. This section sets the tone for the rest of the book, by introducing basic cognitive mechanisms that the authors believe explain how mathematics is invented and understood. These include grounding metaphors, which yield basic, directly grounded ideas, and linking metaphors, which yield abstract ideas.
Chapters 5 – 7 study the linking metaphors that determine algebra, logic, and sets, and chapters 8 – 11 deal with “The Embodiment of Infinity”. The concept of infinity underlies most of modern mathematics including various number systems (integers, rationals, real numbers, and complex numbers). It appears in many different guises from points at infinity in projective geometry to cardinal and ordinal infinities in Cantor’s theory. The authors introduce a “basic metaphor of infinity” (BMI) that is supposed to account for our understanding of all these concepts. Chapter 11, “Infinitesimals”, is perhaps the most mathematically interesting part of the book. The authors present the hyperreal numbers of Robinson and Keisler, which include infinitesimal and huge quantities and provide be an intuitive and direct way of dealing with calculus. They also introduce a system of their own invention, the granular numbers, which is a subset of the hyperreal numbers that seems to be easier to use.
Chapters 12 – 14 continue the discussion of the infinite by critiquing the program of modern analysis that was pioneered by Dedekind and Weierstrass and continues to this day. While the authors profess the highest admiration for the intellectual achievements of these men and their followers, the subtext here is that their (Dedekind’s and Weierstrass’) purpose in separating analysis from its roots based in an intuitive geometric understanding was to obfuscate the subject and make mathematics the preserve of a specially trained elite. I disagree. My understanding is that the development of technology was beginning to make the traditional conceptions of mathematics inadequate. For example, on page 307 the authors approvingly cite James Pierpont’s (1899) list of “prototypical properties of a curve” including that it is continuous and has a tangent. With this definition, it would be difficult to have a consistent theory of Fourier series necessary for the analysis of radio waves or to solve differential equation with a driving function given by a step function in electrical engineering. The Mandelbrot set and related constructions are rooted in the real world (Mandelbrot’s prototypical example is the coastline of England.) and these “monstrous” sets seem to describe nature better than classical curves and regions.
Chapters 15 and 16 contrast the author’s implications for a philosophy of mathematics that is grounded on the human mind –“embodied mathematics”– with other philosophies that either posit mathematics existing outside the real world (Platonism and “the romance of mathematics”) or see mathematics the manipulation of essentially meaningless strings according to given rules (formalism) or as a cultural construct. I will describe my own view in a separate posting.
The last section of the book is a “case study” in four parts shows that the methods developed in the book can be used successfully to teach mathematics in a way that focuses on meaning. The subject is Euler’s famous equation e^(i*pi) = -1. This is quite good, though I might do a few things differently
E5. Duck, Duck, Goose
This problem was sent to me (as is) by Walter Carter of Seattle.
Some children have made up a simple version of the game “Duck, Duck, Goose”. In this game a group of people stands in a circle, and the person who is “it” taps the first person on the shoulder and says “duck”. The next person is tapped and called “duck” The next person is tapped and called “goose”, and the process is repeated. Every person who is called “goose” must sit down when they are tapped.
If there are a million people in a circle, and they are labeled sequentially from 1 to 1,000,000, and the tapper starts at person 1 going around and around until only one person is left standing, then what is that last person’s number?
Review of the Number Sense
The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997
This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.
The book is organized into nine chapters:
Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.
Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.
Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.
Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.
Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.
Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.
Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.
Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.
Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.
A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.
It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.
I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.
I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.
Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.
In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.
Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.
This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.
The book is organized into nine chapters:
Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.
Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.
Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.
Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.
Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.
Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.
Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.
Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.
Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.
A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.
It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.
I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.
I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.
Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.
In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.
Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.
Physical Models for Non-Euclidean Geometry
I strongly believe in the use of physical models, whenever possible, to introduce mathematical concepts. For example, when teaching non-Euclidean geometry to high school teachers, I like to have them create triangles on actual physical spheres, using rubber balls, push pins, and rubber bands to create geodesics. It is easy to “discover” that the sum of the angles in spherical triangles is greater than 180 degrees, and that the excess of a triangle (the sum of the angles minus 180 degrees) is additive and hence proportional to the area of the triangle. These demonstrations can be easily done for specialized triangles, so the student becomes familiar with the geometric fact before thinking about how it might be proven.
What I would like to do next is to show that the sum of the angles in a triangle on a surface of negative curvature is less than 180 degrees. This leaves me with two problems.
(1) How can I make (or obtain) a physical model of a simple saddle surface for experimentation by students. Ideally, models should be cheap enough so that I can supply each pair of students with a model to work with.
(2) How can students draw geodesics on such a surface? Rubber bands are not going to work here, because a band stretched between two points on the surface will not necessarily lie on the surface. This problem is sort of mathematical, because I think a good understanding of the nature of geodesics should lead to discovering a way of having students create them on a surface of negative curvature.
I am aware of some very good software that uses the Poincare disk model to do geometry on the hyperbolic plane, and I plan to use the software when I teach. But I want students to have real physical experience first.
Anyone have any ideas?
What I would like to do next is to show that the sum of the angles in a triangle on a surface of negative curvature is less than 180 degrees. This leaves me with two problems.
(1) How can I make (or obtain) a physical model of a simple saddle surface for experimentation by students. Ideally, models should be cheap enough so that I can supply each pair of students with a model to work with.
(2) How can students draw geodesics on such a surface? Rubber bands are not going to work here, because a band stretched between two points on the surface will not necessarily lie on the surface. This problem is sort of mathematical, because I think a good understanding of the nature of geodesics should lead to discovering a way of having students create them on a surface of negative curvature.
I am aware of some very good software that uses the Poincare disk model to do geometry on the hyperbolic plane, and I plan to use the software when I teach. But I want students to have real physical experience first.
Anyone have any ideas?
Visual Calculus
I found a fascinating page: VisualCalc. This is a talk by Tom Apostol about Visual Calculus, a technique for finding the area bounded by curves without using traditional calculus developed by an Armenian mathematician living in California, Mamikon A. Mnatsakanian, which has been espoused by Apostol. Some of the results using this method would be very difficult if not impossible to uncover with traditional methods. The starting point is the following simple (but neat) problem, solved by Mamikon (as he calls himself) when he was 15:
Problem: A line segment is drawn tangent to the inner of two concentric circles, terminating at the outer circle. The length of the segment is 2a. What is the area of the annulus?
Answer: pi*a^2. It is rather counterintuitive that the result is independent of the radius of the inner circle.
Solution: Let the radius of the smaller and larger circles by r and R, respectively. The area of the annulus is pi*(R^2 - r^2). Draw the obvious right triangle with legs of length r and a, and hypotenuse of length R. Apply the Pythagorean Theorem.
Mamikon noted that if he knew in advance that the answer was independent of r, he could let r = 0, and the tangent segment would become a diameter of the larger circle, establishing the result another way. This led him to a rather breathtaking extension of the result.
Theorem 1. Let C be a smooth convex oval. Move a vector v (of fixed length) around the oval (with the tail on the curve) so that it is always tangent to the curve (at its tail). Then the area swept out by the vector is pi*|v|^2. [I'm not sure what the exact hypothesis is, but this is the basic idea.]
Proof idea: Let S be the set of translates of the vectors v(t), with a common tail formed as v(t) goes around the oval. Then S is a circle of radius |v|.
Problem: A line segment is drawn tangent to the inner of two concentric circles, terminating at the outer circle. The length of the segment is 2a. What is the area of the annulus?
Answer: pi*a^2. It is rather counterintuitive that the result is independent of the radius of the inner circle.
Solution: Let the radius of the smaller and larger circles by r and R, respectively. The area of the annulus is pi*(R^2 - r^2). Draw the obvious right triangle with legs of length r and a, and hypotenuse of length R. Apply the Pythagorean Theorem.
Mamikon noted that if he knew in advance that the answer was independent of r, he could let r = 0, and the tangent segment would become a diameter of the larger circle, establishing the result another way. This led him to a rather breathtaking extension of the result.
Theorem 1. Let C be a smooth convex oval. Move a vector v (of fixed length) around the oval (with the tail on the curve) so that it is always tangent to the curve (at its tail). Then the area swept out by the vector is pi*|v|^2. [I'm not sure what the exact hypothesis is, but this is the basic idea.]
Proof idea: Let S be the set of translates of the vectors v(t), with a common tail formed as v(t) goes around the oval. Then S is a circle of radius |v|.
Tri-Color chessboards
When coloring a checkerboard, the basic requirement is that squares that are full-neighbors (horizontally or vertically) have different colors. Clearly, there are exactly two ways of coloring an n x n checkerboard with two colors (black and red, say). Once a color has been selected for the lower left corner, all remaining square colors are forced. I wondered how many different ways one could color an n x n checkerboard with three colors. This led me to consider two problems:
(1) How many ways are there to color an n x n checkerboard, using at most 3 colors?
(2) How many ways are there to color a 3k x 3k checkerboard, using equal numbers of red, blue, and white squares?
Bridget Tenner of dePaul University immediately came up with the answer to problem (1) by searching in Neal Sloane's wonderful Online Encyclopedia of Integer Sequences, using "3-color" as a search string. The answer is given here as a special case of A078099 (for m x n checkerboards), which is defined recursively. The sequence grows very quickly: it is 3 times a sequence beginning 1, 6, 82, 2604, 193662, 33865632, 13956665236.
A sequence to answer question (2) does not seem to appear in OEIS, so this may be an open question.
(1) How many ways are there to color an n x n checkerboard, using at most 3 colors?
(2) How many ways are there to color a 3k x 3k checkerboard, using equal numbers of red, blue, and white squares?
Bridget Tenner of dePaul University immediately came up with the answer to problem (1) by searching in Neal Sloane's wonderful Online Encyclopedia of Integer Sequences, using "3-color" as a search string. The answer is given here as a special case of A078099 (for m x n checkerboards), which is defined recursively. The sequence grows very quickly: it is 3 times a sequence beginning 1, 6, 82, 2604, 193662, 33865632, 13956665236.
A sequence to answer question (2) does not seem to appear in OEIS, so this may be an open question.
Three Books on Riemann Hypothesis
My first review will be of three semi-popular books about the Riemann hypothesis: Prime Obsession by John Derbyshire, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus Du Sautoy, and The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics by Karl Sabbagh.
Recently there has been a spate of semi-popular books about the Riemann Hypothesis. This is doubtless due in part to the fact that several of the most famous problems of modern mathematics such as the Four-Color Map Theorem, Fermat's Last Theorem, and the Poincare Conjecture have now been solved, leaving the Riemann Hypothesis as the most famous problem standing. However, writing a semi-popular book about the Riemann Hypothesis is an intimidating mission. Unlike the Four-Color Map Theorem and Fermat's Last Theorem, it is difficult to explain to an educated layperson what the theorem states, or why it is important. Even the statement of Poincare Conjecture is easier to comprehend.
John Derbyshire's Book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, is the best of the lot. He sets himself the daunting task of explaining virtually all of the major mathematical ideas needed to understand the statement of the RH, its relation to the distribution of prime numbers, and some of the major methods that have been used to attack the problem, in a book designed for an otherwise educated person who is ignorant of mathematics from high school algebra on. It sounds to me that this goal must have been imposed the publishers, because whatever the talents of the expositor, it is prima facie impossible to bring anyone but a latent mathematical genius on such a trip in the confines of a single 422-page book. However, what Derbyshire does, and does brilliantly, is to explain the RH to someone who has understood two years of college calculus, or the equivalent. The reader who has experience with integrals and infinite series should be able to follow the exposition.
The Reimann Hypothesis and its relation to the distribution of primes belongs to the branch of mathematics called analytic number theory. This subject is not easy to write about. I took a reading course in analytic number theory in graduate school. I was intrigued by the subject, but became discouraged when I found the text, by a famous researcher in the field (who shall remain nameless) riddled with errors. I ended up going into another specialty. Now that I have read Derbyshire's book, I'm tempted to read more. In addition to the mathematical exposition, Derbyshire quickly and deftly sketches the political and social milieu and the personalities involved in the development of the RH and the search for its solution.
I bring up one quibble because it relates to the first chapter, and might cause a reader to give up. Derbyshire introduces the harmonic series (and its divergence) by asking the reader to imagine constructing a bridge out of playing cards. It turns out that the n-th card from the top of this bridge can extend 1/(n – 1) card length from the card above it, so that the span of the entire n-card bridge is 1 + 1/2 + … + 1/n. I've seen this before, and it is cute, but it is not easy. There are easier ways to introduce the harmonic series.
Marcus du Sautoy's book, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, covers much of the same territory as Derbyshire's book, but goes into somewhat less mathematical detail. Du Sautoy is a professor of mathematics at Oxford, and an excellent writer. I recommend this book for the poetry of the language and the vividness of the stories of the mathematicians involved in the story. It is wonderful to read a book by a first-rate mathematician who is also a first-rate storyteller.
In Karl Sabbagh's book, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, the author, like John Derbyshire, attempts to explain the RH to the mathematically unsophisticated reader. In this case, the mathematical basics are covered in a series of appendices, called "Toolkits". This book was something of a disappointment. First, the author comes across as a journalist rather than as an advanced amateur mathematician (like Derbyshire) or a professional mathematician (like du Sautoy). The writing has more of a superficial feel to it, where more tends to be made of the physical appearance or personal idiosyncrasies of mathematicians rather than their ideas. In addition, Sabbagh spends much of the book conversing with and about Louis de Branges, who has claimed to have a proof of the Riemann Hypothesis. It is true that de Branges is a respected mathematician who solved an important long-standing problem, the Bieberbach Conjecture. However, very few mathematicians credit his claims to have, or be close to, a proof of the Riemann Hypothesis. Sabbagh is obviously charmed by de Branges, and spends, in my opinion, far too much time on this player who seems to deserve at most a short footnote in the story.
Recently there has been a spate of semi-popular books about the Riemann Hypothesis. This is doubtless due in part to the fact that several of the most famous problems of modern mathematics such as the Four-Color Map Theorem, Fermat's Last Theorem, and the Poincare Conjecture have now been solved, leaving the Riemann Hypothesis as the most famous problem standing. However, writing a semi-popular book about the Riemann Hypothesis is an intimidating mission. Unlike the Four-Color Map Theorem and Fermat's Last Theorem, it is difficult to explain to an educated layperson what the theorem states, or why it is important. Even the statement of Poincare Conjecture is easier to comprehend.
John Derbyshire's Book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, is the best of the lot. He sets himself the daunting task of explaining virtually all of the major mathematical ideas needed to understand the statement of the RH, its relation to the distribution of prime numbers, and some of the major methods that have been used to attack the problem, in a book designed for an otherwise educated person who is ignorant of mathematics from high school algebra on. It sounds to me that this goal must have been imposed the publishers, because whatever the talents of the expositor, it is prima facie impossible to bring anyone but a latent mathematical genius on such a trip in the confines of a single 422-page book. However, what Derbyshire does, and does brilliantly, is to explain the RH to someone who has understood two years of college calculus, or the equivalent. The reader who has experience with integrals and infinite series should be able to follow the exposition.
The Reimann Hypothesis and its relation to the distribution of primes belongs to the branch of mathematics called analytic number theory. This subject is not easy to write about. I took a reading course in analytic number theory in graduate school. I was intrigued by the subject, but became discouraged when I found the text, by a famous researcher in the field (who shall remain nameless) riddled with errors. I ended up going into another specialty. Now that I have read Derbyshire's book, I'm tempted to read more. In addition to the mathematical exposition, Derbyshire quickly and deftly sketches the political and social milieu and the personalities involved in the development of the RH and the search for its solution.
I bring up one quibble because it relates to the first chapter, and might cause a reader to give up. Derbyshire introduces the harmonic series (and its divergence) by asking the reader to imagine constructing a bridge out of playing cards. It turns out that the n-th card from the top of this bridge can extend 1/(n – 1) card length from the card above it, so that the span of the entire n-card bridge is 1 + 1/2 + … + 1/n. I've seen this before, and it is cute, but it is not easy. There are easier ways to introduce the harmonic series.
Marcus du Sautoy's book, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, covers much of the same territory as Derbyshire's book, but goes into somewhat less mathematical detail. Du Sautoy is a professor of mathematics at Oxford, and an excellent writer. I recommend this book for the poetry of the language and the vividness of the stories of the mathematicians involved in the story. It is wonderful to read a book by a first-rate mathematician who is also a first-rate storyteller.
In Karl Sabbagh's book, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, the author, like John Derbyshire, attempts to explain the RH to the mathematically unsophisticated reader. In this case, the mathematical basics are covered in a series of appendices, called "Toolkits". This book was something of a disappointment. First, the author comes across as a journalist rather than as an advanced amateur mathematician (like Derbyshire) or a professional mathematician (like du Sautoy). The writing has more of a superficial feel to it, where more tends to be made of the physical appearance or personal idiosyncrasies of mathematicians rather than their ideas. In addition, Sabbagh spends much of the book conversing with and about Louis de Branges, who has claimed to have a proof of the Riemann Hypothesis. It is true that de Branges is a respected mathematician who solved an important long-standing problem, the Bieberbach Conjecture. However, very few mathematicians credit his claims to have, or be close to, a proof of the Riemann Hypothesis. Sabbagh is obviously charmed by de Branges, and spends, in my opinion, far too much time on this player who seems to deserve at most a short footnote in the story.
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