Math Digest

Summaries of Articles about Math in the Popular Press

Edited by Allyn Jackson, AMS
Contributors: Mike Breen (AMS), Claudia Clark (freelance science writer), Lisa DeKeukelaere (Brown University), Annette Emerson (AMS)

Davis traces the attempts to make logic itself the subject of inquiry, from Aristotle, to George Boole, to Gottlob Frege, to Bertrand Russell, and states that "during the 20th century, as logic developed into a vibrant branch of mathematics, tension remained between those two aspects of the field: logic as part of mathematics and logic as the foundation of mathematics." The book under review is never actually reviewed, but this piece provides a nice introduction to Alfred Tarski, one of the great logicians of the 20th century, who proved his famous theorem on the "nondefinability of truth," who "always seemed to work on formulations of logical matters that could hope to have the same elegance found elsewhere in mathematics," and who was a great teacher.

--- Annette Emerson

The Lambert W function, defined as "the inverse function associated with the equation We^W = x," is a rising star in the cluttered sky of mathematical equations. With applications ranging from ballistic projectiles to delay differential equations and the omega constant, some say it should join logarithms and exponents in the fundamental mathematical toolbox. The function is difficult to evaluate for two reasons. First, there is no closed form of the solution: One must first guess W based on x, then adjust the guess accordingly and iterate. Second, for certain x there are multiple W which solve the equation. Though it was originally postulated by Euler, the function bears the name of the relatively obscure Lambert because "naming yet another function after Euler would not have been useful." Its presence in the program Maple played an important role in its path to stardom.

--- Lisa DeKeukelaere

The New York Sunarticle quotes Sylvain Cappell, professor at the Courant Institute of Mathematical Sciences, New York University, on Peter D. Lax's achievements: "Besides being the dominant figure in applied mathematics in his time, he's also one of the world's central figures in pure mathematics." Lax, who is also a professor at Courant, is to receive the prestigious 2005 Abel Prize "for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions" (The Norwegian Academy of Science and Letters announcement). The article captures some personal background about Lax, along with quotes from colleagues and from Lax himself.

The New York Times piece--a conversation with Lax--includes his account of his family coming to America to escape the Nazis when he was 15, going to Stuyvesant high school, working at Los Alamos National Laboratory, speculating on what his mentor John von Neumann would think of today's computerized world, and citing what his own most significant contributions have been: work on shock waves, the Lax-Phillips semigroup in scattering theory, and analysis with his student Dave Levermore of "what happens to solutions of dispersive systems when dispersion tends to zero." When asked "has mathematics become too complex for anyone to understand all of it?" Lax says yes, noting that mathematics is a broad subject. But he notes that "it is also true that as mathematics develops, things are simplified and unusual connections appear. Geometry and algebra, for instance, which were so very different 100 years ago, are intricately connected today."

--- Annette Emerson

Thinking about infinity can be like trying to interpret abstract art: You are looking at a conglomeration of tangible pieces, but you're never quite sure what the whole thing amounts to. Singh reports that in The Infinite Book, John D. Barrow helps pin down the facts about this elusive concept and that his writing gives new life to well-worn examples, addressing topics from the Schwarzschild radius of a black hole to connections between Fawlty Towers and the Infinite Hotel. Barrow also keeps readers entertained with a collection of shocking and amusing anecdotes. Infinity may always be a bit abstract to most people, but Singh says this book provides readers with some of the principles and motivation needed to appreciate the myriad exhibits in the museum of mathematics.

---Lisa DeKeukelaere

Scientific American online reports on research by Bruce Malamud (King's College London) and colleagues in which they analyzed over 88,000 wildfires in the U.S. between 1970 and 2000. They found that "small fires occur most frequently and the largest ones are the least likely to occur" and that "going from east to west, the ratio of small fires to large ones decreases." The summary states that their results allow "for the classification of wildfire regimes for probablistic hazard estimation in the same vein as is now used for earthquakes." The findings---categorized "frequency-area statistics, power-law distribution, Bailey ecoregion divisions, U.S. Department of Agriculture Forest Service, and probabilistic hazard"---were published online before print in the Proceedings of the National Academy of Sciences ("Characterizing wildfire regimes in the United States," 21 March 2005).

--- Annette Emerson

These articles concern a result by Karl Mahlburg, a graduate student of Ken Ono at the University of Wisconsin, regarding partitions. Freeman Dyson characterized the result as, 'Beautiful and totally unexpected.' The historical background of the result traces to Ramanujan and goes through Dyson, George Andrews, Frank Garvan, Oliver Atkin, and Ken Ono. Ramanujan made an original discovery (1919) about partitions of numbers dealing with the primes 5, 7, and 11, which Dyson later thought could be explained by a function he called the "crank" (a function that was yet to be discovered). Later George Andrews and Frank Garvan found the crank function which did indeed imply Ramanujan's discovery. In 2000, after looking at Ramanujan's workbooks, Ken Ono further generalized Ramanujan's discovery to primes bigger than three, using modular functions. His was an analytic proof of a deep generalization. Now his graduate student has provided a combinatorial proof that ties Ono's generalization with the crank function. Mahlburg's work has been submitted to the Annals of Mathematics. The Wisconsin State Journal ran the story on its front page.

See also: "Pieces of Numbers," by Erica Klarreich. Science News, 18 June 2005, pages 392-393.

--- Mike Breen

The largest known prime number is now 2^25,964,951-1, a number that is 7,816,230 digits long. (According to the article the number would completely fill 58 issues of the Science News.) The number was found through the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project, on the computer of Martin Nowack, a German eye surgeon. A Mersenne prime is one of the form 2^p-1. For such an expression to be prime, p must be prime, but p being prime is not enough to ensure that 2^p-1 is prime (for example, p = 11). The previous largest known prime was discovered through GIMPS in May. Over 2000 years ago Euclid gave an elegant proof that there are an infinite number of prime numbers.

--- Mike Breen

This issue of Nature includes a special series of articles by writers, visual artists, a composer, and two scientists on their connections to art and science and their thoughts on these connections. Three of the articles specifically mention mathematics as inspiration: In "Fiction informed by science," A.S. Byatt lets the reader know how mathematics enlightened her and plays a role in her four-part series of novels. She recalls her discovery that in snails the "Fibonacci spiral was an example of a Platonic order---a sense that an invisible mathematical order informed all our physical accidental world." In "A tale of two loves," Alan Lightman (physicist and novelist) notes that as a child he was drawn to mathematics: "When my maths teachers assigned homework, I relished the job. I would save my maths problems for last, right before bedtime--like bites of chocolate cake awaiting me after a long and dutiful meal of history and Latin....I loved the abstractions, letting x's and y's stand for the number of nickels in a jar or the height of a building in the distance...And when you had found that answer, you were right, unquestionably right." In later years and as a novelist he "realized that the ambiguities and complexities of the human psyche are what give fiction, and perhaps all art, its power," and now thinks that the sciences and arts contribute to each other, are always in tension, and offer humans different---sometimes synthesized---ways of looking at our world. In "From science in art to the art of science," Martin Kemp writes that "shared intuitions about the natural world drive the pursuits of artists and scientists," and includes photographs and descriptions of mobiles, kinetic and geometric sculptures, etc. of some 20th century artists.

--- Annette Emerson

Without really getting into the mathematics of the new scoring system for competitive figure skating (used at the World Figure Skating Championships in Moscow in March 2005), Powers reports that the skaters and audiences find the new scoring system "numbing" and "bewildering." In an effort to avoid the corruption and collusion of the 2002 Winter Olympic judges, the new system does not link marks to judges and breaks down the old technical and artistic scores into scores for separate sequences and elements for each. The new scoring may force skaters to change their programs to fit the new method. While admitting that the new scoring system may lessen subjective influences, Powers concludes that "the new system is numbers stacked upon numbers, with base values and grades of execution and factored program components and deductions, all of which adds up to scores like 113.93. It's a system in which a key-punch error can cost you a medal if nobody catches it. Compared with the old 6.0 system, this is a quadratic equation."

--- Annette Emerson

Knuth leads the interviewer around his home, pointing out that mathematics was used to design the kitchen and still informs nearly every aspect of his life and lifetime of work in computer science. He notes that computer programming is not set in stone, that mathematical analysis and refinements---and the ideas and work of many individuals---are constantly improving programs. The NPR web page describes Knuth as "legendary in the computer science world for writing a series of must-have reference books called The Art of Computer Programming. Part cookbook, part textbook, part encyclopedia, these books are also considered by many to be technical and personal works of art." Knuth provides some gems of personal philosophy, including his remark that "efficiency" is fun to think about. Mathematicians and math publishers are of course familiar with his development of TeX, the program that revolutionized typesetting of mathematics. Listen to the informal tour of Knuth's home and mind on the NPR website.

--- Annette Emerson

National Public Radio's Jacki Lyden interviews mathematician Daina Taimina (Cornell University) about the hyperbolic planes she crochets. She explains how she started modeling the shapes with paper, then tried knitting them, but found that crochet retains a more rigid structure that students and others can see, touch and turn to understand the geometrical concept of a hyperbolic plane. Taimina's models are on view on the Institute for Figuring website, and a detailed explanation of her works is in an article by Taimina and her husband David Henderson, "Crocheting the Hyperbolic Plane."

See also:
"Professor Lets Her Fingers Do the Walking," by Michelle York, The New York Times, 11 July 2005.

--- Annette Emerson

Michelle, daughter of physicist Richard Feynman, provides a personal view of the man as father, scientist, teacher, and celebrity, and shares many photographs and quotes from letters. Among the letters quoted is one Feynman wrote to high school student Frederich Hipp in 1961: "To do any important work in physics a very good mathematical ability and aptitude are required. Some work in applications can be done without this, but it will not be very inspired. If you must satisfy your `personal curiosity concerning the mysteries of nature' what will happen if these mysteries turn out to be laws expressed in mathematical terms (as they do turn out to be)? You cannot understand the physical world in any deep of satisfying way without using mathematical reasoning and facility..." The article sidebars include a chronology of Feynman's achievments, and Feynman's own words throughout show his personality, humor and insights.

--- Annette Emerson

Click here for a list of links to web pages of publications covered in the Digest.