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A monthly survey of math news
Written in stone. The equations giving the hyperbolic structure for the complement of the Figure-8 knot, and its volume in the hyperbolic metric, are engraved on the base of the sculpture "Figure-8 Knot Complement CMI" by Helaman Ferguson. The piece was unveiled by the sculptor
at the public inauguration on May 10 of the Clay Mathematics Institute, a new institution in Cambridge, Massachusetts, dedicated to furthering "the beauty, power, and universality of mathematical thinking."
"Mathematical genius Kurt Gödel" is the subject of an article in the June 1999 Scientific American. John W. Dawson, Jr., the logician who catalogued Gödel's papers at the Institute for Advanced Study, tells the story of Gödel's life and his struggle with mental illness. Dawson gives an account of Gödel's most famous discoveries on completeness and incompleteness. This is the mathematician who proved that there are true statements about the natural numbers which are not provable: for any reasonable set of axioms which describe the natural numbers there must exist statements which are true but which cannot be deduced from those axioms. "Gödel ... saw his incompleteness theorems not as demonstrating the inadequacy of the axiomatic method but as showing that the derivation of theorems cannot be completely mechanized. He believed they justified the role of intuition in mathematical research." Web resources on Gödel include a page at The Exploratorium and an interesting piece Kurt Gödel in Blue Hill by Peter Suber, philosophy professor at Earlham College.
Unreadable but irresistible. Sales of Principia Mathematica have soared recently after it was voted one (number 23 to be exact) of the 100 greatest nonfiction books of the 20th century. This is reported in the May 31 New Yorker, in a "Talk of the Town" item by John Cassidy. And unnecessary: "People don't need to read it because the important things in it have been done more clearly elsewhere," according to NYU mathematical philosopher Hartry Field, quoted by Cassidy. A good reference on Russell and Whitehead and the context of their work is Stanley Burris at the University of Waterloo. Cassidy concludes, referring to Gödel's incompleteness theorems (which came some 20 years later and showed that Russell and Whitehead's goal of a complete axiomatic derivation of mathematics was impossible), "Logic, even in the hands of figures as brilliant as Whitehead and Russell, has its limits. Book marketing, it appears, doesn't."
"I've been in schools where ... the results of the math team are announced with the same pride as the results of the football team." -- Glenda Lappan, NCTM President, quoted in the May 24 Boston Globe. The piece, "For math whizzes, victory equals respect" by Hermione Malone, was prompted by a Massaschsetts team of eighth-graders winning first place at the Mathcounts national championship in Washington, and two Massachusetts high school students placing among the eight top contestants in the USA Mathematical Olympiad and thereby winning a trip to the International Mathematical Olympiad in Romania next July. The teachers see the competitions as a way to start to redress US students' lag with respect to their ``economic counterparts'' by making it "OK for kids to be smart in math." Evagrio Mosca, the Mathcounts team coach, is quoted as saying that competition "takes the stigma away from enjoying math ..." The one student interviewed, IMO contestant Paul Valiant of Milton Academy, mentioned as rewards the learning experience and the chance to meet other competitors: "It's a good group of friends."
"Brain's Math Machine Traced to 2 Circuits" was the New York Times' take on a report in the May 7 Science by Sebastien Dehaene, Elizabeth Spelke, and their collaborators at INSERM (Orsay, France) and at MIT. The report, "Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence," demonstrates that there are at least two different loci in the brain involved in arithmetic, and that the two loci do different things:
Exact calculation is accompanied by a "large and strictly left-lateralized activation in the left inferior frontal lobe," while approximate calculation tasks activate a region in the bilateral inferior parietal lobule which is not involved in exact calculation. More details in this month's a href="/featurecolumn/archive/mi1.html">column.
Math on the Web in MathML soon? Pages like this which include mathematical equations have a problem: only the most elementary expressions can be coded in HTML. Ralph Youngen's Toward a Mathematical Markup Language appeared in the Notices in October, 1997. Now the Mozilla project has begun work on MathML. The promise is that Netscape Navigator 5 running its Gecko layout engine will be 100% compliant to the standards of the World Wide Web Consortium, including MathML.
SUNY at Stony Brook