On computers and math, by Allyn Jackson
In mathematics, it's people who write proofs and solve complicated problems. But today, computers are becoming sufficiently powerful to carry out proofs and even to come up with new mathematical conjectures. Two articles in New Scientist touch on the impact of computing power on today's mathematics. The first article discusses how researchers are translating highly complex proofs into a form that computers can check in a highly precise way---by confirming the rudimentary logic of each step of the proof. Among the results whose proofs have been checked in this way are the four-color theorem, the Feit-Thompson theorem, and the Kepler conjecture. Some mathematicians are going even further, by trying to revamp the very foundations of mathematics to make them more amenable to computers. Could computers then take off and create mathematics of their own---mathematics that is too complex for humans to understand? Perhaps, but most people working in this area do not think this is the point of using computers in mathematics. The article quotes Vladimir Voevodsky of the Institute for Advanced Study: "The future of mathematics is more a spiritual discipline than an applied art. One of the important functions of mathematics is the development of the human mind."
The second article, an interview with Simon Colton, mainly discusses software he wrote called HR, which is designed not to do particular calculations but rather to be creative. HR has come up with its own mathematical conjectures, including some well known to humans, such as Goldbach's Conjecture. Colton told New Scientist that computers will really be creative only when they are able to write their own software. But, he says, "writing software is one of the most difficult things that people do". Colton also wrote software called The Painting Fool, which creates portraits. He notes that mathematicians will accept a computer as being creative if it repeatedly comes up with interesting mathematical results. But art is different. "When you like a painting, you are celebrating the humanity that went into it," he says.
Read more about this theme in the Special Issue on Formal Proofs, AMS Notices, December 2008; and in the article "Voevodsky’s Univalence Axiom in Homotopy Type Theory," AMS Notices, November 2013.
See "Our number's up: Machines will do maths we'll never understand," by Jacob Aron (subscription required). New Scientist, 26 August 2015; and "The Art of Programming": Interview with Simon Colton, by Douglas Heaven. New Scientist, 29 August 2015.
--- Allyn Jackson (Posted 9/14/15)
On an NFL Mathlete, by Mike Breen
Above: John Urschel, giving a talk at the NSA. Right: Urschel on the field. Images courtesy of the Baltimore Ravens. |
In this Sports Illustrated article, author Emily Kaplan wants to answer the question, "Why would a mathematician delay his PhD to be an offensive lineman for the [Baltimore] Ravens?" The lineman, John Urschel, doesn't feel compelled however to explain his love of football and adds that there is nothing wrong with being good at more than one thing. Urschel has a stellar academic history from grade school--where his teacher first thought that he had "processing problems" then realized he was bored--to college at Penn State--where he had to fight athletic advisers so that he could take advanced math courses--to now, when he has earned a master's degree from Penn State and already published four papers. He is very competitive and really enjoys the physicality of football but he's not at all arrogant. Kaplan calls him "an overachiever with humility." Unfortunately, Urschel suffered a concussion during the pre-season and had to avoid football for two weeks. The most recent news is that he now feels fine and is back playing.
See "The (NFL) Mathlete," by Emily Kaplan. Sports Illustrated, 24 August 2015, page 38-43.
--- Mike Breen (Posted 9/15/15)
On Neil Sloane, by Lisa DeKeukelaere
The OEIS Movie
Quanta Magazine interviews Neil Sloane, the Welsh mathematician who created the Online Encyclopedia of Integer Sequences (OEIS). Sloane began collecting sequences on index cards as a graduate student in the 1960s to help him with his research on neural networks, and since then he has transformed his collection into an online repository with more than 170,000 sequences that celebrated its 50th anniversary last year. He continues to curate the list, deciding carefully whether submitted sequences are too arbitrary or too specialized. Sloane discusses some of his favorite sequences, including a formula for calculating the error term for a certain method of estimating pi that was discovered because of OEIS. He notes that he is working with a German repository to enhance OEIS so that users can search formulas, and he highlights OEIS’s ability to foster collaboration.
See "The Connoisseur of Number Sequences," by Erica Klarreich. Quanta Magazine, 6 August 2015.
--- Lisa DeKeukelaere
On a multiplication game, by Claudia Clark
In this article, Knudson introduces the reader to Bojagi, an online game that involves simple mathematics and visual reasoning. The player is presented with a 10 by 10 grid containing several 1 by 1 squares with numbers in them. The goal of the game is to fill the grid by drawing a rectangle around each square so that the following three conditions are met: (1) each rectangle contains exactly one number; (2) each rectangle's area is the number it contains; and (3) there are no overlapping rectangles. Knudson likes the game for several reasons: "It is easy to learn and understand... And it's a really good game for [third or fourth grade] children…To be good at it, you must know your multiplication tables, but you also must be able to realize any given number as a product of two others. What's more, you may need to know many different ways of factoring an integer." He also likes the visual nature of the puzzle: It may remind people who claim to hate math of the fun of working with arithmetic and numbers. He also discusses some of the more sophisticated mathematics behind the game, specifically partitions of integers, but points out that this understanding is not necessary to play it.
To play the game, go to http://bojagi-gotmath.rhcloud.com. You can solve an existing puzzle by selecting List, or create a new puzzle by selecting Create. Enjoy!
See "This 'Simple' Multiplication Game Will Help You Rediscover the Joy of Math," Kevin Knudson, Forbes, 2 August 2015.
--- Claudia Clark
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