Bad Mathematics: A Trillion Dollar Problem
My Favorite Excerpts (via John F. McGowan @ Math Blog)
This portrayal of the near magical power of mathematics is common in science fiction, especially popular movies and video, serious as well as comedic. The classic science fiction movie The Day The Earth Stood Still (1951) features a famous scene in which the alien visitor Klaatu solves a problem in celestial mechanics on the blackboard of Professor Jacob Barnhardt, supposedly revealing the secret of interplanetary (perhaps interstellar) travel, a somewhat curious act for the representative of an interplanetary civilization supposedly fearing atomic attack from the barbaric Earth men. The original Star Trek television series featured several episodes in which either the science officer Mr. Spock or the ship’s super-intelligent computer solved some new, never before encountered problem by performing some mysterious calculations, all by the end of each less than one hour episode. The 2003 disaster movie The Core, which has the dubious distinction of some of the worst and most inaccurate physics in any major movie, features several scenes where the scientists perform some complex calculation, sometimes in their heads in seconds, and solve an otherwise fatal problem (e.g. the end of the world). The television series Numb3rs features mathematicians who help the FBI solve otherwise unsolvable criminal cases through the magic power of mathematics.
How realistic are these portrayals? Based on the history of mathematics, they are far from realistic. For example, in The Absent Minded Professor, Professor Brainard is stumped for three months — only three months — before making his breakthrough. Most major breakthroughs similar to the fictional flubber have taken years, usually at least five years. Very often, the inventor or discoverer was stumped, on the wrong track, for most of that period; in this respect, the portrayal in The Absent Minded Professor is somewhat accurate.
It is likely that someone who engaged in large amounts of deliberate practice in mathematics could perform extremely well on mathematical tests, exams, and other competitive measures so long as these tests involved calculations or derivations that had been practiced. The problem is that by their nature inventions and discoveries involve problems that have never been solved by anyone. There is no way to practice in this way. Deliberate practice is very time consuming. Ericsson argues that most experts engage in 10,000 hours of deliberate practice, typically over ten years. In part this is derived from original studies of chess where champion chess players have almost always spent at least ten years of intensive study and practice before reaching the International Grand Master level. Ericsson has found similar patterns in many other fields. It is quite possible that deliberate practice at this level can substantially reduce the time available to study basic concepts and to develop the conceptual reasoning skills frequently used in invention and discovery.
Third, many working scientists, engineers, and mathematicians are unaware of the actual history of the inventions or discoveries in their field. To the extent that they have studied this, they are usually relying consciously or not on stories in textbooks or popular science accounts or even word of mouth that are often quite inaccurate on close examination. Working scientists, engineers, and mathematicians are busy with their work and don’t have time to study the past. A number of accounts of major inventions in US history textbooks (such as the Wright brothers and the airplane) are highly misleading. They are probably intended to inspire students and promote patriotism.