Convalescing from an operation, I’ve been reading The Girl Who Played With Fire, a thriller by Stieg Larsson, who as well as being a novelist and investigative journalist was also a revolutionary socialist.
It’s a good thriller, but also includes what has plausibly been called “the silliest fictional mathematics” ever, the silliest ever version of the myth that mathematics is all about inexplicable, given-by-nature, eccentric flashes of inspiration. As the Polish-Canadian mathematician Izabelle Laba has explained:
The worst thing about the series [of stories, of which TGWPWF is no.2] is the mathematical interludes in The Girl Who Played With Fire. We’re told that Lisbeth Salander [the heroine] is also a puzzle-loving math genius who solves Fermat’s last theorem, or thinks she does, in a passage that [the Cambridge mathematician] Tim Gowers singled out for attention some time ago [as “the silliest piece of fictional mathematics I have ever come across”].
Mind you, I’m all for having more novels and movies with strong, resourceful and mathematically talented heroines. I just wish that the math part weren’t so far off the mark…
Salander comes to mathematics by way of puzzles: Rubik’s cube, intelligence tests in magazines, every logical puzzle that she can lay her hands on. She has always been good at solving them, but was not aware of their mathematical side until sometime between the end of TGWTDT and the start of TGWPWF. Mathematics, to her, is “a logical puzzle with endless variations”, a meta-riddle where the goal is to understand the rules for solving numerical or geometric puzzles.
Salander’s primary resource is a book called Dimensions in Mathematics by a Dr. L. C. Parnault (Harvard University Press), a 1,200 page book that’s allegedly considered the bible of mathematics. Quite unsurprisingly, neither Dr. Parnault nor the book in question exist in real life, but Larsson tells us that Dimensions is a book about “the history of mathematics from the ancient Greeks to modern-day attempts to understand spherical astronomy”. It’s supposed to be pedagogical, entertaining, gorgeously illustrated and full of anecdotes. Salander is fascinated by a theorem on perfect numbers – one can verify it for as many numbers as one wishes, and it never fails! – and then advances through “Archimedes, Newton, Martin Gardner, and a dozen other classical mathematicians”, all the way to Fermat’s last theorem.
Unwilling to look at the “answer key”, she skips the section on Wiles’s proof, and tries to figure it out for herself… [Andrew Wiles’s proof, with Richard Taylor, of Fermat’s last theorem, completed in 1995, is lengthy and uses 20th-century techniques which require years of postgraduate study to acquire: click here for an idea of it. There is no way that a brilliant amateur could find it in a flash of inspiration, any more than she could design a self-driving truck in a similar flash].
This is all easy to mock, but unfortunately it seems to be a pretty accurate reflection of what mathematics means to most people… [Gardner, who died in 2010, was a (brilliant) writer of popular mathematical puzzles, not a working, let alone a “classical”, mathematician]… With all due respect to Gardner and his work, I have a problem with the image of mathematics as the art of puzzle solving.
Sure, mathematics involves logical arguments and so do mathematical puzzles. An appreciation of that does offer some insight into what we do. Regrettably, it can also lead to the notion that we get paid for playing with Rubik’s cube and solving crossword puzzles and newspaper-style intelligence tests. It’s the equivalent of a blind person touching an elephant’s trunk and concluding that elephants look like snakes.
In case any non-mathematicians are reading this: logical puzzles convey no sense whatsoever of how vast the subject actually is or how much work it takes to learn the craft. They can’t, for the simple reason that they’re created for entertainment. Their target audience can’t be expected to take a calculus class, never mind advanced graduate courses, before they can even understand the statement of the question. This already narrows it down to simple Euclidean geometry, basic combinatorics, possibly some manipulation of numbers, and eliminates most of mathematics as we know it.
You’d never learn for example that analysis, PDE or ergodic theory even exist, let alone how much accumulated knowledge there is in each of these areas. You wouldn’t get any good picture of contemporary geometry or combinatorics, either. The puzzles you’re left with may be tricky and entertaining, but they’re at best peripheral to mainstream mathematics.
The difficulty of math puzzles is usually calibrated so that the readers would have a good shot at solving them within a short time, usually ranging from a few minutes to an hour or two. A really hard puzzle is one that takes more than a few hours. No wonder that Salander was disappointed when she couldn’t solve Fermat’s theorem within a couple of days, or that she would expect a short solution with no background required.
In real-life mathematics, we don’t have a Ceiling Cat to set up problems for us and control their level of difficulty. Advisors can sometimes do that for their graduate students, to a very limited extent, but mostly we’re left stumbling in the dark, not knowing whether there even is a solution or whether we’re asking the right question in the first place. Learning to navigate this is possibly the hardest part of becoming an independent researcher.
Then there’s Dimensions of Mathematics. The very idea that mathematics should have a “bible” looks like a continued misunderstanding of the nature and scope of the subject. However, Larsson’s description is more reminiscent of any number of popular math books, except for the length.
If I had to suggest a real-life book for Larsson to use instead, it might be a collection of national or international Math Olympiad problems with solutions. It would not be a bible of anything, but it should present a challenge to someone like Salander at about the right level.
Olympiad problems only require normal high school background, which Salander should be able to catch up on. (Did I mention that she had dropped out of school?) They can be very hard, immeasurably more so than logical puzzles in popular magazines, but not necessarily out of reach for an extremely smart newcomer to mathematics who’s willing to put in the time and effort.
[Or the Stanford Mathematics Problem Book, of which there is a copy in the CoLA Maths Library. The ingenuity and imagination and persistence required for these problems is something which every research mathematician has to learn, as well as learning the more advanced techniques developed by other mathematicians’ collective ingenuity and imagination over the centuries. It is also useful in many other areas of thought.
And sometimes a mathematician does – through knowing a lot of possible ways to approach problems, through being imaginative and persistent about trying them out, and through being quick and precise in checking them – make a simple-once-you-see-it breakthrough which produces a quicker, easier proof of something which previously required complicated and abstruse reasoning – John Conway’s proof of Morley’s Miracle, for example.]