Million Dollar Mathematics and the Cleverness of Crowds

On 24 May, 2000, during a meeting held at the College de France in Paris, the challenge was made. A million dollars would be paid to anyone offering a proof or a counterexample to any of seven mathematical conjectures. The offer was made by Arthur Jaffe, co-founder (with Landon Clay) of the Clay Mathematics Institute. In the same year, British publisher Tony Faber offered a million dollars to anyone who could prove that every even integer greater than 2 can be expressed as the sum of two (not necessarily different) prime numbers, e.g. 74 can be expressed as the sum of the prime numbers 31 and 43. This is the ‘Goldbach Conjecture’, first formally proposed in a letter dated 7 June, 1742, from Prussian mathematician Christian Goldbach to Leonhard Euler. The purpose of the Faber offer was to publicize ‘Uncle Petros and Goldbach’s Conjecture’, a novel on the Faber and Faber publishing list by Apostolos Doxiadis. What is especially interesting about all these million dollar prizes is that they have the potential to incentivize and focus the minds of a diverse set of amateur and professional mathematicians around the world. In other words, to tap into the wisdom of the crowd. So nine years later, where are we with Goldbach and his primes? Faber and Faber have sold quite a few books. That much is known and could have been predicted. It has also been shown by computer that Goldbach’s Conjecture is true for all even numbers up to 1,200,000,000,000,000,000. Even so, we are no closer to a formal proof than we were when the prize was announced. How about the conjectures identified by the Clay Mathematics Institute? Better news here. Of the seven one has been proved, the one that states in simple terms that ‘a sphere is a sphere is a sphere’, i.e. no matter what you do to it other than tearing it (punch it, pinch it, kick it, twist it, squeeze it, squash it, poke it, inflate it, deflate it), it remains a sphere, even when the surface of the sphere is in three dimensions (the surface of an ordinary sphere is two-dimensional, of course, though it encloses a three-dimensional volume). Put more rigorously, the conjecture can be expressed as ‘Every simple connected, compact three-dimensional manifold (without a boundary) is a three-dimensional sphere.’ Got it? Anyway, this is the Poincare conjecture, named in honour of French polymath Jules Henri Poincare, and proved in 2003 by Grigori Perelman, though he’s declared that he doesn’t want the prize. The other six conjectures remain unresolved. For the record, these relate to the Hodge Conjecture, the Birch and Swinnerton-Dyer Conjecture, the Riemann Hypothesis, Yang-Mills Theory, the Navier-Stokes Equations and the P versus NP Problem. So here we have a market the size of the world and a total of 8 million dollars worth of liquidity, and only one of the conjectures has been proved, and that by someone who shuns the crowd and hasn’t even bothered to pursue the formal conditions required to pick up the prize. So we are left with the obvious question. If crowds are so clever, why can’t they solve a million dollar maths challenge?