Are Mathematicians In Jeopardy?
Can a mathematician program beat humans?
Watson is the IBM program that just convincingly won on “Jeopardy!” last night. By the way Bill Murdock of IBM, who is on the Watson project, is speaking today at Tech.
Today I am wondering if we are all about to become obsolete as mathematicians.
February 10, 1996: Deep Blue, the IBM chess playing machine, beat Garry Kasparov, who was then the world champion, in the first game of their first match. Kasparov came back to win that match, but his number was up the following year: on May 11, 1997, Deep Blue trounced him in 19 moves to win the second match. Since Kasparov is often considered to be the strongest player ever, this was a major achievement. Deep Blue was massively parallel and used 480 special purpose chips, yet it is believed to have been weaker than a program you can run today on your laptop. For example Rybka, a commercial program, has a projected Elo rating just over 3200 on a dual-core system, while Kasparov’s top rating was 2851. In chess this rating differential means that Kasparov would be fortunate to score 20% versus Rybka.
February 16, 2011: Watson, the IBM question answering machine, beat the two top Jeopardy! champions of all time: Ken Jennings and Brad Rutter. A guess or prediction might be: in less than ten years there will be a laptop program that will be even better at answering general knowledge questions than Watson. Actually the program by then might not even need a laptop but will be able to run on some other personal device—will we even have laptops in ten years?
Could this be next—note I tabbed IBM’s breakthroughs as occurring every fifteen years and six days:
February 22, 2026: Thomas J., the IBM automated mathematician, has just claimed to have solved the Riemann Hypothesis. Experts have started to look at the paper that Thomas J. has submitted to the math archive. The paper is over 100 pages, but is well written says one of the top number theorists in the world, Peter Smith. He goes on to explain that while the proof is not yet checked it contains a number of very interesting new ideas
IBM’s Next Project?
I am serious that Thomas J. could be a real project. Note I am not suggesting that they create a perfect mathematician, anymore than they created a perfect chess player or a perfect “Jeopardy!” player. Deep Blue made mistakes and lost games. It did not solve chess. It did not have to solve chess or play perfectly. Watson also buzzed in late sometimes or even made incorrect answers. It did not solve Jeopardy!. It also did not have to.
I think the big insight in both these brilliant projects and achievements is the point that a program can be immensely powerful even if it is imperfect. In the past people have thought about automated math programs, but I believe that most ideas were around formal proof systems. What intrigues me is that perhaps the goal is quite different: build a math program that makes mistakes, can give false proofs, can make errors. Just like real mathematicians. But such a machine could still be of immense value and importance.
For Ken Regan the contest’s most iconic moment came with the first “Final Jeopardy!” question, in a category titled “U.S. Cities.” The question itself did not specify a U.S. city, and Watson responded “Toronto?????” with five question marks signifying its uncertainty. Apparently it’s not the case that Watson did not take the category title into account. According to CMU professor Eric Nyberg as quoted here,
“A human would have considered Toronto and discarded it because it is a Canadian city, not a U.S. one, but that’s not the type of comparative knowledge Watson has.”
Whatever type it has, it was imperfect, yet that may have made it more powerful.
How To Do This?
I really do not know how IBM, or anyone, could build a mathematician program. I guess if I thought I did I would go off and start building one. But I can imagine several components that could be part of such a machine:
Vast knowledge: Clearly such a machine could have stored all math papers every written. All of the archives, all of every journal, all papers, all books. Some rating system might be needed to be careful about mistakes and false proofs. But unlike humans the machine could easily have stored all theorems, lemmas, and definitions ever written. Note that one project for doing this failed, but the two reasons given there might be helped by a Watson-like change of view.
Vast experimental ability: Many of the greatest mathematicians have been great calculators. Clearly such a machine could easily try vast numbers of examples in an attempt to make good conjectures, and to rule out false theorems. If it is working on a group theory problem, why not try it instantly on all groups of order less than a million?
Vast tools: Clearly such a machine could use certain formal tools. I do not believe it must create a formal proof, but it can use certain formal tools. If it needs the sum of some complex series it should use the best known tools available.
The machine also need not be viewed as stand-alone. The article then quotes Professor James Hendler of RPI,
“A human working with Watson can get a better answer. Using what humans are good at and what Watson is good at, together we can build systems that solve problems that neither of us can solve alone.”
Is this a possible goal? Note the one key insight that I feel the need to repeat: Thomas J. must not be perfect. It must not only put out correct formal proofs. It must be allowed to put out papers like we do. And these must be right often enough to be interesting, but they need not be perfect.