Computers effective in verifying mathematical proofs

When mathematicians prove theorems in the traditional way, they present the argument in narrative form. They assume previous results, they gloss over details they think other experts will understand, they take shortcuts to make the presentation less tedious, they appeal to intuition, etc. The correctness of the arguments is determined by the scrutiny of other mathematicians, in informal discussions, in lectures, or in journals. It is sobering to realize that the means by which mathematical results are verified is essentially a social process and is thus fallible. When it comes to central, well known results, the proofs are especially well checked and errors are eventually found.

Nevertheless the history of mathematics has many stories about false results that went undetected for a long time. In addition, in some recent cases, important theorems have required such long and complicated proofs that very few people have the time, energy, and necessary background to check through them. And some proofs contain extensive computer code to, for example, check a lot of cases that would be infeasible to check by hand. How can mathematicians be sure that such proofs are reliable?

To get around these problems, computer scientists and mathematicians began to develop the field of formal proof. A formal proof is one in which every logical inference has been checked all the way back to the fundamental axioms of mathematics. Mathematicians do not usually write formal proofs because such proofs are so long and cumbersome that it would be impossible to have them checked by human mathematicians. But now one can get "computer proof assistants" to do the checking. In recent years, computer proof assistants have become powerful enough to handle difficult proofs.

Only in simple cases can one feed a statement to a computer proof assistant and expect it to hand over a proof. Rather, the mathematician has to know how to prove the statement; the proof then is greatly expanded into the special syntax of formal proof, with every step spelled out, and it is this formal proof that the computer checks. It is also possible to let computers loose to explore mathematics on their own, and in some cases they have come up with interesting conjectures that went unnoticed by mathematicians. We may be close to seeing how computers, rather than humans, would do mathematics.

Four new articles in the December 2008 issue of Notices of the American Mathematical Society explore the current state of the art of formal proof and provide practical guidance for using computer proof assistants. If the use of these assistants becomes widespread, they could change deeply mathematics as it is currently practiced. One long-term dream is to have formal proofs of all of the central theorems in mathematics. Thomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin to "the sequencing of the mathematical genome".

The four articles are:

1. Formal Proof, by Thomas Hales, University of Pittsburgh
2. Formal Proof---Theory and Practice, by John Harrison, Intel Corporation
3. Formal proof---The Four Colour Theorem, by Georges Gonthier, Microsoft Research, Cambridge, England
4. Formal Proof---Getting Started, by Freek Wiedijk, Radboud University, Nijmegen, Netherlands