Computers Effective In Verifying Mathematical Proofs

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Computers Effective In Verifying Mathematical Proofs

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Computers Effective In Verifying Mathematical Proofs

ScienceDaily (Nov. 7, 2008) New computer tools have the potential to
revolutionize the practice of mathematics by providing far more-reliable
proofs of mathematical results than have ever been possible in the
history of humankind. These computer tools, based on the notion of
"formal proof", have in recent years been used to provide nearly
infallible proofs of many important results in mathematics.

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

Adapted from materials provided by American Mathematical Society, via
EurekAlert!, a service of AAAS.
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