Abstract
The art of formalization consists of writing a proof in an artificial mathematical language in such a way that a computer program can construct and validate a formal version of that proof that is completely certain to be free of errors. These programs are called proof assistants or interactive theoremprovers. Formalization has both applications in mathematics and in computer science. Note that this quest for certainty comes at a price. First, there are some foundational issues. Does the logic of the system “make sense”, i.e. is it consistent? Is it powerful enough, i.e. can we derive a proof of any true statement? These problems were at the root of the creation of a new field of mathematics, Mathematical Logic, at the beginning of the 20th century, with key contributions by Frege, Russell and Godel among others. Second, the system on which the proofs are done is a computer system and any bug in this system taints the benefit of having a formal proof. Finally, one needs to take special care of the formal definitions that have been introduced. They have to match the informal understanding of the notions that the proof is about. Consider, for example, the simple problem of formally proving that there can be at most three Friday the 13ths in a single year. Central to this proof will be to correctly formalize what a year is. In recent years highly non-trivial proofs have been formalized, showing the power of current formalization technology. Examples are, in mathematics:
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