once it was established that automated proof was not keeping all its promises, mathematicians conceived a less ambitious project, namely that of proof checking. When one uses an automated theorem proving program, one enters a proposition and the program attempts to construct a proof of the proposition. On the other hand, when one uses a proof-checking program, one enters both a proposition and a presumed proof of it and the program merely verifies the proof, checking it for correctness.

Although proof checking seems less ambitious than automated proof, it has been applied to more complex demonstrations and especially to real mathematical proofs. Thus, a large share of the first-year undergraduate mathematics syllabus has been checked by many of these programs. The second stage in this project was initiated in the nineties; it consisted in determining, with the benefit of hindsight, which parts of these proofs could be entrusted to the care of software and which ones required human intervention. The point of view of mathematicians of the nineties differed from that of the pioneers of automated proof: as we can see, the idea of a competition between man and machine had been replaced with that of cooperation.

One may wonder whether it really is useful to check mathematical proofs for correctness. The answer is yes, first, because even the most thorough mathematicians sometimes make little mistakes. For example, a proof-checking program revealed a mistake in one of Newton's demonstrations about how the motion of planets is subjected to the gravitational attraction of the sun. While this mistake is easily corrected and does not challenge Newton's theories in the slightest, it does confirm that mathematical publications often contain errors. More seriously, throughout history, many theorems have been given myriad false proofs: the axiom of parallels, Fermat's last theorem (according to which, if n ≥ 3, there exist no positive integers x, y and z such that xn + yn = zn) and the four-color theorem (of which more in Chapter 12) were all allegedly “solved” by crank amateurs, but also by reputable mathematicians, sometimes even by great ones.