Abstract

In the first half of the 1960s, Paul Abrahams implemented a Lisp program for the machine verification of mathematical proofs [1]. The program, named Proofchecker, “was primarily directed towards the verification of textbook proofs, i.e., proofs resembling those that normally appear in mathematical textbooks and journals”. Abrahams did not succeed. If, so Abrahams, “a computer were to check a textbook proof verbatim, it would require far more intelligence than is possible with the current state of the programming art”. Therefore, so Abrahams, “the user must create a rigorous, i.e., completely formalised, proof that he believes represents the intent of the author of the textbook proof, and use the computer to check this rigorous proof”. Abrahams points further out that “it it a trivial task to program a computer to check a rigorous proof; however, it it not a trivial task to create such a proof from a textbook proof”. Abrahams was right. In all later projects, proofs had to be written in a formal language in order to verify them. One well-known example is the the Automath project: van Benthem-Jutting formalised a whole textbook of Landau, the ‘Grundlagen der Analysis’, into a formal language, aut-qe [8]. A second example is the Mizar project [5]: proofs have

Citations