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Extra resources for The Nature of Mathematical Proof
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The bigger the library, the more mathematics one can deal with in a reasonable time. As an example, in Nijmegen we formalized a proof of the Fundamental Theorem of Algebra (see Geuvers et al. 2001) and it took a team of three people two years. At the same time Harrison formalized the same theorem all by himself (as described in Harrison 2001) and it only took him a few days. The main 9 In this proof ‘limit_in1 f D l x0’ is the Coq notation for limx/x0 f ðxÞZ l where x ranges over the set D 4R. 10 Actually, this is a mixture of Mizar and Coq.
Ber formal unentscheidbare Sa Go¨del, K. 1931 U ¨tze der Principia Mathematica und verwandter Systeme. Monatshefte fu ¨r Mathematik und Physik 38, 173–198. ) Translated and commented in Go ¨del (1986) Another English version based on course notes by Kleene and Rosser is in Go ¨del (1965). Go¨del, K. 1965 On undecidable propositions of formal mathematical systems. In The undecidable: basic papers on undecidable propositions, unsolvable problems and computable functions (ed. Martin Davis), pp.
Such checking can be readily automated—and, indeed, it often has been. But even in the age before computers, postgraduate students could have carried out the necessary checks as an apprentice exercise. Mathematical proofs are subjected to a lot of checking, so we can conclude that proofs containing persistent errors are unlikely to be Hilbertian. If a Hilbertian proof contained an error, it would surely be quickly detected and corrected. This may lead us to ask, what are the alternatives to Hilbertian proof presentation?