Why Proof? What is a Proof?

1. The Hilbert-Gentzen Thesis This paper is concerned with real proofs as opposed to formal proofs, and specifically with the ultimate reason of real proofs ('Why Proof?') and with the notion of real proof ('What is a Proof?'). Several people believed and still believe that real proofs can be represented by formal proofs. A recent example is provided by Macintyre who claims that " one could go on to translate " all " classical informal proofs into formal proofs of some accepted formal system " , where such translations " do map informal proofs to formal proofs " (Macintyre 2005, p. 2420). This view is to a certain extent implicit in Frege – to a certain extent only, because according to Frege in a sense " every inference is non-formal in that the premises as well as the conclusions have their thought-contents which occur in this particular manner of connection only in that inference " (Frege 1984, p. 318). Anyway, the view that real proofs can be represented by formal proofs is explicitly stated by Hilbert and Gentzen. For Hilbert claims that formal proofs are " carried out according to certain definite rules, in which the technique of our thinking is expressed " (Hilbert 1967, p. 475). These are " the rules according to which our thinking actually proceeds ". They " form a closed system that can be discovered and definitively stated ". Similarly, Gentzen claims that formal proofs in his natural deduction systems have " a close affinity to actual reasoning " (Gentzen 1969, p. 80). They