Gödel's conflicting approaches to effective calculability

Abstract

Identifying the informal concept of effective calculability with a rigorous mathematical notion like general recursiveness or Turing computability is still viewed as problematic, and rightly so. In a 1934 conversation with Church, Gödel suggested finding axioms for the notion of effective calculability and "doing something on that basis" instead of identifying effective calculability with λ-definability; that identification he found "thoroughly unsatisfactory". He introduced in his contemporaneous Princeton lectures (Gödel 1934) the class of general recursive functions through the equational calculus, but was not convinced at the time that this mathematical notion encompassed all effectively calculable functions. (See (Davis 1982) and (Sieg 1997).) Gödel articulated different and conflicting approaches to the underlying methodological issues during the three decades from 1934 to 1964. The significant shifts in his position underline the difficulty of the problems surrounding the Church-Turing Thesis. In (1936) and (1946) he emphasized that the importance of the notion of general recursive function is largely due to its absoluteness. Yet he also claimed in (193?) that the analysis of the manner in which the calculation of number theoretic functions proceeds leads to the characteristic features of the equational calculus; thus, it provides a "correct definition" of effectively calculable function. In (1951) he calls Turing's reduction of the "concept of finite procedure to that of a machine with a finite number of parts" the most satisfactory way of arriving at a precise definition of the former concept. Finally, in (1964) Gödel saw, quite emphatically, Turing's work as providing a correct analysis of mechanical procedures (thus also of effective calculability) and a proof of the fact that the analyzed notion is equivalent to that of a Turing machine. Eight years later Gödel detected "a philosophical error in Turing's work" (of 1936) and attributed to Turing the claim that "mental procedures cannot go beyond mechanical procedures". Turing, however, did not maintain such a claim when reducing mechanical procedures (carried out by a human computer) to machine computations. A deepened analysis of Turing's reduction can serve, ironically, as a springboard for the methodological approach Gödel had recommended in 1934, but never followed up, namely an axiomatic characterization of computability. © Springer-Verlag Berlin Heidelberg 2006.