Pénzes István (szerk.): Műszaki nagyjaink 6. Matematikusok, az oktatás, a gépészet és a villamos vontatás alkotói, kiváló lisztvegyészek (Budapest, 1986)
Dr. Ádám András - Dr. Dömösi Pál: Kalmár László
Constr ■activity in Mathematics AN ARGUMENT AGAINST THE PLAUSIBILITY OF CHURCH’S THESIS LÁSZLÓ KALMÁR Bolyai Institute, The University, Szeged, Hungary 1. In his famous investigations on unsolvable arithmetical problems [1], Church used a working hypothesis, viz. the identification of the notion of effectively calculable functions with that of general recursive (or, equivalently, ^-definable) functions. This working hypothesis is known under the name Church’s thesis. It has several equivalent forms — e.g. Turing’s identification of the notion of effectively calculable functions with that of functions computable by means of a Turing machine [10], or Markov’s principle of the normalizability of algorithms [6] or [7] — which are generally accepted in the investigations on unsolvable mathematical problems. In tne present contribution, I shall not disprove Church’s thesis. Church’s thesis is not a mathematical theorem which can be proved or disproved in the exact mathematical sense, for it states the identity of two notions only one of which is mathematically defined while the other is used by mathematicians without exact definition. Of course, Church’s thesis can be masked undei a definition: we call an arithmetical function effectively calculable if and only if it is general recursive, venturing however that once hi the future, somebody will define a function which is on the one hand, not effectively calculable in the sense defined thus, on the other hand, its value obviously can be effectively calculated for any given arguments. Similarly, in defining a problem, containing a parameter which runs through the natural numbers, to be solvable if and only if its characteristic function a) is general recursive, one takes the risk that somebody in the future will solve a problem *) *) The characterestic function of a problem with parameter is defined as the function taking the value 1 or 0 for a value of the pr-rameter according as “yes” of “no” is the correct answer to the particular case of the problem corresponding to this value of the parameter. 72 5. ábra A Church-tézishez hozzászóló tanulmány címlapja [A. 61]
