Equivalence of Problems (An Attempt at an Explication of Problem)

Axiomathes 23 (4):617-631 (2013)
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Abstract

On the one hand, Pavel Tichý has shown in his Transparent Intensional Logic (TIL) that the best way of explicating meaning of the expressions of a natural language consists in identification of meanings with abstract procedures. TIL explicates objective abstract procedures as so-called constructions. Constructions that do not contain free variables and are in a well-defined sense ´normalized´ are called concepts in TIL. On the second hand, Kolmogorov in (Mathematische Zeitschrift 35: 58–65, 1932) formulated a theory of problems, using NL expressions. He explicitly avoids presenting a definition of problems. In the present paper an attempt at such a definition (explication)—independent of but in harmony with Medvedev´s explication—is given together with the claim that every concept defines a problem. The paper treats just mathematical concepts, and so mathematical problems, and tries to show that this view makes it possible to take into account some links between conceptual systems and the ways how to replace a noneffective formulation of a problem by an effective one. To show this in concreto a wellknown Kleene’s idea from his (Introduction to metamathematics. D. van Nostrand, New York, 1952) is exemplified and explained in terms of conceptual systems so that a threatening inconsistence is avoided

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References found in this work

Introduction to metamathematics.Stephen Cole Kleene - 1952 - Groningen: P. Noordhoff N.V..
Introduction to mathematical logic.Alonzo Church - 1944 - Princeton,: Princeton University Press. Edited by C. Truesdell.
Introduction to mathematical logic..Alonzo Church - 1944 - Princeton,: Princeton university press: London, H. Milford, Oxford university press. Edited by C. Truesdell.
Quality and concept.George Bealer - 1982 - New York: Oxford University Press.

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