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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Notes
Compare the seminal article by Medvedev in (1955).
Actually, Kolmogorov´s proposal became a component of an interpretation of intuitionistic logic (BHK interpretation).
Mancosu (1998), 329.
See, e.g., Zalta, E. (1988).
and restored, when necessary.
One of the possible notations.
So (C), the construction (an abstract procedure), is not identical with the expression under (C).
As for types, see below.
Composition (unlike Closure) can be (v-)improper, i.e., it can construct nothing.
Again: Precise definitions can be found, e.g., in DJM (2010).
Saying ´type of calculate´ we mean here (and in similar cases) ´type of the object denoted by calculate´.
Trivializing the construction we lose any interest in performing the procedure. We just identify the procedure itself.
Similarly as the famous “Great Fact” in mathematics: we can calmly state that each true mathematical sentence denotes the same truth-value, being aware that what does distinguish particular sentences in mathematics is their meaning rather than their denotation.
See again Medvedev (1955).
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Acknowledgments
I am grateful to the anonymous reviewers for their valuable comments. This paper has been supported by Grant Agency of Czech Republic Project No. P401/10/0792.
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Materna, P. Equivalence of Problems (An Attempt at an Explication of Problem). Axiomathes 23, 617–631 (2013). https://doi.org/10.1007/s10516-012-9201-4
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DOI: https://doi.org/10.1007/s10516-012-9201-4