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An Intuitionist Reasoning Upon Formal Intuitionist Logic: Logical Analysis of Kolmogorov’s 1932 Paper

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Abstract

Two dichotomies are considered as the foundations of a scientific theory: the kind of infinity—either potential or actual-, and the kind of organization of the theory—axiomatic or problem-based. The original intuitionist program relied on the choices of potential infinity and the problem-based organization. I show that the logical theory of Kolmogorov’s 1932 paper relied on the same choices. A comparison of all other theories sharing the same foundational choices allows us to characterize their common theoretical development through a few logical steps. The theory illustrated by Kolmogorov’s paper is then rationally re-constructed according to the steps of this kind of development. One obtains a new foundation of intuitionist logic, which is of a structural kind since it is based on and developed according to the structure of the above mentioned two fundamental choices. In addition, Kolmogorov’s illustration of his theory of intuitionist logic is an instance of rigorous reasoning of the intuitionist kind.

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Notes

  1. Notice that each modal word is equivalent to a DNP; e.g. “it is that…” is equivalent to “It is not true that it is not…”. This equivalence is confirmed by the S4 translation of model logic into IL These words will be pointwise underlined, while the two negative words of a DNP will be underlined continuously.

  2. An application of PSR changing the final predicate of a PO theory constitutes a move by an author that is of a subjective and intuitive nature, because no logical argument of a specific kind of logic can justify it. In my opinion it represents Beth’s “unavoidable appeal of a form of intuitive knowledge of Mathematics” ([1], p. 645). Moreover, it is the ultimate justification of Brouwer’s appeal to a subjective viewpoint of mathematics and logic. It represents in a PO theory that intuitive element which according to Goedel’s theorems cannot be captured by an AO theory.

  3. Their debate on the foundations of mathematics is illustrated in the light of the two dichotomies in [7].

  4. Here and in the following quotations the word “intuitionistic” will be replaced by the word “intuitionist”.

  5. These papers have been already studied by [5].

  6. In this Sect. the reference to a page of Kolmogorov’s 1932 paper will be indicated by means of a number in round brackets.

  7. I will refer to the text in [17]. For a further translation from German, see J. Mckinna’s site: inf.ed.ac.uk/McKinna. (I thank Miriam Franchella for suggesting it to me).

  8. Since 1987 texts of OP theories have been analyzed by means of DNPs: classical chemistry, S. Carnot’s thermodynamics, L. Carnot’s mechanics, Lagrange’s mechanics, Lobachevsky’s theory of parallels, Galois’ theory, etc. For the methodology and the results see [6].

  9. In order to simplify the following analysis, I will omit a proposition including only a modal word, such as “can”, “must”, “should”, “necessary”, etc. Rigorously, it is equivalent to a doubly negated proposition, but the recognition of a true DNP is more difficult and moreover it may be considered, as in fact it is in some cases, a mere colloquial way of speaking. The following belong to this category: “without misunderstandings” (p. 329), “no difficulties” (p. 330), and “cannot lead to any misunderstandings” (fn. no. 6).

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Acknowledgements

I acknowledge Professor David Braithwhaite for the numerous corrections on my English text.

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  1. Department Physical Sciences, University “Federico II” of Naples, Via Benvenuti 5, 56011, Calci, Italy

    Antonino Drago

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Drago, A. An Intuitionist Reasoning Upon Formal Intuitionist Logic: Logical Analysis of Kolmogorov’s 1932 Paper. Log. Univers. 15, 537–552 (2021). https://doi.org/10.1007/s11787-021-00292-3

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