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5 Dadaczyński J.

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The Earliest Form of Hilbert’s Formalism

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Dadaczyński J.

nr 3

117-129

The aim of this paper is to describe and analyze the first (1922) of a long series of Hilbert’s works in which he presented the mature version of formalism. His formalism in 1922 can be called mature because it is characterized by an explicit introduction of metamathematics. Hilbert distinguishes several levels of mathematics (not just two, as one may think - formalized mathematics and metamathematics): the level of meaningless arithmetical signs, labelled I-Z in this paper, the level of arithmetic with content (inhaltliche Arithmetik), II-T, which describes signs from level I-Z, the level of formalized mathematics, II-F (Hilbert postulated a full formalization of mathematics), and the level of metamathematics with content, III-MM, which describes signs from level II-F. Hilbert emphasized that the relation of III-MM to II-F is the same as the relation of II-T to I-Z (description, investigation). In this way, he tried to characterize metamathematics. He expected that in III-MM a consistency proof of II-F could be built, which was the aim of Hilbert’s formalism. This paper discusses Hilbert’s first proof of an auxiliary metamathematical theorem. It is indicated that, on level III-MM, Hilbert assumed a part of arithmetic from level II-T and the classical logic. Although in 1922 he did not distinguish explicitly between the finite and infinite mathematics and between the real and ideal mathematics, such a division was implicit in his study. This allows us to assume that already in 1922 Hilbert had an idea of a finitistic consistency proof of infinitistic mathematics, announced a few years later. It appears, therefore, that already in 1922 he had a very clear idea of formalism, which was presented in detail in the middle of the decade. Hilbert was also aware in 1922 that Brouwer’s objections would eventually force him to explain the issue of logical foundations of classical mathematics.

The Early Hilbert’s Ontology of Mathematics

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Dadaczyński J.

tom 26

nr 4

75-88

Hilbert’s views on the ontology of mathematics changed significantly between 1891 and 1904. Although his contributions to the foundations of mathematics in the years 1899-1904 paved the way for his later program of formalism, in the ontology of mathematics he was then still far from methodological nominalism associated with his mature formalism. Paradoxically, Hilbert’s original position in the ontology of mathematics (from 1891) was that of conceptualism combinedwith constructivism. These two views were the philosophical basis for Brouwer’s intuitionist attacks on Hilbert’s account of the foundations of mathematics in the 1920s.

On Some Inspirations of Hilbert’s Formalism

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Dadaczyński J.

tom 26

nr 3

99-112

Hilbert’s program of formalism was undoubtedly a result of many mathematical, logical, and philosophical factors. The aim of this paper is to indicate some rarely highlighted contexts. One important goal of Hilbert’s program was to prove the consistency of arithmetic. The paper shows that Hilbert did not begin the study of this issue only after the discovery of Russell’s paradox. The issue of the consistency of the arithmetic of real numbers was associated with the discovery - by Burali-Forti and Cantor - of the first set-theoretical antinomy, i.e. the antinomy of the greatest ordinal number. Hilbert, already in 1899, asked whether the set of real numbers - to use Cantor’s terminology - was a consistent collection. He then raised the issue of the consistency of the arithmetic of natural numbers in 1904, after the discovery of Russell’s paradox. Fundamental for Hilbert’s mature program of formalism was the distinction between the finitistic and the infinitisticmathematics. The paper points out that the source of this distinction can be found in Brouwer’s proof-theoretical and constructivist criticism of certain theorems of the classical logic. So significant was the criticism that Hilbert had to take it into account in his formalistic reconstruction of classical mathematics. The result was precisely his distinction between the finitistic and the infinitistic mathematics.

Bernard Bolzano: pierwsze (historycznie) matematyczne ujęcie pojęcia kontinuum

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Dadaczyński J.

nr 64

195-199

Book review: Lukas Benedikt Kraus, Der Begriff des Kontinuums bei Bernard Bolzano, Beiträge zur Bolzano-Forschung, vol. 25, Academia Verlag, Sankt Augustin 2014, pp. 112.

Recenzja książki: Lukas Benedikt Kraus, Der Begriff des Kontinuums bei Bernard Bolzano, Beiträge zur Bolzano-Forschung, vol. 25, Academia Verlag, Sankt Augustin 2014, ss. 112.

Giuseppe Veronesego podstawy matematyki

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Dadaczyński J.

2013

nr 53

53-92

The aim of this paper is to analyze Veronese’s philosophical principles of mathematics. He tried to begin the construction of mathematics (geometry) with the concept of the Thinking Subject and the phenomenon of thinking, which is discussed in detail. It is very likely that this idea had an impact on Hilbert’s concept of the Mathematical Subject. Some similarities between Veronese’s view and the intuitionistic thinkers are also shown.