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The Croatian Reception of Hilbert's Axiomatic Method

The reception of Hilbert's axiomatic method can be traced in Croatian authors from 1907 to the recent university textbooks. It includes the problems of the axiomatization of geometry and real numbers, as well as the foundational and metatheoretical questions of axiomatic systems. We give a selection of authors and problems. The first author in Croatia to present Hilbert's axiom system for geometry was Vladimir Varićak (1907). For him, the non-Archimedean geometry is the ``most important Hilbert's conception''. He further elaborated the concepts of non-Pascalean and non-Pythagorean geometry, and in (1908) refers to Hilbert in the presentation of the general equation of a straight line in the hyperbolic plane. In addition, Varićak applies three-dimensional Lobachevsky's geometry to the relativity theory (1910-1924). Fran Mihletić (1912) refers to Hilbert in a discussion on the sufficient and necessary instruments in the construction and problem solving in elementary geometry, and proposes his Dividers Axiom. He puts Hilbert's theory of the equivalence of polygons into the context of the logical building of mathematics (Cantor, Russell). Mate Meršić (1914) criticizes Hilbert's axiomatization of geometry (axioms should be founded on truths, which in turn guarantee the consistency of the system), and requires a deeper foundation of geometry in the ``algorithmic'' functioning of human mind. Danilo Blanuša's proof of embeddability of hyperbolic plane in the Euclidean 6-dimensional space (1955) is related to the negative result of Hilbert's Theorem (1901). In addition, Blanuša (1965) refers to Hilbert's axiomatization of geometry in his discussion of the provability of Desargues' theorem. Stanko Bilinski (``On the foundations of axiomatics'', 1956) deals with independency, completeness, and consistency as the fundamental principles of axiomatics, without explicitly mentioning Hilbert. He explains different conceptions of the principle of completeness, and presents its interpretation by means of isomorphy. Stjepan Mintaković describes Hilbert's axiomatization of geometry in a separate chapter of his booklet (1962), with the main aim to give insight in the work of Lobachevsky and its consequences for the axiomatic foundation of geometry. In Vladimir Devidé's proof-theoretical work, the preferred requirements are, besides Hilbertian independency and consistency, e.g., concision (axiomatization of natural numbers, 1955), ``complementarity'' of axioms (classical propositional logic axiomatization, 1964, proved equivalent to Hilbert-Bernays 1934), avoidance of induction and ordinals (proof of the well-ordering theorem, 1963, 1967). A generalization of Devidé's axiomatization of natural numbers with respect to the successor concept (immediate/general) is proposed by Z. Šikić (1989, 1991). Technical and philosophical impact of Gödel's incompleteness theorems and of the subsequent development of mathematics and logic on Hilbert's Programme is commented on, for example, by V. Devidé (1975, 1991), M. Mihaljinec (1977), and Z. Šikić (1989). Hilbertian and similar axiomatizations of geometry and real numbers are included in the currently used university textbooks, for example, in B. Pavković and D. Veljan, Elementary Mathematics I, Zagreb, 1992, 2004, and B. Červar, G. Erceg and I. Lekić, Basic Geometry, Split, 2014.

D. Hilbert, axiomatic method, non-Archimedean geometry, natural numbers, consistency, independency, completeness, V. Varićak, M. Meršić, V. Devidé, S. Bilinski,

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