Original scientific paper
The Concept of the Infiniti mysteria in Bošković’s Geometrical Investigations
Ivica Martinović
; Institut za filozofiju, Zagreb, Hrvatska
Abstract
When Bošković first mentioned the concept ‘the mysteries of the infinite’ (Infiniti mysteria) in his treatise De maris aestu (1747), he asserted that it was necessary to include the mysteries of the infinite into the investigation of geometric transformations. At that time, on the basis of the demonstration in his early treatise De natura et usu infinitorum et infinite parvorum (1741), he already had some experience in disputing the actual infinite in geometry. Therefore he based his study of the mysteries of the infinite on the philosophical assumption of the existence of the infinite.
While forming the theory of geometric transformations in his treatise De transformatione locorum geometricorum (1754), he gave a large meaning to this concept: all the manifestations of the potential and actual infinite. Only with his treatise De continuitatis lege (1754) did he start to make a strict distinction between mystery and absurdity in the understanding of the geometric infinite, and from that time on he recognized the mysteries of the infinite only in those geometric quantities and transformations in which the potential infinite occurs, on condition that the principle of continuity was preserved.
Two confirmations of Bošković’s understanding of the Infiniti mysteria in this specific way are to be found in his correspondence in the 1760s: his valuable epistolary treatise written in Constantinople from 20 December 1760 to 26 February 1761 for the young Giovan Stefano Conti, and Bošković’s exhaustive reply to the Swiss scholar Le Sage of 8 May 1765.
On the contrary, absurdity always follows from the assumption of the actual infinite, and it is ascertained during the process in which the structure of bijection and relationship ‘part-whole’ are used, that is, both aspects which strongly mark Bernard Bolzano’s paradoxical conception of the relationship between infinite sets, and Richard Dedekind’s mathematical definition of the infinite system.
In his model for ascertaining absurdity, Bošković always uses the relations between geometric quantities as representatives of the relationships between infinite quantities. The turning point which was prepared by Bolzano in his Paradoxien des Unendlichen (1851), and achieved by Dedekind and Georg Cantor, took place in another mathematical field, namely, in the set approach to the real numbers. These two points, the use of the same mathematical contents, such as the structure of bijection and the relationship ‘part-whole,’ on the one hand, and the difference between the Euclidean geometric approach and the set approach on the other hand, determine the place of Ruđer Bošković in the historical process of forming the exact, mathematical definition of the infinite.
Keywords
Ruđer Bošković; Bernard Bolzano; geometry; theory of geometrical transformations; the actual infinite; the potential infinite; Infiniti mysteria
Hrčak ID:
154370
URI
Publication date:
15.10.2015.
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