Hilbert’s arithmetisation of geometry
DOI:
https://doi.org/10.26806/fd.v10i1.269Abstract
In this article, we describe how David Hilbert (1862–1943) understood the arithmetisation of geometry in the book Grundlagen der Geometrie from 1899. First, we introduce Hilbert’s forerunners from the same period who were either asking for changes in the foundations of geometry or implemented them by axiomatic-deductive method, and we do not omit the work included in Hilbert’s previous lectures. Further, we try to explain the contents of the first two chapters of the book and present the context which is necessary to understand them. We present the implicit definitions of the elementary notions and relationships in the axioms and Hilberts division of the axioms into groups. We focus more on the axioms of continuity in context with the problem of the consistency. As an aside, we describe the construction of the arithmetical model of the axioms of geometry that Hilbert uses for the consistency proof. At the end, we strive to show Hilbert’s main intentions for writing the book and we mention some of the implications of his treatment
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