Abstract
We present and discuss a change in the introduction of Hilbert’s Grundl agender Geometrie between the first and the subsequente ditions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the connection between the notions of independence and simplicity.
Highlights
In this brief note we would like to discuss a philological observation that results from comparing the introductions of the various editions of Hilbert’s Grundlagen der Geometrie. This observations deals with the independence of the axioms that Hilbert proposed for the foundations of geometry
The philological observation we intend to discuss originates from collating the introduction of the Festschri, that is the rst edition of the Grundlagen der Geometrie presented for the unveiling of a statue of Gauss and Weber in Gottingen in, and those of the subsequents editions
The establishment of the axioms of geometry and the investigation of their relationships is a problem which has been treated in many excellent works of the mathematical literature since the time of Euclid. This problem is equivalent to the logical analysis of our perception of space. This present investigation is a new attempt to establish for geometry a complete, and as simple as possible, set of axioms and to deduce from them the most important geometric theorems in such a way that the meaning of the various groups of axioms, as well as the signi cance of the conclusions that can be drawn from the individual axioms, come to light
Summary
In this brief note we would like to discuss a philological observation that results from comparing the introductions of the various editions of Hilbert’s Grundlagen der Geometrie. This observations deals with the independence of the axioms that Hilbert proposed for the foundations of geometry. The issue of independence was discussed already by Hilbert’s contemporaries. The groups of axioms of the Grundlagen der Geometrie were mutually independent, this was not the case for the single axioms that constitute a particular group; as shown for example by E. In discussing the issue of independence, we will outline the theoretical importance of this notion in Hilbert’s foundational work, together with its role in the application of the axiomatic method
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