Geometry, number and structural axiomatics

GIOVANNINI, EDUARDO N.

Viena

Workshop; International Workshop: Structural Methods in Nineteenth-Century Mathematics; 2017

University of Vienna

Hilbert?s axiomatic investigations into the foundations of Euclidean geometry are often viewed as one of the clearest and most influential examples of the introduction of structural methods in modern mathematics. According to this view, Hilbert?s early structuralist position can be easily appreciated in his novel description of the subject?matter of mathematical theories as ?schema or scaffolding of concepts?, as it is illustrated for example in his famous letter exchange with Frege: ?it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes?. Now, as can be also noted in this often quoted passage, Hilbert?s early mathematical structuralism was, in a certain sense, an immediate consequence of his new view on the nature of the primitives of a mathematical theory. As is well known, Hilbert? main methodological and philosophical thesis was that the primitive terms of a mathematical theories should be conceived schematically, and therefore not fixed to a given interpretation. Roughly speaking, his structuralist conceptions of mathematical was a response to the philosophical problem of explaining the nature of mathematical primitives. The main aim of this talk will be then to complement this view by analyzing how Hilbert?s well known early structuralist conception of mathematical theories is connected to one the most important results achieved in Foundations of geometry, that is, the arithmetic of segments of Streckenrechnung. In fact, Hilbert showed that, once the operations of sum and product of line segments have been defined in an adequate and purely geometrical way, it is possible to use the classical theorems of Desargues and Pascal to prove that these operations satisfy all the properties of an ordered field. This purely geometrical construction of a set of segments, which satisfies all the properties of an ordered field, allowed him to reconstruct the classical Euclidean theory of proportions and similar triangles, to which he finally resorted to perform an internal arithmetization of geometry.