reader, who is free to replace it in his mind by another interpretation that satisfies the conditions stated by the unproved propositions. And since the propositions, from the deductive point of view, do not state facts, but conditions, we cannot consider them genuine postulates. Padoa went on to say: ...what is necessary to the logical development of a deductive theory is not the empirical knowledge of the properties of things, but the formal knowledge of relations between symbols. Mathematicians' congress Padoa spoke at the 1900 International Congress of Mathematicians with his title "A New System of Definitions for Euclidean Geometry". At the outset he discusses the various selections of primitive notions in geometry at the time: The meaning of any of the symbols that one encounters in geometry must be presupposed, just as one presupposes that of the symbols which appear in pure logic. As there is an arbitrariness in the choice of the undefined symbols, it is necessary to describe the chosen system. We cite only three geometers who are concerned with this question and who have successively reduced the number of undefined symbols, and through them (as well as through symbols that appear in pure logic) it is possible to define all the other symbols. First, Moritz Pasch was able to define all the other symbols through the following four: 1. point 2. segment (of a line) 3. plane 4. is superimposable upon Then, Giuseppe Peano was able in 1889 to define plane through point and segment. In 1894 he replaced is superimposable upon with motion in the system of undefined symbols, thus reducing the system to symbols: 1. point 2. segment 3. motion Finally, in 1899 Mario Pieri was able to define segment through point and motion. Consequently, all the symbols that one encounters in Euclidean geometry can be defined in terms of only two of them, namely 1. point 2. motion Padoa completed his address by suggesting and demonstrating his own development of geometric concepts. In particular, he showed how he and Pieri define a line in terms of collinear points. References Bibliography A. Padoa (1900) "Logical introduction to any deductive theory" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 118–23. A. Padoa (1900) "Un Nouveau Système de Définitions pour la Géométrie Euclidienne", Proceedings of the International Congress of Mathematicians, tome 2, pages 353–63. Secondary: Ivor Grattan-Guinness (2000) The