 Poincaré’s geometrical conventions and his ontological relationalism are nothing more than two different aspects of his structural approach, as can be seen by considering two parallel debates at the turn of the 20th century on the nature of geometry—one between Henri Poincaré and Bertrand Russell and the other between David Hilbert and Gottlob Frege. Poincaré and Hilbert argue for a new conception of geometrical systems, while Frege and Russell defend a traditional way of looking at these systems. The debate takes up the questions: Which geometry is true? What is the nature of definitions and axioms? To deal with the multiplicity of geometries, questions about their truth, and the meaning of their primitive terms, Poincaré and Hilbert propose a radically new view of geometry. They hold that all that we can say about the meaning of ‘point’, ‘straight line’, ‘distance’, etc. is that which we have stated in the axioms or principles of the system, and that geometry is not a set of truths about some previously known objects. Thus Poincaré formulates a new view of geometric theories, that geometry does not express true or false propositions and that there are no special objects which geometry studies. Rather, geometry is just a system of relations that can be applied to many kinds of objects.