THE ROLE OF MATHEMATICS IN PHYSICAL SCIENCES

The problem whether mathematics is a ``series of consequences of arbitrary axioms" or a branch of natural sciences and of theoretical physics was a subject of discussion since Hilbert (who was a pursuer of Descartes and a precursor of Bourbaki) and Poincaré (the founder of modern mathematics, topology, chaos theory and dynamical systems). I will speak essentially about some examples, showing the cardinal differences of view points between the axiomaphiles and the naturalists already on some basic concepts, as derivatives and limits, theorems of existence and uniqueness, optimization and control theory, the unsolvability of certain problems and the measure of complexity of certain others. Despite the basic role played by mathematics in all sciences, there are still eminent mathematicians believing (or, perhaps, affirming to believe) that mathematics has nothing to share with our world. This stance is especially dangerous in our time, as a new obscurantism, mainly observed in the antiscientific reform of education (in most First World's countries) menaces the bases of our whole culture.