Andrej Ule

In the article are presented the main philosophical explanations of the application of mathematics on the real world (Plato, Aristotle, rationalists, empiricists, Kant, Frege, Husserl, Carnap etc.). They indicate some typical triangular structure of relationships where the mathematical structures somehow correspond to the forms of reality, and thus are possible though something third what bound them. The attempts to solve the question of the application of mathematics by the dispensability of mathematics (e.g. Field) do not success because they do not explain the big success of mathematics in science. However they call our attention to the meaning of transpositions of some empirical contents on the level of theirs mathematical representations. That is neither an abstraction nor an idealisation but is the mapping from the empirical into a formal language. The opposite attempts to identify mathematics and the fundamental structure of reality also do not explain the success of mathematics because of the obvious ontological differences between the mathematical and the real objects. I return finally to Plato and show how he denies the reduction of numbers to some sets. We have to distinguish the idea of numbers and the various aspects of it, for example the cardinal and the ordinal numbers. According to this platonic insight only the idea of numbers exists. Its aspects are only some partial reflections of this idea in our mathematical theories and in the application of mathematics.