A GENETIC APPROACH TO AXIOMATICS

The genetic method is often regarded as a counter-current to the New Math and its exaggeration of formal and axiomatic mathematics. As a consequence axiomatics has been considerably reduced (nearly deleted) in school mathematics, whereas university mathematics is mostly still presented in a rigid deductive way. This discrepancy leads many freshmen to considerable difficulties, as we all know. In this paper I will propose a synthesis between genetic and axiomatic method. In particular the axiomatic method is not only a method but also an interesting and very important subject of teaching and research itself: a milestone in the development of mathematics (Euclid), its philosophical background (Aristoteles), its purpose (Zenon), its consequences (construction with compass and ruler). Axiomatics as a model for representation of topics of mathematics (and other sciences) up to now, axiomatics as a destination of a process, not a starting point. Examples of how to cope with axiomatics at school and at university will be discussed in this paper. 1. Preliminary Remarks In the fifties / sixties of the last century one of the main goal of the New Math was to reduce the gap between mathematics at school and at universities. School mathematics tried to imitat