  We explore the rational, formal and non-formal criteria of consistency,
non-triviality and redundancy in the mathematical research now a days. We
develop a paradigmatic discussion by analysing the different conceptions of
those criteria, from the logic-formal ones to the non formal ones (but still
rational criteria).We illustrate the discussion with concrete examples obtained
form the mathematical reseach, particularly from the results that were
published in the last 50 years in the mathematical theory of deterministic
dynamical systems.
