Starting from the scenario Euclid–Bolyai–Einstein
Our aim is to propose several itineraries which follow the scenario having as a first step Euclid’s Fifth Postulate; as a second step the Bolyai–Lobachevsky’s non-Euclidean geometries and as a third step Einstein’s relativity theory. The role of Euclid’s fifth postulate is successively assumed by Archimedes’ axiom; Zermelo’s choice axiom; Cantor’s continuum hypothesis; von Neumann’s foundation axiom for set theory; Church–Turing thesis and Turing’s computability; the validity of classical logic under the form of the principles of identity, non-contradiction and excluded middle; the macroscopic universe assumed as form of existence, including the fact that humans are macroscopic beings; human language and semiosis assumed as the unique way to approach understanding and communication; the old Greek ideal of order, symmetry and harmony, adopted for representation of the world in Art and Poetry. The common denominator of these starting steps is the fact that they all correspond to the sensorial-empirical perception of the world. We point out the strong interaction among the steps corresponding, in these different itineraries, to Einstein’s relativity theory. We infer in this way the need of a transdisciplinary approach.