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In joint work with Hans-Niels Jahnke (Universität Duisburg-Essen), we investigate the issue of justification of axioms in mathematics, from ancient Greek geometry to current debates on set theory, category theory and the foundations of mathematics. The aim of the talk is not to give a complete history of the phenomenon but to highlight its relevance (not sufficiently taken into account in the existing literature, in our opinion) by focussing on some particular cases. We take a look at Proclus’s discussion of Euclid’s axioms and postulates (especially, but not exclusively, the parallel postulate), at how Archimedes and much later Klein discuss the archimedean axiom, and finally at Penelope Maddy’s account of axioms of set theory, inspired by Zermelo’s remarks on the axiom of choice. The last case leads us to similar considerations concerning the role of category theory in the foundations of mathematics.
