Plato and the foundations of mathematics

Abstract

Plato is well-known in mathematics for the eponymous foundational philosophy Platonism based on ideal objects. Plato's allegory of the cave provides a powerful visual illustration of the idea that we only have access to shadows or reflections of these ideal objects. An inquisitive mind might then wonder what the current foundations of mathematics, like e.g. Reverse Mathematics and the associated Goedel hierarchy, are reflections of. In this paper, we identify a hierarchy in higher-order arithmetic that maps to the Big Five of Reverse Mathematics under the canonical embedding of higher-order into second-order arithmetic. Conceptually pleasing, the latter mapping replaces uncountable objects by countable 'codes', i.e. the very practise of formalising mathematics in second-order arithmetic. This higher-order hierarchy can be defined in Hilbert-Bernays' Grundlagen, the spiritual ancestor of second-order arithmetic, while the associated embedding preserves equivalences. Also, in contrast to Kohlenbach's hierarchy based on discontinuity, our hierarchy can be formulated in terms of (classically valid) continuity axioms from Brouwer's intuitionistic mathematics. Moreover, the higher-order counterpart of sequences is provided by nets, aka Moore-Smith sequences, while the gauge integral is the correct generalisation of the Riemann integral. For all these reasons, we baptise our higher-order hierarchy the Plato hierarchy.