  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.
