There is a wide-ranging set of analogies around the (inherently paradoxical) idea of the "field with one element", one of which is that the integers are expecte

There is a wide-ranging set of analogies around the (inherently paradoxical) idea of the "field with one element", one of which is that the integers are expected to behave like the ring of polynomials over a field with one element. Another example is that the symmetric group on n elements is expected to behave like the group of invertible n x n matrices. See this page for a fairly full list of examples: https://ncatlab.org/nlab/show/field+with+one+element https://ncatlab.org/nlab/show/field+with+one+element