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In some cases he's rational, irrational, but not real: https://en.wikipedia.org/wiki/Math_fab_Mathonwy

Yes. But does irrationality imply non-reality?

I work with several individuals who are consistently irrational and frustratingly real.

Math can also be imaginary, but this also seems to be a feature of the universe.

Not for a numberphile.

Only if it's Schrödinger's math

This is a legit topos.