 Duality between incompleteness/intuitionism and inconsistency/paraconsistency has at least two aspects. First there is the above topological (open/closed) duality. Second there is Routley * duality. The Routley Star * of a set of sentences S, is defined as S* =df {A: ~A is not in S}. Discovered by the Routleys (1972) as a semantical tool for relevant logics, the * operation dualises between inconsistent and incomplete theories of the large natural class of de Morgan logics. For example, starring Peano arithmetic gives a deductively-closed inconsistent complete non-trivial classical arithmetical theory containing all of PA, which presents an interesting challenge for the Godel incompleteness results, see Mortensen (2013). Both kinds of duality interact as well, where the * gives distinctive duality and invariance theorems for open set and closed set arithmetical theories. On the basis of these results, it is fair to argue that both kinds of mathematics, intuitionist and paraconsistent, are equally reasonable.