 The primary question for the philosophy of mathematics is surely what is mathematics. Duality operations like topological duality or Routley* reinforce the point that incomplete/inconsistent duals are equally reasonable as examples of mathematics. From this point of view, disputes about which of intuitionist or classical or inconsistent mathematics to accept seem pointless; they are all part of the subject matter of mathematics. This point is made effectively by Shapiro (2014, in contrast see his 2002). Shapiro’s distinctive position has other ingredients: mathematics as the science of structure, and mathematical pluralism implying logical pluralism (on logical pluralism see also Beall and Restall 2006); but we do not take these up here.