According to the received view, formalism – interpreted as the thesis that mathematical truth does not outrun the consequences of our maximal mathematical theory – has been refuted by Gödel's theorem. For the theorem seems to show that for any putative maximal theory there is at least one sentence – its Gödel sentence – which we can recognize as true and yet is not provable in it. In support of this claim, proponents of the received view usually invoke an informal argument for the truth of the Gödel sentence, an argument which is supposed to reconstruct our reasoning in seeing its truth. Against this, Field (1994, 1998a, 1998b) has argued in a series of papers that the principles involved in this argument –...

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