Fregean hierarchies and mathematical explanation

Michael Detlefsen
University of Notre Dame
There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider two models for such orderings, (i) an implicational model and something I call (ii) a mosaic model. -/- Both models fail to do what such orderings should do. Moreover, in both cases, the trouble arises because the class of truths to be ordered is taken to be closed under various weakening implications. This is a result of their employing a global conception of logic.
Keywords Aristotle  Bolzano  Frege  objective orderings of truths  Leibniz  proof and the objective ordering of truths  mosaic model of truth-orderings  implicational model of truth-orderings  grounding proofs
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DOI 10.1080/02698598808573327
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