Topics in Philosophy of Mathematics - A

Potentialist and generative approaches to the philosophy and foundations of mathematics

Aristotle famously claimed that the only coherent form of infinity is potential, not actual. However many objects there are, it is possible for there to be yet more; but it is impossible for there actually to be infinitely many objects. Although this view was superseded by Cantor’s transfinite set theory, even Cantor regarded the collection of all sets as “unfinished” or incapable of “being together”. In recent years, there has been a revival of interest in potentialist and other generative approaches to the philosophy and foundations of mathematics, according to which an ontology of mathematical objects is successively “generated”, or accounted for, in an incompletable “process”. The course provides a survey of such approaches, older as well as newer, including (i) Aristotle’s view of infinity; (ii) Cantor’s conception of the transfinite and absolute infinity; (iii) the iterative conception of sets; (iv) potentialism in constructive mathematics; (v) recent potentialist and generative approaches; (vi) connections with the hierarchical conception of reality.

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In presence

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