Button, Tim;
(2020)
Mathematical internal realism.
In: Conant, J and Chakraborty, S, (eds.)
Engaging Putnam.
(pp. 157-182).
De Gruyter: Berlin, Germany.
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Abstract
In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. This also allows us to understand why “Models are not lost noumenal waifs looking for someone to name them,” but “constructions within our theory itself,” with “names from birth.”
| Type: | Book chapter |
|---|---|
| Title: | Mathematical internal realism |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1515/9783110769210-007 |
| Publisher version: | https://doi.org/10.1515/9783110769210-007 |
| Language: | English |
| Additional information: | This version is the version of record. For information on re-use, please refer to the publisher's terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL SLASH UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of Arts and Humanities UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of Arts and Humanities > Dept of Philosophy |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10115470 |
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