Ontology and the Foundations of Mathematics
Talking Past Each Other

The philosophical positions of realism and anti-realism are difficult to distinguish, nowhere more so than in the philosophy of mathematics.

Penelope Rush (Author)

9781108716932, Cambridge University Press

Paperback / softback, published 10 February 2022

75 pages
22.9 x 15 x 0.4 cm, 0.09 kg

'… extremely thought-provoking … Rush's tenacity in pressing [ontological access problem] questions about the relevance of objecthood and independence is unique, unsettling, unrelenting, and effective.' Nicholas Danne, Metascience

This Element looks at the problem of inter-translation between mathematical realism and anti-realism and argues that so far as realism is inter-translatable with anti-realism, there is a burden on the realist to show how her posited reality differs from that of the anti-realist. It also argues that an effective defence of just such a difference needs a commitment to the independence of mathematical reality, which in turn involves a commitment to the ontological access problem – the problem of how knowable mathematical truths are identifiable with a reality independent of us as knowers. Specifically, if the only access problem acknowledged is the epistemological problem – i.e. the problem of how we come to know mathematical truths – then nothing is gained by the realist notion of an independent reality and in effect, nothing distinguishes realism from anti-realism in mathematics.

1. What are we Talking about?
2. Inter-translatability
3. Two Access Problems
4. Independence
5. Justification.

Subject Areas: Philosophy of science [PDA], Philosophy of mathematics [PBB], Mathematics [PB], Mathematics & science [P], Philosophy [HP]