An Alleged Tension between Non-classical Logics and Applied Classical Mathematics

TL;DR

The paper examines alleged conflicts between non-classical logics and classical mathematics in scientific applications, specifically addressing quantum logic, and argues that no such tension exists, offering broader philosophical insights.

Contribution

It reconstructs Williamson's argument, refines its scope to quantum logic, and demonstrates the absence of tension between classical mathematics and non-classical logics in scientific contexts.

Findings

No tension between classical mathematics and quantum logic in modeling quantum phenomena.

Williamson's key premise fails for quantum logic and other non-classical logics.

Broader implications for the relationship between applied logic and applied mathematics.

Abstract

Timothy Williamson has recently argued that the applicability of classical mathematics in the natural and social sciences raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We first reconstruct his argument and present its restriction to the case of quantum logic (QL). Then we show that there is no problematic tension between the applicability of classical mathematical models to quantum phenomena and the endorsement of QL in the reasoning about the latter. Once we identify the premise in Williamson's argument that turns out to be false when restricted to QL, we argue that the same premise fails for a wider variety of non-classical logics. In the end, we use our discussion to draw some general lessons concerning the relationship between applied logic and applied mathematics.