Math and Logic

Chia-Hua Lin (University of South Carolina)
 
Mathematical formalisms that are constructed for inquiry in one disciplinary context are sometimes applied to another, a phenomenon that I call ‘tool migration.’ Philosophers of science have addressed the advantages of using migrated tools. In this paper, I focus on the issue of epistemic risks associated with tool migration and argue for a study of tool migration. Using the migrations of game theory as examples, I identify three kinds of epistemic risks related to applying a mathematical construct outside the discipline where it was established as a feasible research tool. These include mischaracterization (i.e., characterizing a new phenomenon with inappropriate assumptions), misinterpretation, (i.e., using ill-fitted background contexts to interpret the result), and misjudgment (i.e., incorrectly rejecting or accepting a novel use of a migrating research tool). I then develop an analytic framework for better understanding the epistemic activities involved in tool migration, especially where in the process epistemic risks may arise and how scientists cope with similar risks.

Alex Koo (University of Toronto)
The Enhanced Indispensability Argument (EIA) for mathematical realism supposedly represents an improvement over its Quinean predecessor due to the replacement of confirmational holism with Inference to the Best Explanation (IBE) to infer realism. The assumption is that IBE can be used to infer that mathematical entities exist. I argue that this only works if we adopt a naive form of IBE, which Bas van Fraassen has shown to be unjustified. Scientific realists instead endorse a mature form of IBE, but I will show that this form cannot lead to mathematical realism, and thus the EIA is not enhanced at all.

Philippos Papayannopoulos (University of Western Ontario)
'Algorithm' is traditionally accepted as an informal-though-precise concept. In the domain of integers, all offered explications of it (Turing machines, recursive functions, etc.) are extensionally equivalent. Nevertheless, the situation is different in real numbers, and we have incompatible explications, based on different conceptualizations of 'computation'. Two different traditions have emerged, trying to understand the essence of computation and claiming to found scientific computing; however, they prove incompatible results. I focus on how 'algorithm' is conceptualized in both and argue against the received view that the informal concept is precise, suggesting that its 'open texture' is responsible for the emergence of incompatible traditions. I submit that the situation suggests a conceptual difference between 'algorithms' and 'effective procedures', and between an absolute and a relative sense of 'algorithm', and close by discussing some implications for topics in the philosophy of science and physics.

Neil Dewar (Ludwig-Maximilians-Universität München)
This paper considers the following question: what is the relationship between supervenience and reduction? I investigate this formally, first by introducing a recent argument by Christian List to the effect that one can have supervenience without reduction; then by considering how the notion of Nagelian reduction can be related to the formal apparatus of definability and translation theory; then by showing how, in the context of propositional theories, topological constraints on supervenience serve to enforce reducibility; and finally, how constraints derived from the theory of ultraproducts can enforce reducibility in the context of first-order theories.

Holger Andreas (University of British Columbia)
The present paper aims to complement causal model approaches to causal explanation by Woodward, Halpern and Pearl, and Strevens. It centres on a strengthened Ramsey Test of conditionals: α ≫ γ iff, after suspending judgment about α and γ, an agent can infer γ from the supposition of α (in the context of further beliefs in the background). It has been shown by Andreas and Günther that such a conditional can be used as starting point of an analysis of ‘because’ in natural language. In what follows, we shall refine this analysis so as to yield a fully fledged account of (deterministic) causal explanation.