This will be a talk in the Centre for Philosophy and the Sciences Lunch Forum. Handout available here.
In this talk, I want to introduce two themes from my research in the foundations of mathematics, and discuss with the group relationships with the philosophy of science more broadly. My focus will be on generating discussion rather than presenting an argument per se.
The first theme is the idea that there are kinds of inference similar to those found in the philosophy of science at play in the justifications of axioms. In this regard we’ll discuss the idea that there is something like prediction and confirmation involved in certain justifications. I’ll ask what the data might be here, and how mathematical “data” might be similar to or different from our normal scientific data. We’ll also discuss the idea that when working with mathematics we ofteninfer to the best conception’. I’ll raise some questions for the group regarding pluralism and open texture in this context.
The second theme is that we often use extensions of a given universe of discourse in set-theoretic mathematics. This has implications for debates on absolute generality—if we can always transcend a given universe of discourse then absolute generality fails. The interesting fact in the context of set theory is that often we use an `external’ perspective in order to prove facts about the universe. This raises interesting questions about how seriously we should take the use of these resources. I’ll ask the group whether there are analogous methodologies and debates in the philosophy of science more widely (for example regarding multiverse theory and quantum mechanics).