The Logic and Metaphysics Workshop will meet on December 2nd from 4:15-6:15 in-person at the Graduate Center (Room 4419) for a talk by Sam Burns (Columbia).

Title: Formalizability and mathematical rigor

Abstract: Mathematicians do not generally prove theorems via formal derivations. Given that formal derivations are the contemporary ideal of mathematical rigor, this raises questions as to how informal proofs can be rigorous. Responding to this worry, derivationists claim that an informal proof is rigorous if it can be routinely translated into a formal derivation. In this talk I raise some concerns about derivationism as a universal claim about mathematical rigor. I break the derivationist thesis into two parts: a claim about the formalizability of the theorems themselves, and a claim about the formalizability of mathematical inferences. I then discuss some case studies that call into question the plausibility of each part of the derivationist thesis. Based on these case studies, I suggest that a contextualist account of mathematical rigor best coheres with mathematical practice, thereby rejecting the claim that (complete) formalizability is a desideratum in all mathematical contexts.

The Logic and Metaphysics Workshop will meet on December 9th from 4:15-6:15 in-person at the Graduate Center (Room 4419) for a talk by Justin Clarke-Doane (Columbia).

Title: Intuition and observation

Abstract: The motivating question of this talk is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles philosophy. We conclude by distinguishing two ideas that have long been associated — realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective insofar as the mathematical, logical, and evaluative hypotheses that they presuppose fail to be.  Another is pragmatism.  Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones.

Note: This is joint work with Avner Ash (Boston College).

The Logic and Metaphysics Workshop will meet on November 18th from 4:15-6:15 in-person at the Graduate Center (Room 4419) for a talk by Claudio Calosi (Venice) and Damiano Costa (Lugano).

Title: Pluralisms in gunky worlds

Abstract: The possibility of gunk, namely the possibility that an entity possesses an infinitely descending chain of smaller and smaller parts, has famously been used by Schaffer (2010) to argue in favour of priority monism, namely the view that the whole universe is the fundamental concrete entity on which any of its parts depends. In this paper, we present and explore different principled ways of being a priority pluralist in gunky worlds, thus deflecting the gunk argument. Some of these ways turn out to be examples of middleism, i.e. the view that the fundamental level is that of middle-sized and mereologically intermediate objects. Hence, they don’t only effectively deflect the gunk threat to pluralism, but they also catalyse any argument in favour of the middleist position.