SEMINAR - Finished

We will continue to read and discuss the sixth chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.

 It is quite obvious that mathematics involves social activities, being a science carried out collectively by mathematicians who, under normal circumstances, interact with each other. But contrary to what one would expect, this rather trivial fact is rarely considered as important for its subject matter, mostly because it has some undesired ontological consequences. An attempted solution for this tension was developed by Julian Cole's institutional account of mathematics, named Practice-Dependent Realism. In this talk, Cole's account will be evaluated and its lights and shadows assessed concerning the ontological problem that he seeks to solve. I'll argue that his institutional account, although failing in delivering a sufficient ontological account of mathematics, still opens an important linguistic route for explaining its practice.

In this talk I explore the possibility of understanding mathematical beliefs as structured in a web of interconnected hyperintensional beliefs. The goal is to make sense of explanation in mathematics, a domain that is traditionally seen as lacking modal distinctions. The view is inspired by Quine’s web of belief, which is modified to make sense of “thick” explanatory relations between beliefs. Each belief that is held by a mathematician is characterized by a set of exact truthmakers, i.e. the mathematical facts (in a possible mathematical universe) that would make just those statements true. Beliefs are connected by relations of inference and support, which are functions of the truthmakers of the connected beliefs. The explanatory power of (i) statements, (ii) entailment relations between statements and (iii) proofs of statements is then determined by their capacity to provide a fruitful or efficient rewiring of a part of the web of beliefs.

The upshot of all this is that a model is provided for some aspects of pragmatics in mathematics. The Gricean principle of quantity requires, for example, that when mathematicians say that their Theorem A follows from Theorem B and Lemma C, they mean something stronger than merely that the A is a logical consequence of some background theory together with B and C (by the monotonicity of logical consequence, C could, with that weak interpretation, have absolutely nothing to do with A). They mean that B and C really contribute to proving A. In our model the pragmatic meaning of “A follows from B and C” in a mathematical paper is the fact that belief A is connected to belief B and C in the web of beliefs held by the author of the paper (and presumably shared with the reader of the paper). 

We will continue to read and discuss the fifth chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.

Neste seminário, abordarei o tema da tradução entre lógicas desde a perspectiva do desenvolvimento de software. Isso em vista, apresentarei algumas das principais características do GNU/Bison, software de código aberto voltado para a criação de compiladores, através de três projetos: (i) o primeiro voltado para a tradução de sentenças da linguagem proposicional em juízos da linguagem da Conceitografia de Frege, (ii) o segundo, voltado ao cálculo de tablôs e (iii) o terceiro, por fim, dedicado a busca e checagem de contra-modelos intuicionistas para tautologias clássicas. Ao longo desta exposição, pretendo ilustrar como tais ferramentas podem ser úteis na pesquisa em lógica.