Sumários
Round table
3 Maio 2023, 15:30 • Ricardo Santos
This is student
led discussion of various topics in the philosophy of mathematics that were
covered in the course of the semester. The discussion focuses on constructive
proofs: what are they really? We discuss examples of constructive and non-constructive
proofs. We discuss the relation between algorithms and constructive proofs and
we end up discussing software like ChatGPT from a formal mathematical
perspective.
Computability and mathematics
26 Abril 2023, 15:30 • Ricardo Santos
We begin by
briefly discussing axiomatic systems and the role of proof in generating
mathematical knowledge. We move from proof to the concept of an algorithm,
using intuitionistic logic as a case study. We introduce the idea of a Turing
machine and we present several examples of Turing machines. We briefly
introduce the Church-Turing thesis drawing on the students’ previous
acquaintance with recursive function.
Ante Rem Structuralism II & Modal Structuralism
19 Abril 2023, 15:30 • Ricardo Santos
Second and final session on ante rem structuralism. In particular, the existence of nontrivial automorphisms’ objection was presented and discussed. Session on Modal Structuralism, with discussion of Hellman & Shapiro’s chapter ‘The Modal Structural Perspective’. Among other things, the following topics were covered: (i) Deductivism; (ii) Problems for Deductivism; (iii) Modal structuralism and how it overcomes Deductivism’s problems; (iv) the modal structuralists’ take on the nature of mathematical entities and mathematical discourse; (v) modal structuralism and potentialism about set theory; (vi) apparent incoherence of the modal structuralists’ notion of ‘logical possibility’.
Ante Rem Structuralism
17 Abril 2023, 15:30 • Ricardo Santos
Session on ante rem structuralism, with discussion of Hellman & Shapiro’s chapter ‘Sui Generis Structuralism’. Among other things, the following topics were covered: (i) what is ante rem structuralism, in particular, the structuralists’ ideology of structures, places, relations and systems; (ii) the ante rem structuralists’ take on the nature of mathematical entities and mathematical discourse; (iii) the Benacerrafian argument for ante rem structuralism; (iv) the vicious circularity objection to the ante rem structuralists’ conception of relations and places; (v) a novel Benacerrafian worry for ante rem structuralism; (vi) the coherence axiom of ante rem structuralists; (vii) apparent meaninglessness of ‘coherence’.