Sumários
Intuitionistic logic (cont.)
20 Março 2023, 15:30 • Ricardo Santos
We continue
the discussion of intuitionistic logic.
We discuss intuitionistic account of negation as the entailment of an absurdity
as well as some proof-theoretic features of the absurdity constant, namely, it
lacking introduction rules and their intuitive motivation given an informal BHK
interpretation of the intuitionistic logical constants. We discuss some
features of intuitionistic natural deduction derivations, including, briefly,
their formal structure and, more extensively, vacuous and multiple discharges
of assumptions. We informally introduce and discuss, with examples,
proof-search heuristics for building natural deduction derivations.
Intuitionistic logic; the BHK interpretation
15 Março 2023, 15:30 • Ricardo Santos
Recapitulate some
generic intuitionistic tenets about mathematical objects and motivate
intuitively the intuitionistic rejection of the principle of the excluded
middle. We briefly revise some elementary notions about logics qua formal
systems, such as the distinction between variables and metavariables and the
substitution invariance property of logical consequence. We move on to present
the rules of a system of natural deduction for intuitionistic logic,
introducing and motivating the idea of using assumptions in deductions. We
discuss briefly the operation of discharging assumptions.
Brouwer’s intuitionism
8 Março 2023, 15:30 • Ricardo Santos
Discussion of
Brouwer’s basic philosophical tenets. We start by pointing out the
dis/similarities between Brouwer and the (so-called) pre-intuitionists. We
discuss Browuer’s two moments of mathematical intuitionism, clarifying the
nature of intuition and underscoring its Kantian origins. We move on to discuss
the idea that mathematical objects are free constructions of the human mind. We
introduce and illustrate the concept of lawful and lawless sequence and explain
how the later form the basis of intuitionistic analysis. We briefly discuss the
revisionist character of intuitionistic mathematics and the fact that it can be
inconsistent with classical mathematics. We end by indicating that the idea of
a construction brings with it substantial logical constraints. We briefly
discuss the expectations related to the upcoming partial examination.
The temptation of constructivism. Pre-intuitionist constructivism
6 Março 2023, 15:30 • Ricardo Santos
Class discussion
of Kroenecker and Poincaré’s views about mathematica and foundationalist
programmes. We retain Kroenecker’s confidence in arithmetic truths and the
transparency of mathematical reasoning over the natural numbers, while pointing
out some difficulties in interpreting his views due to the language he uses.
The idea of mathematics as a mental construction is emphasised. The discussion
of Poincaré focuses on his criticism of foundationalist programmes, in
particular the problem of impredicativity and the circularity introduced by the
attempts to ground mathematical reasoning via
logical/formalistic approaches.