The Blind Spot: Lectures on Logic
European Mathematical Society, 2011 - 537 pagine
These lectures on logic, more specifically proof theory, are basically intended for postgraduate students and researchers in logic. The question at stake is the nature of mathematical knowledge and the difference between a question and an answer, i.e., the implicit and the explicit. The problem is delicate mathematically and philosophically as well: the relation between a question and its answer is a sort of equality where one side is ``more equal than the other'': one thus discovers essentialist blind spots. Starting with Godel's paradox (1931)--so to speak, the incompleteness of answers with respect to questions--the book proceeds with paradigms inherited from Gentzen's cut-elimination (1935). Various settings are studied: sequent calculus, natural deduction, lambda calculi, category-theoretic composition, up to geometry of interaction (GoI), all devoted to explicitation, which eventually amounts to inverting an operator in a von Neumann algebra. Mathematical language is usually described as referring to a preexisting reality. Logical operations can be given an alternative procedural meaning: typically, the operators involved in GoI are invertible, not because they are constructed according to the book, but because logical rules are those ensuring invertibility. Similarly, the durability of truth should not be taken for granted: one should distinguish between imperfect (perennial) and perfect modes. The procedural explanation of the infinite thus identifies it with the unfinished, i.e., the perennial. But is perenniality perennial? This questioning yields a possible logical explanation for algorithmic complexity. This highly original course on logic by one of the world's leading proof theorists challenges mathematicians, computer scientists, physicists, and philosophers to rethink their views and concepts on the nature of mathematical knowledge in an exceptionally profound way.
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A More on the classification of predicates
B Semantic aspects
A Kreisel and functional interpretation
A Type theories
A Exponentials and analytic functions
A Secondorder quantification
Envoi The phantom of transparency
action admits algebra associativity axiom Banach spaces base becomes behaviour bounded calculus carrier Chapter classical closed coherent spaces commutative completeness conclusion conduct connectives consistency contains corresponds daimon define Definition direct element equal equation everything example existence exponentials expression extends fact finite formula function given hence hermitian idea identity immediate implication induces infinite instance interest interpretation introduce intuitionistic logic isomorphism layer limited linear means multiplicative natural negative nets normal form normalisation notion object observe obtained operator partial particular polarity positive possible premise problem projection proof Proposition provable prove quantifiers quantum question reasons reduced relation replace restriction result rule sense sequent solution sort speak structure style tensor Theorem translation true truth typical unique variables weakening write yields