The Blind Spot: Lectures on Logic

Copertina anteriore
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.
 

Cosa dicono le persone - Scrivi una recensione

Nessuna recensione trovata nei soliti posti.

Sommario

A More on the classification of predicates
29
E Undecidability
38
4
51
B Semantic aspects
59
4
67
5
97
A Kreisel and functional interpretation
109
A Type theories
126
the reconstruction
306
A Faithfulness
325
15
331
A Exponentials and analytic functions
340
16
357
17
369
18
403
19
417

E Subtyping
135
9
178
10
195
11
216
12
251
13
272
20
443
21
462
A Secondorder quantification
485
Envoi The phantom of transparency
497
8
503
Copyright

Parole e frasi comuni

Informazioni bibliografiche