# The Blind Spot: Lectures on Logic

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*Jean-Yves Girard*

A publication of the European Mathematical Society

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 Gödel'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.

#### Readership

Postgraduate students and research mathematicians interested in logic and proof theory.