Predicativity and parametric polymorphism of Brouwerian implication

A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten

The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs

Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings. Finally, I defend the Schema and the Theory against a number of objections that have been made

A note on Leibniz' argument against infinite wholes

International audienceLeibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In thisnote, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set theory; (3) This refutation does not cast doubt on the part-whole axiom. Hence, should there be an obstacle to Gödel's wish to integrate Cantorian set theory within Leibniz' philosophy, it will not be this famous argument of Leibniz'

Kant and real numbers

Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a potentially infinite sequence. In this paper I discuss the systematical reasons why in Kant's philosophy this identification is impossible

The proper explanation of intuitionistic logic: on Brouwer's demonstration of the Bar Theorem

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view

Why did Kurt Gödel turn to transcendental idealism?

International audienc

Troelstra's Paradox and Markov's Principle

A prominent problem for the Theory of the Creating Subject isTroelstra's Paradox. As is well known, the construction of thatparadox depends on the acceptability of a certain impredicativity, ofa kind that some intuitionists accept and others do not. After apresentation of the Theory of the Creating Subject and the paradox, Iargue that the paradox moreover depends on Markov's Principle, in aform that no intuitionist should accept. A postscript discusses a newversion of the paradox that Troelstra has proposed in reaction to myargument