Even though Husserl and Brouwer have never discussed each other's work, ideas from Husserl have been used to justify Brouwer's intuitionistic logic. I claim that a Husserlian reading of Brouwer can also serve to justify the existence of choice sequences as objects of pure mathematics. An outline of such a reading is given, and some objections are discussed.

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by. In full topological models * is not generally definable, but over Cantor-space and the reals it can be classically shown that; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate logic.Over [0, 1], the operator * is (...) (constructively and classically) undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness. (shrink)

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by * ≡ ∃Q. In full topological models * is not generally definable but over Cantor-space and the reals it can be classically shown that *↔ ⅂⅂P; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate logic. Over (...) [0, 1], the operator * is undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness. Parameters of the required kind can be obtained as so-called projections of lawless sequences. (shrink)

In this paper we discuss a particular example of the passage from the informal, but rigorous description of a concept to the axiomatic formulation of principles holding for the concept; in particular, we look at the principles of continuity and lawlike choice in the theory of lawless sequences. Our discussion also leads to a better understanding of the rôle of the so-called density axiom for lawless sequences.

(1998). Coherence and the jurisdictions of the tribunal of reason. Social Epistemology: Vol. 12, Real Knowing: Situating Social Epistemology, pp. 227-239.

Brouwer's papers after 1945 are characterized by a technique known as the method of the creating subject. It has been supposed that the method was radically new in his work, since Brouwer seems to introduce an idealized mathematician into his mathematical practice. A newly opened source, the unpublished text of a lecture of Brouwer from 1934, fully supports the conclusions of our analysis that: - There is no idealized mathematician involved in the method;- The method was not new at all;- (...) Brouwer uses an incomplete sequence, also known as choice sequence, in this method, which is special. The method does not take its place in the standard works on choice sequences. (shrink)

The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics.Formal systems NPC, NA, and FIMN for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section (...) 3 this is shown for Kleene's system of intuitionistic analysis FIMand our FIMN. (shrink)

In a seriesof papers beginning in 1944, the Dutch mathematician and philosopherGeorge Francois Cornelis Griss proposed that constructivemathematics should be developedwithout the use of the intuitionistic negation1 and,moreover, without any use of a nullpredicate.In the present work, we give formalized versions of intuitionisticarithmetic, analysis,and higher-order arithmetic in the spirit ofGriss' ``negationless intuitionistic mathematics''and then consider their relation to thecurrent formalizations of thesetheories.

This paper discusses Michael Dummett’s attempt to base the use of intuitionistic logic in mathematics on a proof-conditional semantics. This project is shown to face significant obstacles resulting from the existence of variants of standard intuitionistic logic. In order to overcome these obstacles, Dummett and his followers must give an intuitionistically acceptable completeness proof for intuitionistic logic relative to the BHK interpretation of the logical constants, but there are reasons to doubt that such a proof is possible. The paper concludes (...) by proposing an alternative way of thinking about why one should use intuitionistic logic when doing mathematics. (shrink)

Intuitionistically. a set has to be given by a finite construction or by a construction-project generating the elements of the set in the course of time. Quantification is only meaningful if the range of each quantifier is a well-circumscribed set. Thinking upon the meaning of quantification, one is led to insights?in particular, the so-called continuity principles?which are surprising from a classical point of view. We believe that such considerations lie at the basis of Brouwer?s reconstruction of mathematics. The predicate ?α (...) is lawless? is not acceptable, the lawless sequences do not form a well-circumscribed intuitionistic set, and quantification over lawless sequences does not make sense. (shrink)

According to L.E.J. Brouwer, there is room for non-definable real numbers within the intuitionistic ontology of mental constructions. That room is allegedly provided by freely proceeding choice sequences, i.e., sequences created by repeated free choices of elements by a creating subject in a potentially infinite process. Through an analysis of the constitution of choice sequences, this paper argues against Brouwer’s claim.

According to L.E.J. Brouwer, there is room for non-definable real numbers within the intuitionistic ontology of mental constructions. That room is allegedly provided by freely proceeding choice sequences, i.e., sequences created by repeated free choices of elements by a creating subject in a potentially infinite process. Through an analysis of the constitution of choice sequences, this paper argues against Brouwer’s claim.

ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the infinite (...) descent of classical Fermatian arithmetic or number theory. However, from the point of view of classical logic, the principles of complete induction and transfinite induction, the least number principle and infinite descent are all equivalent. We find here a focal point for a foundational critique that aims for a a clarification of philosophical options in the foundations of logic and mathematics. (shrink)

We show that various fragments of the intuitionistic/constructive theory of the reals are decidable.

Beth models of analysis are used in model theoretic proofs of the disjunction and (numerical) existence property. By glueing strings of models one obtains a model that combines the properties of the given models. The method asks for a common generalization of Kripke and Beth models. The proof is carried out in intuitionistic analysis plus Markov's Principle. The main new feature is the external use of intuitionistic principles to prove their own preservation under glueing.

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those (...) challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms. (shrink)

The collection Virtual Mathematics: the logic of difference brings together a range of new philosophical engagements with mathematics, using the work of French philosopher Gilles Deleuze as its focus. Deleuze’s engagements with mathematics rely upon the construction of alternative lineages in the history of mathematics in order to reconfigure particular philosophical problems and to develop new concepts. These alternative conceptual histories also challenge some of the self-imposed limits of the discipline of mathematics, and suggest the possibility of forging new connections (...) between philosophy and more recent developments in mathematics. This component of Deleuze’s work has, to date, been rather neglected by commentators working in the field of Deleuze studies. One of the aims of this collection is to address this critical deficit by providing a philosophical presentation of Deleuze’s relation to mathematics; one that is adequate to his project of constructing a philosophy of difference and to the exploration of some of its applications. This project developed as a result of encounters with an increasing number of researchers who have been working on the mathematical aspects of Deleuze’s work and the key role that these play in his philosophy. By bringing this work together for the first time, this collection makes an important contribution to the emerging body of work that endeavours to explore the broad range of Deleuze’s philosophy. (shrink)