This paper examines the problem of extending the programme of mathematical constructivism to applied mathematics. I am not concerned with the question of whether conventional mathematical physics makes essential use of the principle of excluded middle, but rather with the more fundamental question of whether the concept of physical infinity is constructively intelligible. I consider two kinds of physical infinity: a countably infinite constellation of stars and the infinitely divisible space-time continuum. I argue (contrary to Hellman) that these do not. (...) pose any insuperable problem for constructivism, and that constructivism may have a useful new perspective to offer on physics. (shrink)

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

Intuitionism’s disagreement with classical logic is standardly based on its specific understanding of truth. But different intuitionists have actually explicated the notion of truth in fundamentally different ways. These are considered systematically and separately, and evaluated critically. It is argued that each account faces difficult problems. They all either have implausible consequences or are viciously circular.

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). it gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-L6f 's theory of types as a formal system for BISH.

At the beginning of the twentieth century, among the foundational schools of mathematics appeared ‘intuitionism’ by Dutchman L. E. J. Brouwer, who based arithmetic on the intuition of time and all mental constructions that could be made out of it. His pupil Arend Heyting was the first populariser of intuitionism, and he repeatedly emphasised that no philosophy was required to practise intuitionism so that such mathematics could be shared by anyone. Still, stimulated by invitations to humanistic conferences, he wrote a (...) series of notes, preserved in the State Archives, Haarlem, about solipsism. In them, he operated a series of theoretical reflections consisting of the stripping away of the patterns that are part of our consciousness and their subsequent progressive re-introduction, in order to understand the formation of our Self as distinct from the natural world and other humans. In 1996, following a stroke, neuroscientist Jill Bolte Taylor experienced the deprivation of certain abilities in her brain and their successive regaining. In her experience there are remarkable similarities with what Heyting had hypothesised about the formation of the self. The purpose of this article is to highlight them and point out how Heyting's theoretical construct was found to be embodied in the brain. (shrink)

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies (...) that Brouwer has invited us to introduce into mathematics. (shrink)

ABSTRACT In ‘The philosophical basis of intuitionistic logic’, Michael Dummett discusses two routes towards accepting intuitionistic rather than classical logic in number theory, one meaning-theoretical and the other ontological. He concludes that the former route is open, but the latter is closed. I reconstruct Dummett's argument against the ontological route and argue that it fails. Call a procedure ‘investigative’ if that in virtue of which a true proposition stating its outcome is true exists prior to the execution of that procedure; (...) and ‘generative’ if the existence of that in virtue of which a true proposition stating its outcome is true is brought about by the execution of that procedure. The problem with Dummett's argument then is that a particular step in it, while correct for investigative procedures, is not correct for generative ones. But it is the latter that the ontological route is concerned with. (shrink)

We present a new English translation of L.E.J. Brouwer's paper ‘De onbetrouwbaarheid der logische principes’ of 1908, together with a philosophical and historical introduction. In this paper Brouwer for the first time objected to the idea that the Principle of the Excluded Middle is valid. We discuss the circumstances under which the manuscript was submitted and accepted, Brouwer's ideas on the principle of the excluded middle, its consistency and partial validity, and his argument against the possibility of absolutely undecidable propositions. (...) We note that principled objections to the general excluded middle similar to Brouwer's had been advanced in print by Jules Molk two years before. Finally, we discuss the influence on George Griss' negationless mathematics. (shrink)

The main ideas behind Brouwer’s philosophy of Intuitionism are presented. Then some critical remarks against Intuitionism made by William Tait in “Against Intuitionism” [Journal of Philosophical Logic, 12, 173–195] are answered.

Friedrich Nietzsche and Luitzen Egbertus Jan Brouwer had strong personalities and freely expressed unconventional opinions. In particular, they dared to challenge the traditional view that considered Aristotelian logic as being absolute and intrinsic to man. Although they formed this opinion in different ways and in different contexts, they both based it on a view of life that considered it as a struggle for power in which logic was a weapon. Therefore, it is interesting to carry out an in-depth analysis on (...) the origins, the core of their opinions concerning logic and the consequences. As a matter of fact, a detailed analysis and a comparison of their contributions to the philosophy of logic have not yet been carried out. In this paper, I intend to present the results of my research.. (shrink)

In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.

After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal algebra and the project of mathesis universalis. In a third part we critically present three lines of research connected to universal logic: logical pluralism, (...) non-classical logics and cognitive science. (shrink)

I argue that the implementation of theDummettian program of an ``anti-realist'' semanticsrequires quite different conceptions of the technicalmeaning-theoretic terms used than those presupposed byDummett. Starting from obvious incoherences in anattempt to conceive truth conditions as assertibilityconditions, I argue that for anti-realist purposesnon-epistemic semantic notions are more usefully kept apart from epistemic ones rather than beingreduced to them. Embedding an anti-realist theory ofmeaning in Martin-Löf's Intuitionistic Type Theory(ITT) takes care, however, of many notorious problemsthat have arisen in trying to specify suitableintuitionistic (...) notions of semantic value,truth-conditions, and validity, taking into accountthe so-called ``defeasibility of evidence'' forassertions in empirical discourses. (shrink)

Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...) in the foundational controversy of the 1920s. In addition, they reveal Weyl's hitherto unknown objections to Becker's detailed attempt (Mathematische Existenz, 1927) to ground the transfinite phenomenologically. (shrink)

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...) that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it. (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.

The ArgumentL. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notoriousGrundlagenstreitcentered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which (...) last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy. (shrink)

This is a review, with historical and critical comments, of a paper by I. E. Orlov from 1928, which gives the oldest known axiomatization of the implication-negation fragment of the relevant logic R. Orlov's paper also foreshadows the modal translation of systems with an intuitionistic negation into S4-type extensions of systems with a classical, involutive, negation. Orlov introduces the modal postulates of S4 before Becker, Lewis and Gödel. Orlov's work, which seems to be nearly completely ignored, is related to the (...) contemporaneous work on the axiomatization of intuitionistic logic. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

In this paper we discuss the difference between (...) proliferations of logic systems, the questions of relativity and universality of logic and the position and interaction of logic with regards to other sciences such as physics, biology, mathematics and computer science. (shrink)

The use of the three labels to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor the...

In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveller from reaching his destination. In this paper it is argued, contra Priest and Yablo, that the paradox must be resolved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New Zeno and the paradoxes (...) of truth, and a constructivist interpretation of Kripke’s theory of truth is given. (shrink)