If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical (...) constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch''s paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett''s meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost. (shrink)

Proceeding of the work in trivalent logic developped under the direction of the professor Hugo Padilla Chacón at the 90's at the National Autonome University of México. Program in RLisp.

Though the divide between reason-based and causal-explanatory approaches in psychiatry and psychopathology is old and deeply rooted, current trends involving multi-factorial explanatory models and evidence-based approaches to interpersonal psychotherapy, show that it has already been implicitly bridged. These trends require a philosophical reconsideration of how reasons can be causes. This paper contributes to that trajectory by arguing that Donald Davidson’s classic paradigm of 1963 is still a valid option.

A comprehensive survey of Martin-Löf's constructive type theory, considerable parts of which have only been presented by Martin-Löf in lecture form or as part of conference talks. Sommaruga surveys the prehistory of type theory and its highly complex development through eight different stages from 1970 to 1995. He also provides a systematic presentation of the latest version of the theory, as offered by Martin-Löf at Leiden University in Fall 1993. This presentation gives a fuller and updated account of the system. (...) Earlier, brief presentations took no account of the issues related to the type-theoretical approach to logic and the foundations of mathematics, while here they are accorded an entire part of the book. Readership: Comprehensive accounts of the history and philosophy of constructive type theory and a considerable amount of related material. Readers need a solid background in standard logic and a first, basic acquaintance with type theory. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic. (shrink)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of (...) these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus. (shrink)

Recently discovered documents have shown how Gentzen had arrived at the final form of natural deduction, namely by trying out a great number of alternative formulations. What led him to natural deduction in the first place, other than the general idea of studying “mathematical inference as it appears in practice,” is not indicated anywhere in his publications or preserved manuscripts. It is suggested that formal work in axiomatic logic lies behind the birth of Gentzen’s natural deduction, rather than any single (...) decisive influence in the work of others. Various axiomatizations are explored in turn, from the classical axioms of Hilbert and Ackermann to the intuitionistic axiomatization of Heyting. (shrink)

I argue that Brouwer''s general philosophy cannot accountfor itself, and, a fortiori, cannot lend justification tomathematical principles derived from it. Thus it cannot groundintuitionism, the jobBrouwer had intended it to do. The strategy is to ask whetherthat philosophy actually allows for the kind of knowledge thatsuch an account of itself would amount to.

It is by now well known that Gödel first advocated the philosophy of Leibniz and then, since 1959, that of Husserl. This raises three questions:1.How is this turn to Husserl to be interpreted? Is it a dismissal of the Leibnizian philosophy, or a different way to achieve similar goals?2.Why did Gödel turn specifically to the later Husserl's transcendental idealism?3.Is there any detectable influence from Husserl on Gödel's writings?Regarding the first question, Wang [96, p.165] reports that Gödel ‘[saw] in Husserl's work (...) a method of refining and consolidating Leibniz' monadology’. But what does this mean? In what for Gödel relevant sense is Husserl's work a refinement and consolidation of Leibniz' monadology?The second question is particularly pressing, given that Gödel was, by his own admission, a realist in mathematics since 1925. Wouldn't the uncompromising realism of the early Husserl's Logical investigations have been a more obvious choice for a Platonist like Gödel?The third question can only be approached when an answer to the second has been given, and we want to suggest that the answer to the first question follows from the answer to the second. We begin, therefore, with a closer look at the actual turn towards phenomenology.Some 30 years before his serious study of Husserl began, Gödel was well aware of the existence of phenomenology. Apart from its likely appearance in the philosophy courses that Gödel took, it reached him from various directions. (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)

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.

I will be concerned with the following question: are there compelling arguments for postulating a distinction between the truth of a statement and the recognition of its truth, when truth is conceived along the lines of a suitable generalization of the intuitionistic idea that it should be characterized as the existence of a proof? I will argue that the distinction is not necessary within the conceptual framework of intuitionism by replying to two arguments to the contrary, one based on the (...) paradox of inference, the other on considerations concerning the content of a statement. (shrink)

Weyl's inclination toward constructivism in the foundations of mathematics runs through his entire career, starting with Das Kontinuum. Why was Weyl inclined toward constructivism? I argue that Weyl's general views on foundations were shaped by a type of transcendental idealism in which it is held that mathematical knowledge must be founded on intuition. Kant and Fichte had an impact on Weyl but HusserFs transcendental idealism was even more influential. I discuss Weyl's views on vicious circularity, existence claims, meaning, the continuum (...) and choice sequences, and the intuitive-symbolic distinction against the background of his transcendental idealism and general intuitionism. (shrink)

Brouwer's criticism of mathematical proofs making essential use of the tertium non datur had a surprisingly late response in logical circles. Among the diverse reactions in the mid 1920s and early 1930s, it is possible to delimit a coherent body of opinions on these questions: (1) whether Brouwer's denial of the tertium non datur meant only the abandonment of this classical law or, beyond that, the affirmation of its negation; (2) whether one or both of these alternatives were logically inconsistent; (...) and (3) whether Brouwer's line of argument was forced to take resort to the very law it was designed to refute. The controversy centred around a series of articles by Marcel Barzin and Alfred Errera who fought against the intuitionistic critique, missed their victory because of conceptual confusions and fallacious reasoning, but emerged unconvinced from the debate in the late 1930s. The controversy is of interest to the historiography of formal logic since it stimulated the clarification not only of the concepts of formal validity, decidability, many-valued systems of logic and non-classical systems generally, but also of the distinction between object and meta-level, and between a formal system and its semantics. Most important, the debate, by putting pressure on the intuitionistic camp to make their ideas more precise, seems to have given the decisive motivation towards Heyting's answer to this demand by his axiomatization of intuitionistic logic in 1930. (shrink)

In the present paper I wish to regard constructivelogic as a self-contained system for the treatment ofepistemological issues; the explanations of theconstructivist logical notions are cast in anepistemological mold already from the outset. Thediscussion offered here intends to make explicit thisimplicit epistemic character of constructivism.Particular attention will be given to the intendedinterpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts thesystem on par with the early efforts of Frege andWhitehead-Russell. This quite (...) recent work, however,has proved valuable not only in the philosophy andfoundations of mathematics, but has also foundpractical application in computer science, where thelanguage of constructivism serves as an implementableprogramming language, and within the philosophy oflanguage.\footnote{Nordstr\"{o}m et al. give an overview of the work in computerscience, whereas Ranta provides an impressiveconstructivist alternative to Montague Grammar usingthe richer type structure of Martin-L\"{o}f in placeof the simple classical type theory of Church.} Mypresentation will be carried out through a contrastwith standard metamathematical work.\footnote{Troelstra and van Dalen give an encyclopedictreatment of the metamathematics of constructivism.}In the course of the development I have occasion tooffer some novel considerations on thenature of proof and inference. (shrink)

Truth-maker analyses construe truth as existence of proof, a well-known example being that offered by Wittgenstein in theTractatus. The paper subsumes the intuitionistic view of truth as existence of proof under the general truth-maker scheme. Two generic constraints on truth-maker analysis are noted and positioned with respect to the writings of Michael Dummett and theTractatus. Examination of the writings of Brouwer, Heyting and Weyl indicates the specific notions of truth-maker and existence that are at issue in the intuitionistic truth-maker analysis, (...) namely that of proof in the sense of proof-object (Brouwer, Heyting) and existence in the nonpropositional sense of a judgement abstract (Weyl). Furthermore, possible anticipations in the writings of Schlick and Pfänder are noted. (shrink)

The notion of cognitive act is of importance for an epistemology that is apt for constructive type theory, and for epistemology in general. Instead of taking knowledge attributions as the primary use of the verb 'to know' that needs to be given an account of, and understanding a first-person knowledge claim as a special case of knowledge attribution, the account of knowledge that is given here understands first-person knowledge claims as the primary use of the verb 'to know'. This means (...) that a cognitive act is an act that counts as cognitive from a first-person point of view. The method of linguistic phenomenology is used to explain or elucidate our epistemic notions. One of the advantages of the theory is that an answer can be given to some of the problems in modern epistemology, such as the Gettier problem. (shrink)

Michael Dummett has interpreted and expounded upon intuitionism under the influence of Wittgensteinian views on language, meaning and cognition. I argue against the application of some of these views to intuitionism and point to shortcomings in Dummett's approach. The alternative I propose makes use of recent, post-Wittgensteinian views in the philosophy of mind, meaning and language. These views are associated with the claim that human cognition exhibits intentionality and with related ideas in philosophical psychology. Intuitionism holds that mathematical constructions are (...) mental processes or objects. Constructions are, in the first instance, forms of consciousness or possible experience of a particular type. As such, they must be understood in terms of the concept of intentionality. This view has a historical basis in the literature on intuitionism. In a famous 1931 lecture Heyting in fact identifies constructions with fulfilled or fulfillable mathematical intentions. I consider some of the consequences of this identification and contrast them with Dummett's views on intuitionism. (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.

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may (...) consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting’s view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference. (shrink)

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed (...) in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier. (shrink)

What is the appropriate notion of truth for sentences whose meanings are understood in epistemic terms such as proof or ground for an assertion? It seems that the truth of such sentences has to be identified with the existence of proofs or grounds, and the main issue is whether this existence is to be understood in a temporal sense as meaning that we have actually found a proof or a ground, or if it could be taken in an abstract, tenseless (...) sense. Would the latter alternative amount to realism with respect to proofs or grounds in a way that would be contrary to the supposedly anti-realistic standpoint underlying the epistemic understanding of linguistic expressions? Before discussing this question, I shall consider reasons for construing linguistic meaning epistemically and relations between such reasons and reasons for taking an anti-realist point of view towards the discourse in question. (shrink)

This paper is an attack on the Dummett-Prawitz view that the principle of bivalence has a crucial double significance, metaphysical and meaning theoretical. On the one hand it is said that holding bivalence valid is what characterizes a realistic view, i.e. a view in metaphysics, and on the other hand it is said that there are meaning theoretical arguments against its acceptability. I argue that these two aspects are incompatible. If the failure of validity of bivalence depends on properties of (...) linguistic meaning, then there are no metaphysical consequences to be drawn. The case for this view is straightforward as long as we are discussing a language different from our own. But it seems that the distinction between failure because of meaning and failure because of reality cannot be applied to our own language, simply because our own language is just what we use to represent reality. I argue that this impression is illusory. In order to draw a conclusion about reality, meaning must be connected with truth in a non-trivial way, and precisely this cannot be done in the language for which the meaning theory itself is correct. (shrink)

After the 1930s, the research into the foundations of mathematics changed.None of its main directions (logicism, formalism and intuitionism) had any longer the pretension to be the only true mathematics.Usually, the determining factor in the change is considered to be Gödel?s work, while Heyting?s role is neglected.In contrast, in this paper I first describe how Heyting directly suggested the abandonment of the big foundational questions and the putting forward of a new kind of foundational research consisting in the isolation of (...) formal, intuitive, logical and platonistic elements within classical mathematics.Furthermore, I describe how Heyting indirectly influenced the abandon?ment of the old directions of foundational research by making out some lists of degrees of evidence that exist within intuitionism. (shrink)

Dans ce texte, je pars de l’analyse intuitionniste de la vérité mathématique, « A est vrai si et seulement s’il existe une preuve de A » comme cas particulier de l’analyse de la vérité en termes de « vérifacteur », et je montre pourquoi Wittgenstein partageait celle-ci avec les intuitionnistes. Cependant, la notion de preuve à l’oeuvre dans cette analyse est, selon l’intuitionnisme, celle de la « preuve-comme-objet », et je montre par la suite, en interprétant son argument sur le (...) caractère « synoptique » des preuves, que Wittgenstein avait plutôt en tête une conception de la « preuve-comme-trace ».In this paper, I start with the intutionist analysis of mathematical truth, « A is true if and only if there exists a proof of A », as a particular case of the analysis of truth in terms of « truth-makers », and I show why Wittgenstein shared it with the intuitionists. However, the notion of proof at work in this analysis is, according to intuitionism, that of « proof-as-object », and I then show, with an interpretation of his argument on the « surveyability » of proofs, that, instead, Wittgenstein had in mind a notion of « proof-as-trace ». (shrink)

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)

This article provides an English translation of a historic discussion on the foundations of mathematics, during which Kurt GÖdel first announced his incompleteness theorem to the mathematical world. The text of the discussion is preceded by brief background remarks and commentary.

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.

A SUMMARY IN ENGLISH [by Editor]The problem is to find out whether mathematical propositions are analytical, and if so, or if not, to what extent.Kant defined the analyticity in terms of Cartesian res extensa, exemplified by “A body is extended”, while he considered, because of such examples, mathematical propositions to be synthetic. The recent studies in set theory by Gödel, P.J.Cohen, etc., indicate, however, that such a proposition as the continuum hypothesis is certainly not “analytic (tautological)” in the strict sense (...) of logicists. Rather a small portion, comparatively speaking, of mathematical propositions must therefore be considered analytic relative to the definitions of fundamental conrepts in logic. Axioms, for example, are per definitionem non-analytic. Other problems that demand reöxaminations are, for instance: 1. Why is there only one unique series of the natural numbers? 2. On which ground is the factual eminence of Euclidean geometry to be founded? etc. The two assertions that mathematical propositions are analytic and that they are synthetic a priori are thus not at all contradictory; they are, to begin with, not even mutually exclusive Possibilities. (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...

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)

There are two different ways to introduce the notion of truthin constructive mathematics. The first one is to use a Tarskian definition of truth in aconstructive (meta)language. According to some authors, (Kreisel, van Dalen, Troelstra ... ),this definition is entirely similar to the Tarskian definition of classical truth (thesis A).The second one, due essentially to Heyting and Kolmogorov, and known as theBrouwer–Heyting–Kolmogorov interpretation, is to explain informally what it means fora mathematical proposition to be constructively proved. According to other authors (...) (Martin-Löfand Shapiro), this interpretation and the Tarskian definition of truth amount to thesame (thesis B). My aim in this paper is to show that thesis A is only reasonable, that thesis Bis false and to answer the following question: what is defined by the Tarskian definition ofconstructive truth? (shrink)

The birth, growth, stabilization and subsequent understanding of a new field of practical and theoretical enquiry is always a conceptual process including several typologies of events, phenomena an...

Prawitz has recently developed a theory of epistemic grounding that differs in many respects from his earlier semantics of arguments and proofs. An innovative approach to inferences yields a new conception of the intertwinement of the notions of valid inference and proof. We aim at singling out three reasons that may have led Prawitz to the ground-theoretic turn, i.e.: a better order in the explanation of the relation between valid inferences and proofs; a notion of valid inference based on which (...) valid inferences and proofs are recognisable as such; a reconstruction of the deductive activity that makes inferences capable of yielding justification per se. These topics are discussed by Prawitz with reference to a very general and ancient question: why and how correct deduction has the epistemic power to compel us to accept its conclusions, provided its premises are justified? We conclude by remarking that, in spite of some improvements, the ground-theoretic approach shares with the previous one a problem of vacuous validity which, as Prawitz himself points out, blocks in both cases a satisfactory explanation of epistemic compulsion. (shrink)

MICHAEL DUMMETT, Elements of intuitionism. With the assistance of Robert Minio. Oxford: Oxford University Press, 1977. xii + 466 pp. No price stated.

I argue against the interpretation of propositions as intentions and proof-objects as fulfillments proposed by Heyting and defended by Tieszen and van Atten. The idea is already a frequent target of criticisms regarding the incompatibility of Brouwer’s and Husserl’s positions, mainly by Rosado Haddock and Hill. I raise a stronger objection in this paper. My claim is that even if we grant that the incompatibility can be properly dealt with, as van Atten believes it can, two fundamental issues indicate that (...) the interpretation is unsustainable regardless: (1) it is hard to determine, without appealing to propositional intentions on pain of circularity, what intention a proof-object should be understood as a fulfillment of; (2) due to a difficult fulfillment dilemma, it is unclear, at best, what the object of an intention corresponding to a proposition is. (shrink)

The aim of this paper is to present a constructive solution to Frege's puzzle (largely limited to the mathematical context) based on type theory. Two ways in which an equality statement may be said to have cognitive significance are distinguished. One concerns the mode of presentation of the equality, the other its mode of proof. Frege's distinction between sense and reference, which emphasizes the former aspect, cannot adequately explain the cognitive significance of equality statements unless a clear identity criterion for (...) senses is provided. It is argued that providing a solution based on proofs is more satisfactory from the standpoint of constructive semantics. (shrink)

While developing his theory of world 3, Popper rejects two claims made by Plato: first, that the inhabitants of world 3, ideas, are a source of ultimate explanation, a divine revelation of truth, and second, that these ideas are unchanging. I will rehabilitate the second claim. Man does not construct world 3 by creating his theories, nor is it a source of ultimate truth. Instead, world 3 is discovered by man, and it destroys some of his theories: destructive Platonism. I (...) discuss the impact of the modified position on the Grundlagenstreit in mathematics, taking into account Gillies’s constructive Aristotelianism. I then turn to the philosophy of computers and discuss how destructive Platonism relates to computers and the Internet. In particular, I criticize Popper’s claim that computers are nothing but elaborate pens as too pessimistic and incautious, and the fashionable “singularity” theory as too optimistic and at the same time too fatalistic. In my closing remarks, as an antidote against both views, I provide a more modest and reasonable speculation about the ultimate goal and meaning of life. (shrink)

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...) resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)