This is the Spanish translation, by María Gabriela Fulugonio, of Arend Heyting’s classical text “Die logizistische Grundlegung der Mathematik”, which was originally presented at the Königsberg’s Symposium on Philosophy of Mathematics in 1930, and finally published in Erkenntnis in 1931.

La littérature témoigne d’une tendance croissante à soutenir l’intuitionisme par la phénoménologie. Le disciple de Brouwer Arend Heyting est considéré comme un précurseur de cette tendance, parce qu’il usait d’une terminologie phénoménologique en vue de définir la négation intuitioniste, en élaborant la première logique intuitioniste. Dans cet article, l’auteur tente d’explorer — en référence aux matériaux inédits conservés aux Archives Heyting — ce qui, dans la pensée de Heyting, est compatible avec la phénoménologie. Dans la conclusion, (...) l’auteur suggère que Heyting et Husserl, pour autant qu’ils pensent tous deux que la conscience doit être le véritable commencement de la connaissance, partagent une même attitude antipsychologiste coexistant avec une tentative de dépassement du solipsisme. Toutefois, le concept phénoménologique de degré d’évidence ne peut être appliqué à l’échelle de l’évidence de Heyting (incluant les petits nombres naturels, les grands nombres naturels, les suites infinies, le quantificateur universel), d’un côté parce qu’on ne voit pas clairement si la seconde est partagée par tous les intuitionistes, et de l’autre parce que le premier présuppose une évidence révisable qui ne cadre pas avec le point de vue de Heyting. En outre, les conceptions de la nature de la mathématique et de la logique et de leurs relations mutuelles défendues par Husserl et Heyting sont essentiellement différentes. D’un point de vue intuitioniste, la mathématique est le domaine de l’évidence, alors que la logique transcrit ses régularités. D’un point de vue phénoménologique, la mathématique demeure en dehors du domaine de l’évidence. La logique apophantique coïncide avec la mathématique (sans que l’une absorbe l’autre), mais la logique transcendantale se situe à un niveau plus élevé. (shrink)

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)

Brouwer’s intuitionism was a far-reaching attempt to reform the foundations of mathematics. While the mathematical community was reluctant to accept Brouwer’s work, its response to later-developed brands of intuitionism, such as those presented by Hermann Weyl and Arend Heyting, was different. The paper accounts for this difference by analyzing the intuitionistic versions of Brouwer, Weyl, and Heyting in light of a two-tiered model of the body and image of mathematical knowledge. Such a perspective provides a richer account (...) of each story and points to a possible connection between the community’s reaction and the changes each mathematician had proposed. (shrink)

This paper presents the content of the unpublished notes that the Dutch mathematician Arend Heyting wrote in different periods of his life on solipsism and that are preserved in Heyting's archive at the University of Amsterdam. Most of the notes are quoted here and translated into English. Their study shows the originality of Heyting's reflections on a subject that was typical of his master, L. E. J. Brouwer, the father of intuitionism.

After sketching the essentials of L. E. J. Brouwer’s intuitionistic mathematics—separable mathematics, choice sequences, the uniform continuity theorem, and the intuitionistic continuum—this chapter outlines the main philosophical tenets that go hand in hand with Brouwer’s technical achievements. It presents his views about general and mathematical phenomenology and shows how these views ground his positive epistemological and ontological positions and his stinging criticisms of classical mathematics and logic. The chapter then turns to intuitionistic logic and its philosophical side. It first sets (...) out the basic meta-logical technical results, then discusses the relevant philosophical views—those of Arend Heyting and Michael Dummett. It concludes by tracing intuitionism’s philosophical and technical roots in Aristotle and Kant. (shrink)

This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers - both new and previously published - it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer's idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the classical (...) realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer's work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics. (shrink)

The classical interpretation of mathematical statements can be seen as comprising two separate but related aspects: a domain and a truth-schema. L. E. J. Brouwer’s intuitionistic project lays the groundwork for an alternative conception of the objects in this domain, as well as an accompanying intuitionistic truth-schema. Drawing on the work of Arend Heyting and Michael Dummett, I present two objections to classical mathematical semantics, with the aim of creating an opening for an alternative interpretation. With this accomplished, (...) I then make the case for intuitionism as a suitable candidate to fill this void. (shrink)

In 1930, Johann von Neumann, together with Rudolf Carnap and Arend Heyting, participated in a conference held in Königsberg, called “Second Seminar on the Epistemology of Exact Sciences”. The idea behind the reunion of these three researchers was to compose a fairly faithful picture of the three main foundational programs of mathematics at the time: formalism, logicism, and intuitionism. The main objective of this paper is to propose an analysis of the text “The Formalist Foundation of Mathematics” presented (...) by von Neumann in this conference. We show how, from von Neumann's perspective, the problem of the consistency of mathematical theories results from the transformation of a genuinely philosophical question into a genuinely logical-mathematical question and how a proof of consistency could be seen as the solution to the foundational problem. In the final part of the text, we present some considerations about Gerhard Gentzen’s consistency proofs and their importance to contemporary proof theory. (shrink)

The classical interpretation of mathematical statements can be seen as comprising two separate but related aspects: a domain and a truth-schema. L. E. J. Brouwer’s intuitionistic project lays the groundwork for an alternative conception of the objects in this domain, as well as an accompanying intuitionistic truth-schema. Drawing on the work of Arend Heyting and Michael Dummett, I present two objections to classical mathematical semantics, with the aim of creating an opening for an alternative interpretation. With this accomplished, (...) I then make the case for intuitionism as a suitable candidate to fill this void. (shrink)

The classical interpretation of mathematical statements can be seen as comprising two separate but related aspects: a domain and a truth-schema. L. E. J. Brouwer’s intuitionistic project lays the groundwork for an alternative conception of the objects in this domain, as well as an accompanying intuitionistic truth-schema. Drawing on the work of Arend Heyting and Michael Dummett, I present two objections to classical mathematical semantics, with the aim of creating an opening for an alternative interpretation. With this accomplished, (...) I then make the case for intuitionism as a suitable candidate to fill this void. (shrink)

L.E.J. Brouwer famously took the subject’s intuition of time to be foundational and from there ventured to build up mathematics. Despite being largely critical of formal methods, Brouwer valued axiomatic systems for their use in both communication and memory. Through the Dutch Mathematical Society, Gerrit Mannoury posed a challenge in 1927 to provide an axiomatization of intuitionistic arithmetic. Arend Heyting’s 1928 axiomatization was chosen as the winner and has since enjoyed the status of being the de facto formalization (...) of intuitionistic arithmetic. We argue that axiomatizations of intuitionistic arithmetic ought to make explicit the role of the subject’s activity in the intuitionistic arithmetical process. While Heyting Arithmetic is useful when we want to contrast constructed objects with platonistic ones, Heyting Arithmetic omits the contribution of the subject and thus falls short as a response to Mannoury’s challenge. We offer our own solution, Doxastic Heyting Arithmetic, or $${\textsf{DHA}}$$, which we contend axiomatizes Brouwerian intuitionistic arithmetic. (shrink)

The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his endnotes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.

In den letzten Jahrzehntel! hat sich das Interesse an der Grund­ legung der Mathematik immer gesteigert. Fanden frtiher die wenigen Forscher, die sich emsthaft mit dieser 'Frage beschaftigten, wenig Be­ achtung, heute ist die Teilnahme sowohl von mathematischer wie von philosophischer Seite fast allgemein. Zu diesem Umschwung hat sieher die CANToRSche Mengenlehre, die gleich nach ihrem Entstehen lebhafte Erorterungen tiber ihre Berechtigung hervorrief, den AnstoB gegeben, und besonders die bei riicksichtsloser Durchfiihrung ihrer Grundgedanken auftretenden Widerspriiche zogen die allgemeine Aufmerksamkeit auf (...) sich. Doch ist die bisweUen noch geauBerte Behauptung, der Zweck -der Grundlagenforschung liege in der Beseitigung der Widersprtiche, verfehlt. In philosophischer und in mathematischer Richtung geht diese weit tiber eine solche Zielsetzung hinaus. Philosophisch untersucht man -das Wesen der mathematischen Erkenntnis, ihre Voraussetzungen und Endziele, ihr Verhaltnis zu anderen Wissensgebieten, insbesondere der Physik, und ihre Abgrenzung gegen diese dem lnhalt und der Methode nacho An diese philosophischen Erorterungen schlieBen sich umfangreiche mathematische Untersuchungen tiber den Aufbau der Mathematik aus den philosophisch gegebenen Voraussetzungen und tiber die Struktur der mathematischen Beweisftihrungen. Einzelne Teilgebiete dieser Unter­ suchungen entwickeln sich schon zu selbstandigen Disziplinen, die sich in ihren Methoden und Problemstellungen von der eigentlichen Grund­ lagenforschung unabhangig machen; ein Beispiel eines solchen neuen Zweiges der Mathematik, der sein Entstehen der Grundlagenforschung verdankt, ist die mathematischp. Logik. Allmahlich haben sich drei J. auptrichtungen gebildet, die je einer eigenen Auffassung tiber das Wesen der Mathematik entsprechen undo je zu verschieden gearteten mathematischen Untersuchungen geftihrt haben. (shrink)

As public research organisations are increasingly driven by their national and regional governments to engage in knowledge transfer, they have started to support the creation of companies. These research based spin-off companies (RBSOs) often keep contacts with the research institutes they originate from. In this paper we present the results of a study of four research institutes within two universities and two non-university public research organisations (PROs) in the Netherlands. We show that research organisations have distinct motivations to support the (...) creation of spin-off companies. In terms of resources RBSOs contribute, mostly in a modest way, to research activities by providing information, equipment and monetary resources. In particular, RBSOs are helpful for researchers competing for research grants that demand participation of industry. Furthermore, RBSOs may be seen as a proactive response by Dutch public research organisations to demands of economic relevance from their institutional environment. RBSOs enhance the prestige of their parent organisations and create legitimacy for public funds invested in PROs. At the same time, most RBSOs do not have a significant impact on the direction of the research conducted at the PROs. (shrink)

This book provides an interdisciplinary examination of international law by addressing four critical questions: How are international legal rules distinctive? How does an investigator determine the existence of a rule of international law? Does international law really matter in international politics? and What effect could the changing nature of international relations have on international law? Using Constructivist theory, Arend argues that international law can alter the identity of states, and, consequently, have a profound impact on state behavior.

We report to which extent elementary particles and the nucleons can be described by an Ansatz that is alternative to the established standard model, and can still yield predicted results that reproduce the observed ones, without using the formalism of quantum mechanics. The different Ansatz is motivated by the attempt to explain known properties of elementary particles as a consequence of an inner structure, in contrast to the approach of the standard model, where the properties are ascribed to point-like particles. (...) Based on the assumption of the existence of photons, the possibility of the creation of fermion and anti-fermion in an interaction of two photons of equal energy is shown. The properties of these created elementary material particles are found to agree with the ones observed. Also the possibility of the creation of a neutron by interaction of two photons of equal energy is shown. In this case, the newly formed neutron rests in the center of the collision system as a combined system of the localized two photons. The created neutron is shown to have the known properties of the neutron, and in addition, to have a definite shape of definite size. The proton is described as the particle formed by decay of the neutron, also owning the observed properties, and in addition a definite shape. For all particles described by the Ansatz, their properties are consequences of an inner structure. The merits of the alternative description as compared to the standard model and the application of quantum mechanics are discussed. (shrink)

Several theoretical publications on the Dirac equation published during the last decades have shown that, an interpretation is possible, which ascribes the origin of electron spin and magnetic moment to an autonomous circular motion of the point-like charged particle around a fixed centre. In more recent publications an extension of the original so called “Zitterbewegung Interpretation” of quantum mechanics was suggested, in which the spin results from an average of instantaneous spin vectors over a Zitterbewegung period. We argue that, the (...) corresponding autonomous motion of the electron should, if it is real, determine non-relativistic spin measurements. Such a direct connection with the established formal quantum mechanical description of spin measurements, into which spin is introduced as a “non-classical” quantity has, to our knowledge, not been reported. In the present work we show that, under certain “model assumptions” concerning the proposed autonomous motion, results of spin measurements, including measurements of angular correlations in singlet systems, can indeed be correctly described using classical probabilities. The success of the model is evidence for the “reality” of the assumed autonomous motion. The resulting model violates the Bell—inequalities to the same extent as quantum mechanics. (shrink)

L’ouvrage d’A. Wenger est la version remaniée de sa thèse de doctorat complétée par certains développements parus dans des revues spécialisées, Clio. Histoire, Femmes et Sociétés, Dix-huitième Siècle… La reprise de travaux antérieurs a entraîné parfois quelques redondances – sur l’onanisme ou la lectrice, par exemple – qu’une lecture plus attentive aurait pu éviter. Cette restriction est mineure au regard de l’intérêt de cette étude dont « l’objectif est de faire bouger les objets d’analyse t...

L’auteur a pris le risque de traiter d’une question qui, aujourd’hui encore, fait polémique : celle du rapport qu’entretient le philosophe et romancier Jean-Jacques Rousseau avec la différence sexuelle. L’un des mérites de l’étude de Laure Challandes est d’argumenter, avec une grande précision, sur la nécessaire frontière qui existe entre les textes théoriques, Lettre à d’Alembert sur les spectacles, Du contrat social, Émile ou de l’éducation, entre autres, et les textes fictifs, parmi lesque...

The purpose of this paper is the clarification of some methodological problems concerning Rawls’ theory of justice. The first part seeks to make more precise Tugendhat’s distinction between 1st-person-theory and 3rd-person-theory. Rawls’ theory fulfills all criteria for 1st-person-theories. In the second part Rawl’s coherence model for the justification of norms („reflective equilibrium“) is critically analyzed and opposed to the hypothetical decision which individuals are to make in the original position (contract model). It is shown that the conception of reflective equilibrium (...) is in various aspects mistaken. In conclusion a problem is indicated which Rawls has not satisfactorily resolved: The veil of ignorance is supposed to guarantee that the decision for the basic principles of social justice is unanimous. Nevertheless it would appear that the individuals in the original position either have too little empirical knowledge in order to make a rational decision, or they have too much knowledge in order to come to an unanimous decision. The veil of ignorance is either too fine or not fine enough. (shrink)

ZusammenfassungDie Arbeit enthält Bemerkungen über den Intuitionismus and über seine Beziehungen zu anderen Gebieten der Grundlagenforschung. Innerhalb der intuitionistischen Mathematik werden, im Anschluss an die Kritik von Griss gegen den Gebrauch der Negation, Evidenzstufen unterschieden, abhängend von der Art, in der bedingte Konstruktionen zugelassen werden. Auch werden gewisse Schwierigkeiten in der Theorie der endlichen Spezies diskutiert. Was die Grundlagenforschung im Aligemeinen betrifft, wird bemerkt, dass sie die klassische Mathematik weitgehend in ihre intuitiven, formalen and platonischen Bestandteile zerlegt hat. Es wird (...) näher eingegangen auf die These von Church in der Theorie der rekursiven Funktionen.RésuméL'article contient des remarques sur l'intuitionisme et sur ses rapports avec les autres directions dans la recherche des fondements. Dans la mathématique intuitioniste, et en connexion avec la critique de Griss contre l'usage de la négation, on distingue des degrés d'évidence, dépendant de l'espèce de constructions conditionnées qui est admise. En outre, on discute certaines difficultés dans la théorie des espèces finies. Concernant la recherche des fondements en général, on remarque que les mathématiques classiques ont été scindées en leurs éléments intuitifs, formels et platonistes. Quelques remarques sur la thèse de Church dans la théorie des fonctions récursives terminent l'article.The paper contains remarks on intuitionism and its relations with other domains of foundational research. Inside the intuitionistic mathematics, in connection with Griss' criticism against the use of negation, different degrees of evidence are distinguished, depending upon the way in which conditioned constructions are admitted. Some difficulties in the theory of finite species are discussed. Concerning the foundational research in general it is observed that it has separated intuitive, formal and platonistic constituents in classical mathematics. Some remarks are made on Church's thesis in the theory of recursive functions. (shrink)