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)

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)

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...

This paper presents the content of the unpublished notes that the Dutch mathematician Arend Heyting wrote in different periods of his life on solipsism and that are preserved in Heyting's archive at the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.

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)

Julius König is famous for his mistaken attempt to demonstrate that the continuum hypothesis was false. It is also known that the only positive result that could have survived from his proof is the paradox which bears his name. Less famous is his 1914 book Neue Grundlagen der Logik, Arithmetik und Mengenlehre. Still, it contains original contributions to logic, like the concept of metatheory and the solution of paradoxes based on the refusal of the law of bivalence. We are going (...) to discover them by analysing the content of the book. (shrink)

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)

Kant è stato il principale riferimento epistemologico di quella ricerca sui fondamenti della matematica che ha avuto in Gottlob Frege il suo massimo esponente. In seguito, tuttavia, al tramonto dei programmi logicista, intuizionista e formalista, l'esigenza di garantire la certezza della matematica passa in secondo piano rispetto al problema della natura degli enti matematici. E in tal modo che, almeno a livello terminologico, Platone subentra a Kant come essenziale referente di una diversa costruzione epistemologica. Terreno fertile della ricerca fondazionale diviene (...) infatti, a partire dagli anni '60, il dibattito fra i sostenitori e gli avversari del 'platonismo matematico': etichetta che, utilizzata per indicare la trascendenza degli enti matematici, ha non di rado banalizzato o reso marginale l'originario pensiero di Platone. Il dibattito è ancor oggi un crogiolo di idee e tendenze, tra le quali emerge, accanto alla necessità di un maggiore approfondimento dei risultati e della reale attività del matematico, l'opportunità di interrogare i testi del filosofo greco. Nella sterminata letteratura contemporanea sui fondamenti della matematica, Platone e Kant si presentano, dunque, come problematici fili d'Arianna per orientarsi, mettere ordine, cogliere una linea evolutiva o anche soltanto originali spunti teorici per nuovi orizzonti. (shrink)

Though E. W. Beth is famous for his contributions to logic aspects of his philosophical reflections and details of its development are almost unknown. In his work four periods can be distinguished: the neo-kantian, the anti-kantian, the anti-irrationalist and the logical one. Within this framework it is possible to individuate a core around which Beth developed his reflections: it is the interplay between philosophy and the sciences. His philosophy was always linked to the sciences in two ways: He steadily checked (...) his philosophical ideas against the current situation in foundational research and he occupied himself with the general question of the relationship between philosophy and the sciences. (shrink)

Though E. W. Beth is famous for his contributions to logic aspects of his philosophical reflections and details of its development are almost unknown. In his work four periods can be distinguished: the neo-kantian, the anti-kantian, the anti-irrationalist and the logical one. Within this framework it is possible to individuate a core around which Beth developed his reflections: it is the interplay between philosophy and the sciences. His philosophy was always linked to the sciences in two ways: He steadily checked (...) his philosophical ideas against the current situation in foundational research and he occupied himself with the general question of the relationship between philosophy and the sciences. (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)

This book summarizes the intense research that the author performed for his Ph.D. thesis , revised and with the addition of an intuitionistic critique of Husserl's concept of number. His starting point consisted of a double conviction: 1) Brouwerian intuitionism is a valid way of doing mathematics but is grounded on a weak philosophy; 2) Husserlian phenomenology can provide a suitable philosophical ground for intuitionism. In order to let intuitionism and phenomenology match, he had to solve in general two problems: (...) 1) the question of the reciprocal indifference that the authors had toward each other's theorizing which indeed they knew; 2) Husserl's general attitude of accepting classical mathematics, which contrasts with the critical attitude of the intuitionists.Moreover, in the specific case focused on by this book—concerning choice sequences, that is, sequences that can be completely lawless and that were admitted as mathematical entities by Brouwer—van Atten had to handle a further problem: such sequences do not fit the characteristic of omnitemporality that the late Husserl seemed to consider a necessary attribute of entities if they were to be considered mathematical. In as far as they are lawless we cannot predict at any moment how they will develop in the future, hence they are not determined at any particular time—they will be determined only in the future.Let us begin with van Atten's first conviction, i.e., that Brouwerian intuitionism is grounded on a weak philosophy. We have to recall that Brouwer provided a mystical attitude for proposing his intuitionism: man can obtain happiness only by keeping himself closed inside the inner Self. Any attempted movement towards the exterior world is a source of pain: in particular language and the sciences. Mathematics, in order to be morally acceptable …. (shrink)

Paul Bernays was both a philosopher and a mathematician. He is famous for his logical-mathematical production (also in collaboration with David Hilbert), while his philosophical works have been given less consideration. The present article is an attempt to reconstruct the way that led Bernays from his early writings on ethics to his final epistemological thought.

A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real content (...) of the foundational schools. Hence, a re-thinking from him would be required about them, otherwise he would risk the same reproaches as post-modernists. (shrink)