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Arend Heyting and Phenomenology: is the meeting feasible?
No full textIn the literature one can see the increasing trend of supporting intuitionism through phenomenology. Brouwer’s pupil, Arend Heyting, is said to be a forerunner of this trend, as he used a phenomenological terminology in order to define intuitionist negation, by elaborating the first intuitionist logic. In this paper, the author tries to explore—with reference to the unpublished material stored in the Heyting archive—how much of Heyting’s general thought is compatible with phenomenology. In the conclusion she suggests that Heyting and Husserl, insofar as they both think consciousness must be the very beginning of knowledge, share a same antipsychologistic attitude which coexists with an attempt to overcome solipsism. Yet, the phenomenological concept of degree of evidence cannot be applied to Heyting’s scale of evidence (including small natural numbers, large natural numbers, infinitely proceeding sequences, the universal quantifier), on the one side because it is not clear if the latter is common and shared by all intuitionists, and, on the other side, because the former presupposes a revisable evidence that does not fit to Heyting’s viewpoint. Furthermore, Husserl’s and Heyting’s conceptions of the nature of mathematics and logic and of their relationship are essentially different. From an intuitionist viewpoint mathematics is the domain of evidence, while logic transcribes its regularities. From a phenomenological viewpoint, mathematics remains outside the domain of evidence. Apophantic logic coincides with mathematics (without either of them absorbing the other), but transcendental logic lies at a higher level
Intuition in Mathematics: from Racism to Pluralism
Get PDFIn 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 (Felix Klein, Henri Poincaré, Ludwig Bieberbach, Arend
Heyting) who interacted with Luitzen Egbertus Jan Brouwer (the father of the intuitionist
foundational school) in order to compare their conceptions of intuition. We will see how
to the same word “intuition” (in German Anschauung ) very different meanings
corresponded: they varied from geometrical vision, to a unitary view of a
demonstration, to the perception of time, to the faculty (shared by everybody) 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
Logic and totalitarianism: Arendt vs. Orwell
Get PDFThe final chapter of Hannah Arendt‘s 1951 book The Origins of Totalitarianism can be used as
a good guide to schematize the construction of a totalitarian government as it was described
in Orwell’s novel 1984. Namely, Arendt points out three steps that a state performs in order to
obtain the murder of the juridical, moral and spontaneous person, and we find the same ones
in Orwell. Still, in spite of this similarity, Hannah Arendt and George Orwell have very different
opinions, if we consider their views about the role of logic in totalitarianism. For Orwell, the
loss of sensitivity to the contradictions is the culmination of the totalitarian regime, which he
sees strewn with them: if you do not notice contradictions in history, then do not feel
compelled to change it. On the contrary, for Arendt it is the fear of contradiction that keeps
people within the tunnel of totalitarianism: when you accept an initial idea (for instance:
“history is a class struggle”), then you have to develop logically all its consequences and you
are no more allowed to escape from them. In this paper, we will consider authors’ reasons in
detail and try to take stock of them
Paul Bernays' philosophical way
No full textThough E.W. Beth is famous for his contributions to logic, aspects of his philosophical reflections and details of its development are most unknown. This article presents them, after distinguishing four periods in his work: the neo-kantian, the anti-kantian, the anti-irrationalist and the logical one. Within this framework it is possible to induviduate a core around which Beth developed his thoughts: 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
Brouwer and Nietzsche : Views about Life, Views about Logic
Get PDFFriedrich 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 (Nietzsche was born in Saxony and lived in the second half of the nineteenth century, beginning his career as a classical philologist before turning to philosophy; Brouwer was a Dutch mathematician who lived from 1881 to 1966), 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 on this subject performed by taking into account the studies conducted by E. Steinhart, O. Schutte and S.D. Hales
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