Hilbert's Legacy in the Philosophy of Mathematics

David Hilbert

Research project funded by the Portuguese Science Foundation FCT
PTDC/FIL-FCI/109991/2009



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Norma Claudia Yunez Naude

Université de Marseille, França

Some epistemological aspects of intuition in mathematical thought

Thursday, October 6th, 14h30, Faculdade de Ciências da Universidade de Lisboa, Room C4.2.07

Seminário Permanente de Filosofia das Ciências, CFCUL

Abstract: My aim in this talk is to explore the connexion of creative mathematical thinking with the origins of mathematical concepts and structures and the ways in which this relationship may provide a better understanding of intuition, as a process of thinking itself. Because the epistemological problem of intuition goes right to the heart of philosophical inquiry, first I will propose to point out and to distinguish different uses of the term intuition in mathematics and philosophy (for instance, Kant, Husserl, Poinacré, Brouwer and Gödel's notions of intuition). Then I will explore intuition through an historical case study: Desargue's projective geometry versus Descartes' algebraic geometry. The epistemological aspects that emerge from this analysis will be discussed by means of the different philosophical notions of intuition. I will finish by suggesting the need for further work.


Olga Pombo

Universidade de Lisboa

Dificulties of intuition by the hand of Poincaré

Friday, October 7th, 14h30, Faculdade de Ciências da Universidade de Lisboa, Room to announce

Seminário Permanente de Filosofia das Ciências, CFCUL


Description

This is an interdisciplinary project in Philosophy and Mathematics, touching on some of the essential history of the overlap of those two subjects. Our aim is to evaluate David Hilbert's impact on the Philosophy of Mathematics today. For a long time, Hilbert's contribution to the Philosophy of Mathematics was reduced to his alleged role as advocate of naive Formalism. This naive picture of Hilbert's philosophy of mathematics has endured for too long. Some ongoing work challenges this simple picture of Hilbert's philosophy on the basis of a handful of well-known publications of Hilbert. But on the basis of a wealth of currently unpublished works, there is still much to do to complete the unfinished picture of Hilbert's Philosophy of Mathematics as he himself conceived it, and much scholarly work remains if we are to have a fully satisfactory understanding of how Hilbert's philosophy influenced further work.

The first part of this project is devoted to the study of (still) unpublished material of David Hilbert. The scholarly labor here is to comment upon and disseminate Hilbert's early contributions, which are available only in unpublished lecture notes kept at the library of the Department of Mathematics at the University of Göttingen. These lecture notes–some of which are rather elaborated–contain valuable material which allow a better understanding of the early history of Mathematical Logic; they also contain many original ideas that can inspire Philosophy of Mathematics even today.

In a second part of this project, we will investigate, both historically and systematically, how Hilbert's ideas contributed–and continue to contribute–to the development of Philosophy of Mathematics.

On the one hand, Hilbert's programme has a quite successful history, despite it's seeming failure by Gödel's results. It was Gerhard Gentzen who first showed how meaningful consistency proofs (although not absolute ones) can be carried out for mathematically interesting formal systems. Even today, on the back of a rich history, the study of relative consistency proofs is a rather challenging enterprise, both technically and philosophically. One of our task is to compile and assess the state of the art of consistency proofs, and to evaluate the body of results with respect to its philosophical significance. This effort will also include technical contributions at the frontlines of research.

On the other hand, many traces of Hilbert in contemporary Philosophy of Mathematics are largely unnoticed, neglected, and undeveloped. It is our aim to investigate this influence, reviewing different topics in Philosophy of Mathematics which have links (at least implicitly) to Hilbert's work.


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For further information, please contact Reinhard Kahle, kahle@mat.uc.pt.
FCT