Research project funded by the Portuguese Science Foundation
FCT

PTDC/FIL-FCI/109991/2009

*Abstract: *

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 But on the basis of a wealth of currently unpublished works, there is still much to do to complete the unfinished picture of Hilbert's Philosophy of Mathematics as he himself conceived it, and much scholarly work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The first part of this project is devoted to the study of (still) unpublished material of David Hilbert. The scholarly labor here is to comment upon and disseminate Hilbert's early contributions, which are available only in unpublished lecture notes kept at the library of the Department of Mathematics at the University of Göttingen.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.

- João Branquinho
- Eduardo Castro
- Fernando Ferreira
- Nuno Jerónimo
- Reinhard Kahle (Principal investigator)
- Josiano Nereu
- Augusto Franco de Oliveira
- Gonçalo Santos
- Wilfried Sieg

- Jesse Alama
- Bruno Jacinto

- Centro de Matemática e Aplicações Fundamentais (CMAF/UL) (host institution)
- Centro de Filosofia (CF/UL)
- Centro de Filosofia da Ciência (FC/UL)

7/10/2011 :,Olga Pombo Universidade de Lisboa ,.Dificulties of intuition by the hand of Poincaré Seminário Permanente de Filosofia das Ciências, CFCUL 6/10/2011 :,Norma Claudia Yunez Naude Université de Marseille, França ,.Some epistemological aspects of intuition in mathematical thought Seminário Permanente de Filosofia das Ciências, CFCUL 28/6/2011 :,Mark van Atten Université Paris 1 ,.Kant and real numbers Seminário Permanente de Filosofia das Ciências, CFCUL 27/6/2011 :,Mark van Atten Université Paris 1 ,.Intuitionism as phenomenology: a critique of Rota 24/6/2011 :,Grigori Mints Stanford University ,.Extension of Epsilon Substitution to Second Order Systems Seminário de Lógica Matemática (SLM), CMAF 18/6/2011 :,Reinhard Kahle CENTRIA and DM, FCT, Universidade Nova de Lisboa ,.A Herança de Hilbert na Filosofia da Matemática 24.º Encontro Seminário Nacional de História da Matemática 13/4/2011 :,Fernando Ferreira CMAF and Universidade de Lisboa ,.The Two Faces of Logic Seminário de Crítica Genética, no âmbito do projecto *As Investigações Filosóficas de Wittgenstein: Reavaliando um Projecto*16/3/2011 :,Augusto Franco de Oliveira CFCUL and Universidade de Évora ,.Os legados de Hilbert na filosofia e fundamentos da matemática Os Novos Desafios da Razão, Ciclo de Conferências 2011 da Cátedra **A RAZÃO**11/3/2011 :,Michael Rathjen University of Leeds ,.Hilbert's second problem revisited

- Wikipedia entry
- MacTutor History of Mathematics entry
- Mathematics Genealogy Project entry
- Hilbert's 1930 Radio Address: Audio file (in German), PDF (German and English)
*Mathematical Problems*: German, English*Foundations of Geometry*: 3rd German edition from 1903, English*Collected Works*: German*Mathematical Notebooks*: excerpts- David Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891-1933, edited by Springer-Verlag
- The Hilbert Bernays Project
- Hilbert's Program: The Stanford Encyclopedia of Philosophy
- The Frege-Hilbert Controversy: The Stanford Encyclopedia of Philosophy

- Gentzen Centenary Workshop, The Quest of Consistency, 12 September 2009, Coimbra, Portugal