In a paragraph of Physics, Aristotle states that in his days the mathematicians themselves stopped feeling the need of the infinite. The Stagirite thus authorises a tendency in the history of thought that repudiates this concept, which nevertheless constitutes a true obsession for science itself. For, as much from the angle of the infinitely large as from that of the infinitely small, the apeiron of the Greeks slipped like a shadow, and sometimes like an essential resource, into all attempts to explain the phenomena.

A labyrinth was the infinite for Leibniz, who although being a co-founder of the calculus called infinitesimal, declares on several occasions that, from the point of view of philosophical rigour: “I do not believe that there are neither truly infinite magnitudes nor truly infinitesimal magnitudes; they are just useful fictions to abbreviate and speak in a general manner”. A delicate labyrinth into which “entering was not given to me” regretted Jorge Luis Borges speaking about the Cantorian infinite that he just contemplated “from Bertrand Russell’s pages.”

The infinite likewise is a labyrinth for cosmologists, confronted today with dilemmas about the geometric structure consistent with the objective Universe. The Universe is unquestionably finite and closed only on the supposition of the objective density of matter being superior to the critical density, and on that of the universe having a positive curvature. For the remaining hypotheses the universe is at least open, although it would be perhaps useful to turn to the old Aristotelian distinction between potential infinite and actual infinite to determine whether it is infinite or not.

The evolution of ideas about the universe is often presented as the history of the incorporation of the infinite to the cosmos. The sequence Aristotle/Newton appears like an almost unavoidable itinerary. To prevent the aporias of the void, it pushes its way the idea that matter should occupy space completely, be extended in all its infinitude. The property of an imagined infinite space was thus attributed to matter. Since Netwon’s death this agreement was challenged. Cosmic matter was first formed by stars, then by stars and nebulae, later by galaxies and finally by a group of stellar bodies as varied as the natural species that developed on our planet. Astronomy leaned on physics to explain the structure of matter.

It is precisely this alliance between cosmos and physics that brings back the problem of the infinite in the universe. A simple transfer from the world of mathematics to physics is not enough to believe that the problem of its interpretation has been removed. The cosmic infinite is much more complex. It is the infinite of physical space, of the interpretation of time, of the reformulation of causality and, above all, of the explanations of how everything happened in a complex that shows restless, evolutionary and dynamic, where bodies like stars appear and disappear, where the galaxies move at a surprising speed, where it seems as if a large part of space cannot be seen, where the look of the terrestrial observer is suspended at an instant of time from where he can travel through the entire history of the universe. To look at the heavens would be to look at the past if this word had the same meaning as in a personal biography. But it does not, precisely because the notion of infinity has stopped being the safe guide it was at the good Newtonian age, the notion imagined, simple, and the receptacle where everything occurred in our world, the reference of space and time.

Returning now to a strictly mathematical field, we confirm that the question of infinity continues being a source of aporias as it was at the time of Aristotle and Leibniz. First of all, the debate from the end of the 19^th century surged in relation to the Cantorian construction of the transfinite numbers. Far from being settled, the discussion has been re-ignited in the last several years. On the one hand, there are the criticisms about the Cantorian infinite formulated by first class mathematicians such as Solomon Feferman. On the other hand, there are the discussions about the problem of the continuum, already dealt with by Cantor and brought up to date very recently by mathematicians like W. H. Woodin.

A complementary problem is that of the infinitely small. Rejected by 19^th century mathematics (to such a degree that Mario Bunge could talk in a debate with Abraham Robinson about Execution and Burial of infinitesimals), Cantor confirmed such repudiation by affirming that a theory of the infinitesimals en acto had nothing to do with the Differential Calculus or with the theory of functions. And nevertheless, in the middle of the 20^th century Abraham Robinson restored the concept of infinitesimal magnitude in the prodigious construction known as Non Standard Analysis. Is then the infinitely small the basis of this vraie metaphysique du Calcul differentiel that D’Alembert ascribed to the notion of limit? The debate remains open…

The infinite should be a relative concept, assures cosmologist Joe Silk. It will also be a source of metaphors in order to be able to express the universe, to enable a notion of origin that allows to talk about before the origin, to tell a story that seems to ramify in all the directions of its meaning, to fulfill the physicists’ dream who desired from the beginning of the 20^th century to find a unique source of explanation that allows to comprehend the world as if it were unique.

back to index / back to the home page

Sections of the congress

1. The concept of infinity in the history of philosophical thought.

2. The limits of the cosmos: infinity and finiteness in the history of cosmology.

3. Contemporary debates on cosmology: the Big Band Model and other alternative proposals.

4 History of theological controversies related to the idea of Infiniteness

5. Limits of thought: infinity in mathematics.

6. Infinity and fundamentals of mathematics: contemporary developments in set theory, alternative approximations to the foundation.

7. Controversies around the notion of infinitesimal magnitude along the history of thought.

8. Philosophical and Mathematical weight of Non-Standard Analysis.

back to index / back to the home page

PROGRAMA / BEHINBEHINEKO EGITARAUA

Special sections

Meeting

“Quantum jumps of light recording the birth and death of a photon in a cavity”.

[Nature 2007 March 15]

Discussion concerning the philosophical and epistemological implications with members of the team from the Laboratoire Kastler Brossel, Département de Physique de l'Ecole Normale Supérieure.

Seminar

“Mathematics and the Infinite in the Asian context”

Coordinator: Joseph Dauben (CUNY & Chinese Academy of Science)

SOME CONFIRMED SPEAKERS

Evandro Agazzi (International Academy of Philosophy of Science, Brussels) Joan Bagaría (ICREA Barcelona) – H. Benis-Sinaceur (CNRS Paris) - Henk Bos (Utrecht University) - Craig Callander (California University, San Diego) ZOU Dahai (I. for the History of Natural Science, Chinese Academy of Sciences) - Solomon Feferman (Stanford University) - Brian Greene (Columbia University, NewYork) - Adolf Grünbaum (Pittsburgh) - Michael Hallett (McGill University. Philosophy) - Michael Hoskin (Cambridge University) - Ignacio Jané (Universidad Barcelona) - Akihiro Kanamori (Boston University) LIU Dun (Institute for the History of Natural Science, Chinese Academy of Sciences, Beijing) - Paolo Mancosu (Berkeley University) - Tim Maudlin (Rutgers University) -Rafael Núñez (Cognitive Science UC San Diego) - Marco Panza (Université Denis Diderot París 7) - Jean-Michel Raimond (Kastler Brossel Laboratoire París) - GUO Shirong (Inner Mongolia Normal University, Huhehot, Inner Mongolia, China).- John Steel (California University) - Paul Teller (California University). HORNG Wann-Sheng (National Taiwan Normal University, Taipei, ROC) - XU Yibao (City University of New York, New York, USA)

back to index / back to the home page

Call for Papers

Papers will be submitted to the Advisory Council of the International Scientific Committee. Only one page abstract should be send before June, 30, 2008. Some selected contributed papers will be included in the proceedings.

back to index / back to the home page

Registration

UPV, UAB & Arteleku students: Free

Students: 35 euros

Audience in general before May, 31, 2008: 70 euros

Audience in general after May, 31, 2008: 100 euros

Send bank transference to this account number:

2101 0001 10 0010789824: Kutxa, Calle Garibay, San Sebastián (Spain)

IBAN code................................ ES98 2101 0001 10 0010789824

BIC (SWIFT) code..................... CGGKES22

Account holder.......................... Congreso Internacional de Ontología

Reference................................. IOC

back to index / back to the home page

Registration form

Send this form to:

Departamento de Filosofía

Universidad del País Vasco

Avda. de Tolosa, 70.

E-20018-SAN SEBASTIAN (España)

Name:

Adress

Phone:

Fax:

E-mail:

Institution:

Presenting a paper: yes no

Section in which you wish the paper to be included:

Title:

back to index / back to the home page