Archives mensuelles : septembre 2014

K-théorie et ses fondations

Readers of th​is list​

​may be interested in the following slightly abridged  version  of remarks on the background and founding of the K-Theory Foundation. The remarks were delivered at the opening ceremony of the first conference to be sponsored by the Foundation. The conference was  titled ‘K-theory and related topics’ and took place in Beijing in August of this year as an ICM satellite conference.  At the conference, the Foundation awarded for the first time its prizes to the most deserving young mathematicians working in K-theory and  related fields  to Joseph Ayoub of the University of Zurich and Moritz Kerz of the University of Regensburg.

 

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​Remarks on the background of the K-theory Foundation

 

Dear Colleagues,

 

Thank you for participating and contributing to this conference.

 

 

 

Special thanks go to the organizers of the conference Professor Guoping Tang and Professor Jianzhong Pan and the sponsors of the conference, the Chinese Academy of Sciences, the University of the Chinese Academy of Sciences, the Chinese National Science Foundation and the K-Theory Foundation, Inc. For the K-Theory Foundation, it is a special occasion. It is the first conference the Foundation is sponsoring and it is the first conference at which the Foundation will award prizes for outstanding work related to K-theory. The ceremony awarding the prizes will follow immediately after the first talk of this morning.

 

 

 

The funds which the Foundation is using to support the conference and to award the prizes were derived from sales of the Journal of K-Theory. There was a time, however, when there was no journal for the subject K-theory and not even a major listing for the subject in the American Mathematical Society’s Mathematical Subject Classification scheme. Let us go back in time, specifically to the year 1982, when this was the case and recall events which led to the creation 5 years later of a journal for K-theory, namely the journal ‘K-Theory’ and then 20 years after that the founding of the ‘Journal of K-Theory ‘ and finally 3 years after that the founding of the K-Theory Foundation. 1982 was a time when K-theory had established itself as a very fashionable topic and had played a major role in solving problems of great interest to mathematicians in various areas of mathematics. Many Fields Medalists of the preceding 20 years had contributed to or used K-theory in their work: Atiyah, Connes, Grothendieck, Milnor, Quillen and Serre. Still the vast majority of mathematicians remained skeptical that K-theory would be more than a passing vogue, a fashionable topic whose interest and application would dwindle, certainly by now.

 

 

 

In 1982, Grothendieck was 52 years old and the average age of a participant at a K-theory conference was between 30 and 40 years old. K-theory was a subject for young people. Against this background, I approached an academic publisher, Reidel, to ask if it might be interested in publishing a journal in a new, but rapidly growing subject of mathematics. I explained that K-theory was a new discipline of mathematics, embracing concepts, methodologies and problems central to many major disciplines of mathematics and that the aim of the proposed journal would be to further the cause of crosspollinization in mathematics by bringing together under one roof work having close conceptual and methodological relationships, work which had hitherto been scattered in the literature. (Further details regarding this are contained in the Editorial to the first issue of ‘K-Theory’.) Reidel’s response was cautious, but there was some optimism, which was supported by my close contact with one of its advisors. To make a long story short, let me simply say that it took 5 years before the first issue of ‘K-Theory’

appeared in 1987, the key development during this period of time being a genuine swing in attitude of the mathematical community towards K-theory, from a fashionable topic to a subject in its own right. Despite misgivings of some colleagues even in K-theory, the journal was an immediate success right from the beginning. In 1990 the journal ‘K-Theory’ played a role in the American Mathematical Society’s admitting K-theory as a major subject index in its Mathematical Subject Classification scheme and K-theory was the first subject the AMS spoke of as interdisciplinary within mathematics.

 

 

 

Many of you might have never heard of the publishing house Reidel, the first publisher of ‘K-Theory’. It was small, but fine. It does not exist anymore.

It was taken over in the late 1980’s by Kluwer Academic Publishers, also seated in the same city as Reidel, namely Dordrecht, Holland. Kluwer was considerably bigger than Reidel.   The commercial aspects of publishing a scientific journal were felt more strongly at Kluwer than at Reidel. This may be attributed to the size of the company. The size aspect was enhanced in 2003-04 when Springer Science & Business Media took over Kluwer. This was exacerbated by the fact that both Kluwer and Springer were migrating at that time from print to print-and-online. It was noticed by editors, authors and readers that the quality of the publication process and of the final product was declining considerably during this time, for example the quality of the typesetting and the timeliness of the appearance of issues of the journal were suffering. Many complaints were received from authors and this prompted in 2004 the idea among editors of the journal to begin looking for a new publisher. At the same time, prices for the journal were increasing systematically.  Some of you will recall that Rob Kirby had been voicing concern regarding high prices charged by commercial academic publishers and had gathered broad agreement among mathematicians on this issue.

Furthermore, the publisher began violating the copyright embedded in the contract with Reidel, which had not been changed for 20 years. Attempts to mutually resolve these problems with the publisher failed.  It was decided to at least resolve the copyright violation issue by filing a law suit against Springer, which was done in 2006. This made moving to a new publisher/distributor of the journal at a later date less problematic. The move to a new publisher happened a year later. To make once again a long story short, let me say that in 2007 a contract was negotiated with Cambridge University Press to bring out a successor to ‘K-Theory’ called the ‘Journal of K-Theory’.

 

 

 

Almost all the editors of the old journal were present on the Editorial Board of the new journal and a couple younger ones were added. However, a new era began with the Journal of K-Theory. The continual complaints the old journal had been receiving regarding shoddy typesetting and delayed issues disappeared overnight, being replaced although not as frequently as the former complaints, by emails of thanks and praise.

 

 

 

In the course of starting the new journal, it was conceived that the new journal should not only publish papers, but support in general the scientific community producing the papers it publishes and in particular use financial gains from publishing the journal for supporting K-theory activities. Accordingly, the K-Theory Foundation was set up in 2010 to carry out the philanthropic mission above. Its first Board of Trustees consisted in alphabetical order of Tony Bak (President), Max Karoubi, Tim Porter, Jonathan Rosenberg (Secretary) and Chuck Weibel (Treasurer). Through the Boards pioneering work, the Foundation received the basis of its current organization as a corporation and the formulation of operational rules for implementing the mission of the Foundation. The latter was accomplished last year by setting up 2 sets of rules, one for supporting conferences in K-theory and related subjects and the other for awarding prizes for outstanding work related to K-theory. This brings us to this moment in this room, to the opening ceremony of the first conference the Foundation is supporting and to the first awarding of the Foundation prizes. Let us extend our thanks and gratitude to the first Board of Trustees for bringing this about.

 

 

Some organizational parts of the K-Theory Foundation still have to be completed, but are on the way.

 

 

 

Finally, words of thanks go to the University of Bielefeld and the Faculty of Mathematics for their support for over 30 years of the projects above and to Cambridge University Press for its contribution as distributor of the Journal of K-Theory.

 

​​​Anthony Bak

 

President of the K-Theory Foundation, Inc.

 

 

 

 

Husserl, Cantor, Hilbert : la grande crise des fondements

http://arxiv.org/abs/1311.1524

Three thinkers of the 19th century revolutionized the science of logic, mathematics, and philosophy. Edmund Husserl (1859-1938), mathematician and a disciple of Karl Weierstrass, made an immense contribution to the theory of human thought. The paper offers a complex analysis of Husserl’s mathematical writings covering calculus of variations, differential geometry, and theory of numbers which laid the ground for his later phenomenological breakthrough. Georg Cantor (1845-1818), the creator of set theory, was a mathematician who changed the mathematical thinking per se. By analyzing the philosophy of set theory this paper shows how was it possible (by introducing into mathematics what philosophers call ‘the subject’). Set theory happened to be the most radical answer to the crisis of foundations. David Hilbert (1862-1943), facing the same foundational crisis, came up with his axiomatic method, indeed a minimalist program whose roots can be traced back to Descartes and Cauchy. Bringing together these three key authors, the paper is the first attempt to analyze how the united efforts of philosophy and mathematics helped to dissolve the epistemological crisis of the 19th century.
Keywords: Set theory, number, axiomatization, geometry, function, infinity.

Near death experience : il est complètement flou!

A ne pas aller voir si l’on aime Bruce Willis…

Au moins ce qui est bien avec Houellebecq (dont je n’ai pas apprécié énormément les œuvres) c’est qu’il ne finira pas complètement gâteux comme Sartre…

J’ai vu sortir de la salle trois spectatrices, avant la première demi heure: dommage pour elles, elles n’auront pas entendu cette répartie de Houellebecq, à un moment il est assoiffé, il boit un peu d’eau d’une piscine privée, le propriétaire ouvre la fenêtre et lui crie « Ne buvez pas cette eau, elle n’est pas potable »

Houellebecq (Paul, dans le film), l’œil torve, laisse passer un long moment de silence puis:

« M’en fous, je suis mort! »

Si j’ai bien compris, Houellebecq a un cancer du poumon, il est visiblement marqué, très amaigri…il est le seul acteur dont on voit entièrement le visage, mis à part un autre au milieu du film…il porte le film sur ses épaules d’un bout à l’autre, grâce à une présence étonnante.

Je ne perdrai pas mon temps à raconter ce film impossible à raconter…disons qu’on ne verra jamais « Near death experience 2″…il est d’ailleurs impossible de réaliser une peuvre analogue, ou d’imiter la performance de Houellebecq.

Disons que c’est une toute nouvelle façon de faire du cinéma, mais qui ne sera jamais répétée…un film qui rend ternes tous les films « normaux ».

Je ne révélerai qu’une péripétie : à un moment, une voix féminine se fait entendre tutoyant Paul et affirmant qu’il est « plein d’amour », tellement qu’il est trop tôt pour lui pour mourir.

C’est profondément vrai, à mon avis…cette voix se fera encore entendre à la fin , après avoir été prise par surprise par le dernier geste de Paul.

Je pourrais ajouter que pour certaines personnes, au demeurant assez rares, il se pourrait bien que ce film constitue une voie vers une sorte de « salut »…je n’éprouve pas le besoin de développer.

La critique des Inrocks vaut son pesant d’or et de fou rire, comme d’habitude:

http://www.lesinrocks.com/cinema/films-a-l-affiche/near-death-experience/

*****

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Category of categories as a model for the platonic world of forms

http://www.math.uga.edu/~davide/The_Category_of_Categories_as_a_Model_for_the_Platonic_World_of_Forms.pdf

application immédiate et mise en oeuvre de l’article précédent : la théorie des catégories est le stade le plus achevé de la mathématique universelle, son algébrisation la plus complète, datant de 1945.

Il n’ y a pas d’ensemble de tous les ensemnles (paradoxe de Russell) , mais il y a la catégorie de toutes les catégories.

 

Langue divine (mathesis), langues humaines

Reportons nous au premier chapitre (« Raison ») du dernier livre de Brunschvicg, termine en novembre 1943 deux mois avant sa mort:

« Héritage de mots, héritage d’idées »

http://classiques.uqac.ca/classiques/brunschvicg_leon/heritage_de_mots_idees/brunschvicg_heritage_mots.doc

Le passage suivant, qui oppose les conventions de l’orthographe aux démonstrations de l’arithmétique, nous introduit au cœur de notre sujet:

« Reportons-nous au moment, presque solennel, dans notre vie, où tout d’un coup la différence radicale nous est apparue entre les fautes dans nos devoirs d’orthographe et les fautes dans nos devoirs d’arithmétique. Pour les premières nous devions ne nous en prendre qu’à un manque de mémoire ; car nous ne savions pas, et nous ne pouvons jamais dire, pourquoi un souci de correction exige que le son fame soit transcrit comme flemme et non comme flamme. En revanche pour les secondes on nous fait honte ou, plus exactement, on nous apprend à nous faire honte, de la défaillance de notre réflexion ; on nous invite à nous redresser nous-même. Notre juge, ce n’est plus l’impératif d’une contrainte sociale, la fantaisie inexplicable d’où dérivent les règles du comme il faut et du comme il ne faut pas, c’est une puissance qui, en nous comme en autrui, se développe pour le discernement de l’erreur et de la vérité.
Cette impression salutaire d’un voile qui se déchire, d’un jour qui se lève, l’humanité d’Occident l’a ressentie, il y a quelque vingt-cinq siècles, lorsque les Pythagoriciens sont parvenus à la conscience d’une méthode capable et de gagner l’assentiment intime de l’intelligence et d’en mettre hors de conteste l’universalité. Ainsi ont-ils découvert que la série des nombres carrés, 4, 9, 16, 25, etc., est formée par l’addition successive des nombres impairs à partir de l’unité : 1 + 3 ; 4 + 5 ; 9 + 7 ; 16 + 9, etc. Et la figuration des nombres par des points, d’où résulte la dénomination de nombres carrés, achevait de donner sa portée à l’établissement de la loi en assurant une parfaite harmonie, une adéquation radicale, entre ce qui se conçoit par l’esprit et ce qui se représente aux yeux.
Les siècles n’ajouteront rien à la plénitude du sens que l’arithmétique pythagoricienne confère au mot de Vérité. Pouvoir le prononcer sans risquer de fournir prétexte à équivoque ou à tricherie, sans susciter aucun soupçon de restriction mentale ou d’amplification abusive, c’est le signe auquel se reconnaîtra l’homo sapiens, définitivement dégagé de l’homo faber, porteur désormais de la valeur qui est appelée à juger de toutes les valeurs, de la valeur de vérité.
 »

Doit on ici se limiter à l’orthographe ?

Il existe certes des « vérités » dites en langues de « logoi », de mots : les livres d’histoire ou de philosophie en sont remplis.

Il existe cependant un domaine, la logique, qui a été conquis au cours des deux derniers siècles par la mathématique, alors que Kant encore estimait qu’elle avait atteint son achèvement complet avec Aristote.

Se peut il qu’il y en ait d’autres dans l’avenir ?

Il est en tout cas évident que seuls les théorèmes (de la mathématique, de la physique mathématique ou de la logique mathématisée) sont des vérités éternelles (pour parler comme Descartes) : universelles et absolues.

Parce que la démonstration est immédiatement vérifiable par tous ceux qui en font l’effort (et font d’abord l’effort d’apprendre les bases).

Peut on en dire autant de propositions comme « La marquise sortit à 5 heures » ou « Au commencement Dieu créa le ciel et la terre » ?

Non, ne serait ce qu’à cause du « halo » de significations diverses que possède n’importe quel mot….