Contenant et contenu

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El Báltico en Helsinki – 2015

“Je m’amusais à regarder les carafes que les gamins mettaient dans la Vivonne pour prendre les petits poissons, et qui, remplies par la rivière, où elles sont à leur tour encloses, à la fois « contenant » aux flancs transparents comme une eau durcie, et « contenu » plongé dans un plus grand contenant de cristal liquide et courant, évoquaient l’image de la fraîcheur d’une façon plus délicieuse et plus irritante qu’elles n’eussent fait sur une table servie, en ne la montrant qu’en fuite dans cette allitération perpétuelle entre l’eau sans consistance où les mains ne pouvaient la capter et le verre sans fluidité où le palais ne pourrait en jouir…”

Este pasaje (Du côté de chez Swann, I, II, p. 166 en Pléiade) evoca la dualidad entre contenedor y contenido pero muy proustianamente genera ambigüedad y hace del contenido el contenedor a su vez – en un juego infinito de percepción: me gustaba mirar las jarras que los chicos ponían en la Vivonne para atrapar pescaditos, y que al ser llenadas por el río, en el cual a su vez estaban sumergidas, a la vez “contenedores” con bordes transparentes como un agua endurecida, y “contenido” sumergido en un contenedor más grande de cristal líquido y corriente, evocaban la imagen de la frescura de una manera más deliciosa y más irritante que como lo hubieran hecho sobre una mesa servida, mostrándolo solamente en fuga en esta aliteración perpetua entre el agua sin consistencia donde las manos no podían captarla y el vidrio sin fluidez donde el paladar no podría gozar…”.

Uno de los temas que me interesan mucho hoy en día por razones puramente matemáticas y también por razones filosóficas conectadas con las anteriores, es la ruptura de la sintaxis, la manera como la semántica a priori determina la sintaxis, la manera como una a su vez determina (o no) la otra. Obviamente el teorema de Presentación de Shelah es un ejemplo de esto, pero hay mucho más (lógicas implícitas, maximalidad, etc.).

Esta descripción de las jarras de Proust, con textura casi fluida del vidrio y casi rígida del agua, y con la alternación entre contenido y contenedor, me parece impresionante.

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El Pilar, Zaragoza – reflejado en el río Ebro – 2017

Unity and Diversity of Logic (Kossak, Villaveces)

We wrote the following essay on the Unity and Diversity of Logic, together with Roman Kossak, a few months ago.

It appeared as a mathematical introduction to the book we edited (also with Åsa Hirvonen and Juha Kontinen) Logic Without Borders (with essays by S. Abramsky, J. T. Baldwin, J. Burgess, X. Caicedo, Z. Chatzidakis, C. Di Prisco, M. Dzamonja, C. Franks, P. Galliani, L. Hella+J. Väänänen, W. Hodges, J. Hubicka+J. Nešetřil, T. Hyttinen, R. Kaye+T. Lok Wong, J. Kennedy, J. Kontinen, S. Lindell+H. Towsner+S. Weinstein, M. Malliaris+S. Shelah, I. Niiniluoto, J. Paris+A. Vencovská, S. Shelah and J. Väänänen.)

A couple of months just spent in Helsinki, with various conferences since May and many mathematical encounters, convinced me more than ever of the importance of the Unity of Logic viewpoint.

Here is the first paragraph of our essay:

What is mathematical logic today? How does it connect with its historical roots? How does it continue to serve as foundations of mathematics, and how does it impact mathematics in general? Does it continue to serve as the foundations of mathematics at all? What distinguishes advanced areas of mathematical logic from other branches of mathematics? What parts of mathematical logic should be considered philosophy, and what parts evolved into independent subdisciplines of algebra, analysis or computer science? The article by Juliette Kennedy in this volume addresses some of these issues directly, as does Jouko Väänänen’s personal account of the development of his interests in mathematical logic. Other articles in the volume might be construed as providing partial responses to these questions, of course not necessarily in a direct way, but through the connections and links they explore, both internally within logic and externally between logic and other disciplines.

You can download the essay from here.

Addenda: Javier Moreno has now read our essay. He seems to find it interesting (he suggested the topic is good for a book!) but found it too short, too dispersed and lacking a unified voice. (All of this I lift from a twitter conversation…)

To this I have to say:

  • first of all, thanks Javier for reading!
  • second, I agree it is too short (but as it was the introduction to a quite long volume, we didn’t want it to become like another article – it should somehow open up the question of unity versus diversity in logic today – but should not have the weight of the real papers collected – we are editors, not authors!)
  • furthermore, I agree: it lacks unity! As it is the product of two minds, of two voices, of two points of view, it has a combination of both. Although we speak quite a lot with Roman (on logic, math, art and many other things), in the subject of our introduction there are points of disagreement (or different perspectives). At some point, the essay was going to be a conversation but it felt a bit overacted – we ended up doing write-and-rewrite of our own sentences, crisscrossing ideas. The result is bound to be pointing in at least two directions… I kind of like it that way at this point…
  • there is a long essay, somehow on the same topic, and definitely recommended to anyone interested in the topic, by Jouko Väänänen, in the volume itself. We asked him to write his own statement, his own “manifesto” on why logic (and not a part of logic, or as is so fashionable, seeing logic as some part of geometry). The text he wrote is a superb piece of intellectual understanding of what logic is today, and may be.
  • finally, I have been writing a longer piece for a volume for the Simplicity meeting – now almost finished. And Roman has written longer pieces on subjects connected to this (and we both have to write the mathematical parts of our joint project with artists Wanda Kossak and María Clara Cortés).

Breaking syntax molds

Of course in Mathematics we have always been quite formalism-free, quite glad to jump from “First Order” to “Second Order” logic without missing a heartbeat, quite glad to dwell in Semantics as our home-base, quite glad to regard the insistence of some logicians on first order formalizations with a little of the same puzzled horror we normally reserve for items like a 19th century corset. Some logicians have actually dug hard into the mathematical (and logical, and philosophical) root of this phenomenon: there are solid intrinsic reasons why we can do away with syntax most of the time. Mathematical logic (especially in the neighborhood of second order model theory, its sister set theory and the formalism-freeness aware abstract elementary classes) has long been able to account for those phenomena.

Excessive insistence on syntax normally seems to be the hallmark of pedantic philosophers, or of computer scientists. Mathematical logicians (especially Model Theorists) usually try to stress the “mathematical” and deemphasize the “logician” in them – often for good reasons, sometimes with a bit of envy of the rest of mathematicians too evident perhaps.

In Gödel’s 1946 paper (for the Bicentennial at Princeton), as Juliette Kennedy has remarked, a framework for breaking syntax is set up, a framework that can be seen to oddly combine the insight obtained from Turing (that helped unify the many previously logic-dependent or syntax-dependent notions of computability due to Gödel, Church and others) with the quest for new axioms for mathematics. Lately (cf. Kennedy’s Dublin lecture) these breaks of syntax have been realized in several domains: (1) the development of “transdefinability hierarchies” (Kennedy, Magidor, Väänänen) where the idea of iterating the construction of L by Gödel is transferred to other logics [and apparently for logics with the cofinality quantifier a robust definition is obtained]  –  the structures obtained are now called C*(L) where L is a logic and some of them exhibit interesting behavior, (2) the development of “symbiotic” notions between set theory and second-order logic (symbioses can be seen as extremely rich adjoint pairs) capturing compactness, amalgamation, Löwenheim-Skolem phenomena and much more (Väänänen), (3) the development of Model Theory in a very syntax-free manner through abstract elementary classes (Shelah opened that door, several people went in [not en masse] and have done hard work in extending Classification and Stability Theory in a way that reveal structural features not depending on first order syntax), extracting structure way beyond the pale of first order logic yet with a robust classification theory, allowing other kind of symbioses with both set theory (large cardinals, forcing) and geometry (sheaves, accessible categories) – albeit with a firm connection to syntax and even first order through the Presentation Theorem – aecs are really versions of projective classes, and this fact underlies much of the strength of this generalization, in spite of its strong freedom from syntax; and (4) a blend of some of the previous with sheaves (this builds on earlier work of Macintyre and of Grothendieck himself, and through several stages has distilled several semantic/syntactic “blends” and limits – Caicedo’s “sheaf-generic models” (a version of ultraproducts, aware of topology, encompassing usual set-theoretic generic models) an important step in this distilling process – more recently, metric versions and awareness of group actions on the sheaves (joint work with colleagues and students in Bogotá: Padilla, Ochoa) and Zilber’s very recent “Weyl sheaf” – lifting classical semantics/syntax through five stages of generalization of the classical “algebra/geometry” duality (quotients of polynomial rings by ideals/algebraic varieties with Zariski topology all the way through C*-algebras/Zariski geometries) onto the Weyl sheaves.

Minimal notes.

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Fragment of a topos: Through a glass, blurredly. June 2014.
  • Aotearoa, next 10 days. To dive into conversation with a mathematical couple (he, an opener of Shelah’s “third way” (according to the Lazy Model Theoretician guide), she, an algebraic geometer). Will try to pay honor to Ruapehu, if time allows it.
  • Back to LPs for sound (now, with USB connection). Warmer? Different sampling?
  • Neighbor is a fan of Mos Def. While I shower I hear his Mos Def – not sure if I like it that way, but at least I know the song (and like it). Being a model theorist, seeing the words Mos Def, I cannot help thinking what kind of definability is he implying in his works. I cannot not see definability whenever it is hinted at, even unwillingly or unawares.
  • Extreme fatigue caused by election nervousness: the state of all around me, myself included. Brutal weariness.
  • Guasca: veredas ondulantes a 2900 metros. Fin de semana idílico, en un paisaje que ya empieza a dejar la Sabana, sin ser realmente el Páramo.
  • Pleasantly surprised by Bovykin’s approach to Paris-Harrington, presented in my logic class by three students. I asked each group (at the start of the semester) to find their own project subjects – they had freedom to discover it, but then had to work on what they chose. This group chose to work on Paris-Harrington (we saw Gödel’s Incompleteness by the middle of the semester and I had been mentioning the more “close to real mathematics” result by Paris and Harrington). They found Bovykin’s approach to be much more readable – one of them seems to have gotten well the role of cuts and minimal initial segments.
  • Backlog in almost everything.
  • (Godement (following Cartan and Grothendieck))’s treatment of classical construction of sheaves veers close to invariant sheaves in model theory.
  • Zalamea (Fernando) has managed to write a letter that is at the same time personal and a work of art – close to the style of his late brother Gustavo. I received that letter, and I feel I have a version of a “dual Zalamea”, a kind of hybrid of the talents of the two brothers. I have tried to photograph it in a way that blurs the personal, yet allows the “tree” structure, the “pictorial” background to be seen and appreciated. No success: my abilities as a photographer, when trying to “mod out” reality, are too limited.
  • In our project topoi (where we have been slower than at the beginning, for good reasons that have to do with internal dynamics of the project itself), we have been trying to capture dynamic topoi  –  “dynamic systems” that like the other topoi, bridge the (almost impossible, yet almost obvious) conversation between two artists and two mathematicians. So far, the project has been building up, slowly really slowly, the new path toward dynamic topoi.
  • I truly liked my other student’s presentation of Hrushovski’s cryptic line in the Covers and EI paper – ninety minutes of Galois theory to cover up a sentence. Beers after seminar were welcome, and good.
  • I see more and more young parents around me (well, not really young – compared to my own parents when they became parents, they are really old, but I mean “young parents” as in “parents of small children”) incredibly concerned about every little detail of their children’s life, to an insane degree. Unhappy with all school systems, unhappy with vaccines in some cases, unhappy with possible germs in schools, they utter sentences such as “how can I trust my dear child to unreliable teachers, horribly bad influence of society, strange people’s children?”. They ponder and discuss and are concerned endlessly about wanting their children to be atheistic and free from bad influence from religion, or wanting their children to be free from violence in movies or TV, or maybe decide to home-school their children because schools are “all so terrible”. Those poor children are the center of too many concerns, and I worry they will react sharply to their parents’ extreme insecurities. While they may not hear their parents’ constant concern being voiced over, they for sure can feel it and may absorb the excess insecurity of our times. Too much helicoptering doesn’t forebode well for those poor children of over-concerned parents.
  • For a month now, have listened to no hip-hop and no rap (except for my neighbor’s Mos Def). Cleansing my own ears and mind is good, to appreciate better. Now Mahler and Bruckner and Schoenberg are filling the void.
  • Like our ancestors, I always feel that when the semester ends I crawl, slowly, out of a boiling pool of water, slowly into open air. While I like very much teaching (and consider it a privilege), I also love being freed from teaching obligations, at the end of the semester. Both feelings seem contradictory, but both are important. עֵת לִפְרוֹץ וְעֵת לִבְנוֹת.

Cohen, sobre Gödel (y la memoria)

Es bien interesante cómo cuenta Paul Cohen (en la parte 2) cuando (siendo estudiante en Chicago) quería encontrar un procedimiento de decisión para las ecuaciones diofantinas, Kleene le dijo que no era posible, y lo llevó al teorema de Gödel. Al principio Cohen trató de encontrar un error, se convenció de la demostración de Gödel y quedó inspirado. La parte de los “muchos universos” versus el programa de universo único también vale la pena. Es bellísimo el excursus literario (yendo a Bacon, a Proust, sobre la verdad de la memoria y la música) .

Habla de intuición matemática (el “flash” que hace que uno vea la demostración, si está de buenas).

Muy inspiradora esta conferencia de Cohen en el centenario de Gödel en 2006 en Viena. Rather early in the game I think forcing occurred in a very very hazy form to me… but I didn’t know what I hadHa! Construction! That is what Gödel’s doing! … My God, this thing is crazy, but it actually seems to work! I thought that the notion of forcing was on a nice edge… it seems to be nonsensical, because you are trying to construct a model where things will be true, you don’t know what the model is, you use what you want to be true as the basis for constructing the model, you don’t know what truth is, and so, conscious … I couldn’t believe it… look, it’s really basically two lines… those of you who know what forcing is … universal versus existential quantifier…

(Gracias a Luis Miguel Villegas por enviar el enlace.)

recap – tarte tatin – stream of

Having no time to compose posts – to com-post, as it were – during these past days, I reduce my comments to some photographic hints of a few moments of these past heady days. No glimpse of a six-day immersion into the world of modular invariants with Tim – quantum and classical – one of the most beautiful times of my mathematical life so far. This is what I want to do, that is what I want to solve. On Sunday mere extreme, but elated, fatigue:

OLYMPUS DIGITAL CAMERADays before, a meeting with our friends F and ME, accompanied by tarte tatin [now made symbolic of all that’s not green jelly=analytic philosophy, tarte tatin is continental, phenomenological, husserlian, merleau-pontyian, perhaps even, in a twisted way, peircian! (I need to add the quote from F’s essay, of course…)] and fishpie – a meeting to reminisce the harshnesses and porosity of the Scottish coastline, all the while visiting beloved movies in our palates and minds, evoking Riemann surfaces and antinomian roots and branches (and so many trees, so many shapes in ME’s incredible collection). An infinite conversation, infinitely branching and porous and twisting and self-reflecting. An evening made of expectations, discoveries, much more unsaid than said, much more implied than stated. Like walking on the upper decks of those abandoned abbeys of Yorkshire and Scotland, made similar to the coastline by cumulated winds and storms, and glimpsing at the abysses, the fall, the vertigo. It was vertigo!

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No time to read: barely 30 pages of Gombrowicz: the fall into immaturity as told in Ferdydurke – one cannot read too fast such topics. It seems to speak of my own (daily) falls into immaturity, my own inner struggles between a mature AV and the many immature versions of AV that co-exist and resurface and bully the mature one as in the novel. The novel creates streams of consciousness, pro-bono.

Of course, a couple of articles for my logic course – among them, one by Schiemer and Reck called Logic in the 1930s: Type Theory and Model Theory. The very least I can do for my students if starting to teach mathematical logic for the n-th time is to renew sources, at least a little bit – and read about the origins (ideally, some original material as well). Juliette’s paper for the Bulletin is of course, another important source.

Teo visited today. MC took him to the big park at Museo del Chicó, to be in the park, to play. And then, afternoon, he stayed here, talking a bit, playing a lot, laughing and looking at labyrinths and building houses and car runs. He is almost impossible to photograph – he moves too fast for the low light of Bogotá, for my lack of flash in the camera.

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Jogo de cintura – jugando con tres periféricos

La expresión común en Brasil jogo de cintura (que apareció en el título de una charla de Leandro Aurichi sobre principios de selección y juegos topológicos – casi sistemas dinámicos topológicos y significa algo así como “saber hacerle el quite a los problemas, no enredarse”) me gustó mucho. Ignoro de dónde viene – la gente a quien pregunté tampoco parecía saber; simplemente les parecía obvio que se usara “ter jogo de cintura” ante situaciones donde conviene no ser rígido. Aunque no es lo mismo, el cutting slack del inglés parece remotamente relacionado con la misma idea.

Ayer me tocó tener bastante jogo de cintura al intentar por primera vez usar openeya para filmar una clase de teoría de modelos que estoy dando y para la cual hubo pedido de Medellín y Pamplona. Intenté transmitir directamente, pero hubo problemas de horario. Decidí entonces filmar con openeya – algo que usan algunos colegas del Departamento de Física para filmar y colgar sus cursos de Teoría Cuántica de Campos.

Openeya está bastante bien pensado: una cámara normal lo filma a uno en resolución baja, mientras un buen micrófono va grabando. Otra cámara de alta resolución va tomando una foto del tablero cada 15 segundos (más o menos el tiempo en que puede aparecer algo relevante, tal vez) – foto que idealmente pueden ampliar los que sigan el curso y ver las cosas escritas en detalle. Los físicos llegaron a Openeya en parte gracias al Instituto de Trieste, donde parece que impulsan bastante este tipo de iniciativas.

Problemas locales:

  1. El tablero blanco es mucho más difícil de fotografiar que el tablero tradicional negro. Aquí no tenemos tableros decentes (problema de todas las universidades bogotanas) sino tableros blancos de marcador. El contraste en las fotos es mucho más bajo que con buenos tableros. Los reflejos de luz son fatales.
  2. Por alguna razón mi computador quería suplantar una de las cámaras buenas que me dieron los físicos por su camarita medio boba incorporada, menos buena. No logré dar bien con el chiste y me tocó reemplazar por un computador más pesado y viejo de los físicos.

De resto, con un poco de jogo de cintura, paciencia de mis estudiantes de Bogotá y algo de buena voluntad, arrancamos. Espero que en Medellín y Pamplona encuentren que las clases les ayudan a aprender. Todavía ando viendo si consigo mejor salón, mejores tableros, y sobre todo si aprendo a hacer esas cosas mejor.

La gran ventaja de openeya es que no depende uno de salas especiales y en principio (módulo tableros clásicos decentes) con llevar dos camaritas y un micrófono uno puede armar un muy buen simulacro de clase para transmitir. Además, openeya procesa todo y genera directamente una página web que uno cuelga y que tiene todo bien sincronizado. Las cosas que financia ese instituto de Trieste.

Ahora hay pedido para hacer lo mismo con nuestros (tres) seminarios de lógica de Bogotá. Uno de los estudiantes ahora se fue a Berlín a trabajar con Amador, pero dice que quiere poder seguir los seminarios de aquí, que están muy interesantes este semestre (estamos estudiando temas que incluyen teoría de modelos sobre haces perversos (con Gabriel Padilla y Gregorio Mijares), categorías modelo y pcf (con Andrés Ángel y Xavier Caicedo), dicotomías conjuntísticas en teoría de modelos, métodos de subestructuras elementales en topología (Ramiro de la Vega), clases de Fraïssé categóricas y teoría de Ramsey (con Gregorio Mijares, Carlos Di Prisco, Gabriel Padilla), etc. etc. Es un semestre de seminarios agradables, activos, llenos de preguntas y pequeños descubrimientos.

Hasta ahora solo uno de esos seminarios se está transmitiendo (el de Andrés Ángel) pero con cámara tradicional, sin openeya. Si aprendo a usar bien openeya, creo que les diré que traten de adoptarlo.

Teoría de modelos sobre haces, ahora en Bogotá

En estos días tiene lugar el Primer Encuentro de Lógica y Geometría – que organizó principalmente Gabriel Padilla, con el profesor Mijares del IVIC como invitado.

En Bogotá el tema de Teoría de Modelos sobre Haces es supremamente importante. En seminarios que tuvieron lugar en la Universidad Nacional en la década de 1980, Xavier Caicedo desarrolló una versión de lógica sobre haces original, distinta de versiones de MacLane, Joyal o Macintyre de ciertas maneras específicas. Aunque es más específica que la versión puramente categórica, es más general y sobre todo más natural que la versión booleana de Macintyre.

Varios estudiantes de Xavier Caicedo exploramos en múltiples tesis de maestría algunos aspectos de la lógica de haces, durante nuestra etapa de formación inicial.

Posteriormente, algunos de nosotros hemos seguido trabajando – al lado de muchos otros intereses – en aspectos de la lógica de haces. El tema ha empezado así a atravesar fronteras (de la matemática y del ámbito geográfico local). Por un lado, Fernando Zalamea en su libro Filosofía sintética de las matemáticas contemporáneas (reseña mía aquí) hace un uso radical, novedoso, de cierta interpretación de la lógica sobre haces, en una propuesta filosófica contemporánea que busca integrar, analizar y a la vez sintetizar, toda una variación continua de conceptos filosóficos que surgen de problemáticas de la matemática contemporánea.

Por otro lado, en trabajo conjunto con Ochoa – y más recientemente a raíz de conversaciones con Boris Zilber en Cuernavaca – he notado cómo muchas preguntas naturales que está haciendo la gente en distintos ámbitos están exigiendo (literalmente) nociones muy cercanas a la semántica de haces. Con Ochoa logramos una generalización de la teoría de Caicedo a ámbitos de lógica continua. Ochoa logra plantear el inicio de haces específicos para problemas de Química Teórica. Trabajos de Schoutens en Álgebra Conmutativa (y en teoría de números) están requiriendo construcciones cercanas a la semántica de haces.

Hoy en día el tema ha llegado a un punto de madurez interesante y a la vez peligroso (si no difundimos más este tema, si no vemos la importancia que tiene para la lógica en Bogotá -primero- y para la lógica en general -más aún en este siglo – corremos el riesgo de ver todo este tema redescubierto en otros lugares). En Helsinki recientemente dí un minicurso sobre el tema (septiembre de 2010 – evento para los 60 años de Jouko Väänänen), en tres sesiones de dos horas. Las primeras dos horas estuvieron dedicadas exclusivamente a la “teoría de Caicedo”. Las últimas a generalizaciones y construcciones más recientes.

Daré una versión (compacta) de ese minicurso en este evento de Lógica y Geometría este viernes y el próximo martes.