samedi 11 septembre 2021

Comment Grothendieck n'est pas devenu physicien...

... mais mathématicien, (faute de moyen matériel ?)

 

Surtout après l’expérience de la guerre et du camp de concentration, en butte à des discriminations et préjugés qui semblaient défier la raison même la plus rudimentaire, ce qui me fascinait surtout dans l’activité mathématique ...c’était ce pouvoir qu’elle donnait, par la vertu d’une simple démonstration, d’emporter l’adhésion même la plus réticente, de forcer l’assentiment d’autrui en somme, qu’il soit bien disposé ou non — pour peu seulement qu’il accepte avec moi les “règles du jeu” mathématique. 
... plus tard, séduit par le prestige soudain de la physique atomique, c’est pourtant pour des études de physique que je me suis d’abord inscrit à l’ Université de Montpellier, avec l’idée de m’initier aux mystères de la structure de la matière et de la nature de l’énergie. Mais j’ai vite compris que si je voulais m’initier à des mystères, ce n’était pas en suivant les cours de la Fac que j’y arriverais, mais en travaillant par mes propres moyens, seul, avec ou sans livres. Comme je n’avais pas le flair, ni l’appareillage, pour apprendre la physique de cette façon-là ; j’ai renvoyé la chose à des temps plus propices, me suis alors mis à faire des maths, tout en suivant "de loin" quelques cours, dont aucun ne pouvait me satisfaire, ni m’apporter rien au delà de ce que je pouvais trouver dans les manuels courants.
Source

 


vendredi 10 septembre 2021

What is the role of Mathematics?

The role of the mathematician is to create concepts and...

 

Let me take one instance, the notion of truth, in order to illustrate what I mean. We all have in mind that something is either true or false. If we attend a debate on politics or another controversial topic, we are prone to say that this guy is right and that guy is wrong and that’s our way of making a judgement. Now, it turns out that probably we should be more advanced at the level of the formalisation of the idea of truth. In fact, there is a mathematical concept which has been created by A. Grothendieck which is the concept of topos and which has, thanks to contributions of F.W. Lawvere, a far more sophisticated notion of truth [read transcript of the video below about topos]. Technically the “truth values” form an object of the topos and this object classifies sub-objects exactly like the characteristic function (which takes values in the two point set “True, False”) of a subset does in the case of the topos of sets. Thus for this simplest topos something is true or false. But as soon as you take a slightly more involved topos, such as the topos of quivers, you get a much more refined notion of truth values and in the case of quivers it involves "making mistakes, corrections, checking" as fundamental parts of the structure. From this example, you witness that mathematics is a factory of concepts which are extremely rich, which are subtle and which of course are hard to grasp by the public in general due to their mathematical precise and involved formulation. This lack of grasp by the public holds at a certain time in the history of civilisation but I believe that in later years these concepts will become common. This sophisticated notion of truth has, by the way, nothing to do with probabilities. It’s a very beautiful and precise notion developed by a great genius of mathematics.to me, this is the role of mathematics: fabricate concepts and facilitate the process by which the public acquires them. That’s it.

Alain Connes mathematcian, interviewed by a theoretical physicist Jay Armas in Conversations on Quantum Gravity




Transcription of an extract in French:




 

mardi 8 juin 2021

Is [doing mathematics, doing physics]?


I try to use the information from physics as very sophisticated computations that physicists do that are tested by experiments and that somehow reveal if one looks at them carefully enough from a matehmatical point of view reveal amazinsingly beautiful structure

 Alain Connes (online.kitp.ucsb.edu/online/strings05/connes/rm/jwvideo.html)


-Do you have a preference for mathematics over physics?
-“My heart lies with both.”

dimanche 18 mars 2018

De quantum natura quidem Tractatus logo-physicus

18 Février 2018 : Bon anniversaire Roland Omnès !


Je découvre par hasard aujourd'hui (avec un mois de retard) que le merveilleux professeur (aujourd'hui émérite) et chercheur reconnu encore actif  avec qui j'ai eu le privilège de découvrir la mécanique quantique en première année de magistère de physique fondamentale d'Orsay vient de fêter cette année ses 2²+3²+5²+7² printemps !

J'espère avoir le temps un jour d'approfondir la lecture de ses dernières réflexions sur l'ambitieux programme de démontrer l'unicité de la réalité physique macroscopique par une approche quantique

Pour évoquer ses qualités de chercheur et de professeur sans verser dans la flagornerie je laisse la parole à un autre physicien qui fait ci-dessous une recension d'un des livres d'Omnès les plus importants directement écrit en anglais.  

The Interpretation of Quantum Mechanics. By Roland Omnès. Princeton University Press, Princeton, New Jersey, 1994, 
This monograph is the first book-length treatment of the consistent histories approach to the interpretation of quantum mechanics which I initiated in 1984, and to which Omn`es (starting in 1987) and Gell-Mann and Hartle (starting in 1990) have made major contributions. While consistent historians do not agree on every detail, there is a common core of ideas which can be summarized as follows. A closed quantum system (the universe, if one is ambitious) is represented by a Hilbert space, and anything that can sensibly be said about it at a particular time is represented by some subspace of this Hilbert space; in other words, there are no hidden variables. A history consists of a sequence of subspaces E1, E2, . . . associated with times t1, t2, . . ., understood as events occurring, or properties which are true, at these times. Provided the history is a member of a consistent family of histories, it can be assigned a probability, and within a given consistent family these probabilities function in the same way as those of a classical stochastic theory (imagine a sequence of coin tosses): one and only one of them occurs, and the theory assigns a probability to each possibility. Inconsistent histories, those which do not belong to a consistent family, are meaningless. The unitary time evolution generated by the Schrödinger equation, without any stochastic or nonlinear modifications, is used both for determining the consistency conditions which define consistent families, and for calculating the probabilities of histories belonging to a particular family. 
Measurements play no fundamental role in the consistent history interpretation; they simply correspond to sequences of events inside a closed system in which the measurement apparatus, along with everything else, is treated quantum mechanically. Thus a possible history for a closed system in which there is an apparatus for measuring the z component of the spin of a particle might include an initial state, a value of S z at a time shortly before the particle reaches the apparatus, and the position of a pointer on the apparatus at some later time. Using conditional probabilities one can show, under suitable conditions, that the particle earlier had the property indicated later by the pointer position. In this and various other ways the consistent history approach replaces the smoky dragons which inhabit textbook and other treatments of “measurement” with precise mathematical and logical rules yielding results which are often much closer to the intuition of experimental physicists than to what one finds in the literature on quantum foundations... 
While it does have defects, I nevertheless admire this book as a bold attempt to give substance to a vision common to consistent historians, namely that our current scientific understanding of the physical world, macroscopic phenomena as well as microscopic, can be linked to a firm foundation of quantum mechanical principles by appropriate and precise rules of sound reasoning. There is no need to add hidden variables to the Hilbert space, or tinker with the Schrödinger equation, or restrict ourselves to talk about “measurements” in order to construct a coherent interpretation of quantum mechanics which overcomes the well-known conceptual problems which have given such trouble to those who have been attempting to understand the subject for the past seventy years. We are indebted to Omnès for showing how much of this program can actually be carried out within the consistent histories framework. Some imperfections are inevitable in a pioneering effort, and those portions of the book which I find problematical are nonetheless useful as indications of what needs to be better understood, or more clearly explained, or perhaps both.
Robert B. Griffiths (Carnegie-Mellon)
(Submitted on 16 May 1995)

Pour goûter toute la saveur de la pensée d'Omnès exposée en termes non techniques, il vaut sûrement mieux le lire quand il écrit dans sa langue maternelle. Etudiant, je me souviens avoir beaucoup apprécié Alors l'un devint deux en particulier son chapitre sobrement intitulé conclusions. (Mon indécrottable romantisme péri-germanique m'inciterait à proposer comme autres titres : Tractatus logo-physicus ou bien De quantum natura pour évoquer le caractère philosophique et analytique mais aussi l'élégance et la simplicité de l'exposition). 
 
Voici ce qu'annonce la quatrième de couverture :  

Que sont donc les mathématiques ? Après Platon et Russell ou Gödel, cette grande question suscite toujours plus de débats et de controverses et elle attend encore sa réponse. Il s’agirait d’un pur jeu formel pour les uns, de l’image d’une réalité immatérielle pour les autres... Et voici que les acquis récents de la physique des particules invitent à explorer une troisième voie. Car les lois qui gouvernent les quarks et autres constituants de la matière, la manière dont elles furent découvertes et les modes de pensée par lesquels il fallut passer sont autant de révélateurs d’une parenté intimement profonde entre physique et mathématique. Pourtant, si l’on pénètre davantage les rapports complexes qui s’exercent entre la physis – la réalité empirique – et le logos – la réalité des formes –, force est de constater aussi qu’un irréductible hiatus les sépare. Alors, l’un devient deux...   
What is mathematics? After Plato and Russell or Gödel, this fundamental question is still the subject of more debate and controversy and is still waiting for an answer. It would be a pure formal game for some, the image of an immaterial reality for others... And now recent achievements in particle physics invite us to explore a third way. Because the laws that govern the quarks and other constituents of matter, the way they were discovered and the ways of thinking through which they had to pass are all revealing of a profound relationship between physics and mathematics. However, if we penetrate more deeply into the complex relationships between the physis - the empirical reality - and the logos - the reality of forms - we must also note that they are separated by an irreducible hiatus. So, one becomes two...

(Translated with the help of www.DeepL.com/Translator)

 

Dans ma thèse j'ai choisi la citation suivante :

Proposition 32 
La connaissance de la physis ne peut provenir que d'une forme d'expérience. La disjonction de nature entre physis et logos se laisse voir dans la philosophie traditionnelle comme l'impossibilité de construire la science par pure induction
Knowledge of physis can only come from a form of experiment. The natural disjunction between physis and logos can be seen in traditional philosophy as the impossibility of building science by pure induction.


dimanche 17 décembre 2017

Adventures in math of a quantum thinker

...once one took String Theory seriously, one soon found a lot of reasons that physicists would have to pay attention to previously unfamiliar topics in more or less modern mathematics. For example, at a basic level, a string moving in spacetime sweeps out a two-dimensional surface with the property of what mathematicians call a Riemann surface. Riemann surfaces are an important topic in the mathematics of the last century, and they became important for physicists primarily because of String Theory. 

From my vantage point, all this made the interaction of physics with more or less contemporary mathematics far more robust and significant. Opportunities to apply physics-based insights to “purely mathematical” problems stopped seeming like exceptions. 

In my own work in the years just after 1984, the development that is most worth mentioning here involves topological quantum field theory. This was partly motivated by hints and suggestions by the mathematician Michael Atiyah, who pointed to mathematical developments that he suggested should be better understood using physical insight. Other hints came from developments in physics.


Each problem here involved applying physics ideas to a problem that traditionally would have been viewed as a math problem, not a physics problem. These were all problems that I would not have seriously considered working on until String Theory broadened our horizons concerning the relations between mathematics and physics. In each case, the aim of my work was to try to show how a problem that naively is “purely mathematical” could be approached by methods of physicists. 

I will just tell you about one of these problems. It involved knots in ordinary three-dimensional space. A tangled piece of string is a familiar thing in everyday life, but probably most of us are not aware that in the 1900's, mathematicians built a deep and subtle theory of knots. By the time I became involved, which was in 1987-8, there was a puzzle, which Atiyah helped me appreciate. The mathematician Vaughn Jones had discovered a marvelous new way of studying knots – for which he later received the Fields Medal. Vaughn Jones had proved that his formulas worked, but “why they worked was mysterious. 
It may be hard for someone who does not work in mathematics or science to fully appreciate the difference between understanding “what” is true and understanding “why” it is true. But this difference is an important part of the fascination of physics and mathematics, and I guess all of science. I will say, however, that the difference between “what” and “why” depends on the level of understanding one has at a given level of time. One generation may be satisfied with the understanding of “why” something is true, and the next generation may take a closer look.  
Anyway, getting back to knots, I was able to get a new explanation of Vaughn Jones's formulas by thinking of a knot as the trajectory followed by an elementary particle in a three-dimensional spacetime. There were a few tricks involved, but many of the ideas were standard ideas of physicists. Much of the novelty was just to apply the techniques of physicists to a problem that physicists were not accustomed to thinking about. 
This work became one of my best-known contributions, among both mathematicians and physicists. But it is also an excellent illustration of something I said in my acceptance speech the other night. No matter how clever we are, what we can accomplish depends on the achievements of our predecessors and our contemporaries and the input we get from our colleagues. My ability to do this work depended very much on clues I got from work of other scientists. In several cases, I knew of these clues because colleagues pointed out the right papers to me or because the work was being done right around the corner from me by colleagues at the Institute for Advanced Study in Princeton. It also helped at a certain point to remember some of what I had learned from Sidney Coleman back when I had been at Harvard, involving yet another insight of Albert Schwarz.



Edward Witten


En 1979, j'étais à une conférence sur les théorie de jauge à Cargèse où l'un des vedettes était ... Ed Witten. Il se trouve qu'on était dans le même hôtel et que j'ai eu donc souvent l'occasion de l'écouter ; et il y a une chose qui m'a beaucoup frappé. Tandis que les autres physiciens ... parlaient toujours en termes de phénomènes, éventuellement en termes de modèles concrets testés sur ordinateurs, Witten jonglait tout le temps avec les théories ellees-mêmes, leur manière d'intéragir, de se compléter ou tout simplement d'exploser. Il me semble qu'il y a un lien direct entre cette manière de penser et la M-théorie d'aujourd'hui. 
In 1979, I was at a lecture on gauge theory at Cargèse where one of the stars was ... Ed Witten. It turns out we were in the same hotel and so I often had the opportunity to listen to him ; and there is one thing that struck me a lot. While other physicists ... always spoke in terms of phenomena, possibly in terms of concrete models tested on computers, Witten juggled all the time with theories themselves, their way of interacting, of complementing each other or simply of exploding. It seems to me that there is a direct link between this way of thinking and today's M-theory. (blogger's translation)

Valentin Poénaru 

mercredi 6 décembre 2017

Physics, Mathematics, Calculation, Experiment


As a mathematician and outsider to the world of physics, I feel that gives me a perspective which to some extent is above the fray which is taking place in theoretical physics. Experience shows that successful physical theories follow a fairly well defined sequence of major steps. The first step is the elucidation of the essential physical ideas in purely physical terms which one needs to describe the theory. The second step is to describe mathematically a system which corresponds with the physical ideas resulting from the first step. The third step is to use the mathematical system resulting in step two to make calculations which make predictions of physically interesting quantities. The fourth step is to experimentally test the predictions resulting from the third step. One might introduce a fifth step which is to modify physical ideas in step one and go through the sequence of steps again in order to make improvements in the theory. In short, the steps which can be considered as a cycle, are  
(1) Physics
(2) Mathematics
(3) Calculation
(4) Experiment 
Some comments on these steps are now in order. It is important that the physical ideas of (1) not be overly influenced by mathematics. In step one, physics must be the main consideration. In (2) on the other hand, the level of rigour should not be too high as it can prevent progress and get in the way of progress to step three. Likewise, in (3), for calculations, similarly sometimes physical ideas can be used to aid in calculation where it would not be permitted in pure mathematics. For instance, if the Weierstrass level of rigour had been required of Newton, it could have prevented the development of Newtonian mechanics. The reason that the requirements of rigour can be relaxed here is due to the final step (4) which will be the final arbiter of success. Notice that this means that if (1), (3), and (4) are omitted, all that remains is sloppy mathematics... 
Maurice J. Dupré, Department of Mathematics New Orleans, LA 70118 
18 September 2013

vendredi 24 février 2017

A propos du (pseudo) paradoxe de Banach-Tarski

Il n'y a guère de paradoxe sans utilité
Leibniz in Lettre à l'Hospital, M. S. II p302


L'axiome du choix permet de casser une boule en un nombre fini de morceaux, puis de réajuster tous ces morceaux pour former exactement une boule de rayon différent! Ce théorème connu sous le nom de paradoxe de Banach-Tarski... semble contredire que ces deux boules ont des volumes différents! Mais ce n'est pas ainsi qu'il faut le comprendre; ce résultat est en effet motivé par son corollaire, à savoir qu'il est impossible de parler du volume d'une partie arbitraire de l'espace, dès lors qu'on impose à la fonction volume de satisfaire aux trois propriétés suivantes : deux parties de l'espace exactement superposables ont même volume, le volume de la réunion d'une famille finie de parties disjointes est la somme des parties de ces parties, et deux boules de rayons différents ont des volumes différents. Ce paradoxe n'en est donc pas un, car les morceaux de boules dont il énonce l'existence sont si irréguliers qu'on ne peut pas parler de leur volume : celui-ci n'est ni nul ni non nul, il n'est tout simplement pas défini. Et bien entendu cette fraction des boules n'a pas de sens physique ...
Gilles Godefroy
Ed. Odile Jacob



The axiom of choice allows you to break a ball into a finite number of pieces, then readjust all these pieces to form exactly one ball of different radius! This theorem known as Banach-Tarski's paradox... seems to contradict that these two balls have different volumes! But this is not the way to understand it; this result is indeed motivated by its corollary, namely that it is impossible to speak of the volume of an arbitrary part of space, if the volume function is required to satisfy the following three properties: two exactly superposable parts of space have the same volume, the volume of the reunion of a finite family of disjoined parts is the sum of the parts of these parts, and two balls of different radii have different volumes. This paradox is therefore not a paradox, because the pieces of balls whose existence it states are so irregular that it is impossible to talk about their volume: it is neither zero nor non-zero, it is simply not defined. And of course this fraction of the balls has no physical meaning...
Translated with www.DeepL.com/Translator