Added scientific polymath Giuseppe Longo’s 32 page paper on (and - especially w/ regards to biology - beyond) Zalamea’s SPCM - Synthetic Philosophy of Mathematics and Natural Sciences: Conceptual Analyses from a Grothendieckian Perspective. (Reflections on “Synthetic Philosophy of Contemporary Mathematics” by Fernando Zalamea) - to the Fernando Zalamea page, and created the Longo page. Longo used to be focused on theoretical computer science and category theory, but his research now revolves around biology.

]]>It is against etiquette, but there is no rule against it. (At the level of ICANN rules, there is no difference between .com, .org, and .net whatsoever.) So it is just part of the scam.

]]>If you ever need to access a file on academia.edu and a download window pops up, just click the little x in the upper right and you’re good to go.

Then you are back to step one. No downloading happens at my firefox after I click x, so you have to press on download again, then you have a new pop up again, click x back to step one and so on. I done it about 10 times in different variants and gave up at the end.

Thanks for the new, non-academia.borg link where I succeeded with the download! By the way, why is (obviously intended to become) a commercial site academia having org appendix ? It is against the etiquette at least…

]]>Yeah, academia.edu is really horrible.

]]>I put up a new non - academia.edu link yestday, but for future reference, if you ever need to access a file on academia.edu and a download window pops up, just click the little x in the upper right and you’re good to go.

]]>The file requires login to download, not to view

I did not succeed that. Any time I try to download it, it pops the subwindow to registrate.

]]>The file requires login to download, not to view, but I replaced the link with another incase anyone wants to download it, and put it in proper link form: A new analytic/synthetic/horotic paradigm. From mathematical gesture to synthetic/horotic reasoning.

creativity at the frontiers

(A phenomenology of ) Creativity is just one among 3 of the main things in Zalamea’s book (the other 2 being a transitory ontology and comparative epistemology), but you’re right to single out creativity b/c zalamea’s current work is centered around mathematical creativity from Riemann and Galois to Grothendieck and Gromov. Zalamea is in an excellent position to deal with mathematical creativity because - while his phd and masters degrees are in mathematics and he is a professor of mathematics - he is also an accomplished art/culture critic/theorist (as was his mother, Marta Traba).

the influence of late Albert Lautman

Lautman is absolutely crucial, but so are C.S. Peirce and Gilles Chatelet. Peirce’s Cenopythagorean Categories* and Pragmaticist Maxim** run throughout Zalmea’s entire oeuve (both mathematical and literary-artistic) and Zalamea uses Peirce and Category Theory to arm himself / his readers with a minimal mathematico-philosophico framework before diving into 13 case studies (Grothendieck, 4 Eidal mathematicians (mathematicians who ascend to the general realm of the idea), 4 Quiddital mathematicians (mathematicians who descend to physics), 4 Archeal mathematicians (mathematicians who identify archetypes behind the transits of mathematics - between concepts and data, languages and structures, mathematics and physics, imagination and reason.)) and finally fleshing out the framework in the final section of his book. Chatelet was a differential topologist / philosopher who wrote a book called “les enjeux du mobile” (which is out of print and nearly $200 … so I think its reasonable to link to a pdf http://www.scribd.com/doc/188664562/Chatelet-gilles-figuring-space-philosophy-mathematics-and-physics ) which is about math’s natural osmoses with physics (+ to a certain extent philosophy), processes on the border of the actual and virtual, the role of metaphor*** in mathematics, the importance of diagrams and the visual, and the role of gesture. (edit: chatelet is also, like jean-yves girard, highly entertaining to read.)

*Aristotle had 12 categories, Kant had 10, I don’t recall how many Hegel had, Peirce had 3 (these are so general as to be no longer concepts so much as general tints or tones which concepts take):

firstness: possibility; ideas, chance ; vagueness, “some” ; “the mode of being of that which is such as it is, positively and without reference to anything else” ; the best (& most beautiful) example of firstness has to be James Turrell’s installation art, where the viewer is immersed in pure fields of color.

secondness: actuality; brute facts ; haecceity, discreteness, “this” ; “the mode of being of that which is such as it is, with respect to a second but regardless of any third.”

thirdness: necessity; habits, laws ; generality, “all” ; “the mode of being of that which is such as it is, in bringing a second and third into relation to each other;

**”Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.”, “Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood”, and/or “The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol.”. The maxim reminds me somewhat of the Yoneda Lemma, and Peirce called his maxim “pragmatICIST” rather than “pragmatIC” in order to distinguish his interpration of it from behaviourist, utilitarian, and psychologistic interpretations. Zalamea takes the maxim to mean that 1. “to know a given sign (the realm of the actual) we must traverse the multiple contexts of interpretation capable of interpreting that sign (the realm of the possible) and, in each context, study the practical imperative consequences associated with each one of those interpretations (the realm of the necessary).” 2. “knowledge, seen as a logico-semiotico process, is preeminently contextual (versus absolute), relational (versus substantial), modal (versus determined), and synthetic (versus analytic)”. In reality though, Zalamea sees knowledge as primarily in between those 2 poles - as illustrated for example by the notion of horotic reasoning or “universal relatives” in what zalamea identifies as a peircian-einsteinian-grothendieckian turn in philosophy, math, and physics.

***this famous andre weil quote illustrates the role of metaphor and the virtual: "nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the research. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another". The virtual is, quoting wikipedia “a kind of surface effect produced by actual causal interactions at the material level” and “a kind of potentiality that becomes fulfilled in the actual”.

]]>I am just repeating your link in link form: Maddalena review Though we should find another link. This is academia.edu, requires login for downloading and if you become listed there it starts using yoru name and your posts in their adds and also spams your mailbox. We should avoid such providers of spam.

how useful are the type of observations in that book for people doing actual mathematical research

One should mention that Fernando Zalamea’s book is based, in addition to erudition and the thinking of the author, on the influence of late Albert Lautman whose work was much about understanding the phenomenon of advanced mathematics and the creativity at the frontiers as opposed to dwelling on particular logical foundation in analytic school (Frege, Russell…).

]]>Since this thread began with a post about Fernando Zalamea’s Synthetic Philosophy of Contemporary Mathematics, I thought I’d ask, how useful are the type of observations in that book for people doing actual mathematical research? Are they too general to be of any use?. I read the book when I was 19 and it was a great introduction to the world of contemporary mathematics (it still is a great introduction) …. but I’m curious to ask what people with more experience think of the general observations contained in it. Also, following David (Corfield) on how Lautman’s philososphical ideas can be formalized … can any of Zalamea’s observations be formalized?

(Section 5 of this paper is a good overview of Zalamea’s book. A new analytic/synthetic/horotic paradigm. From mathematical gesture to synthetic/horotic reasoning. . Of Course, it’s impossible to pack such a dense book into one section of an article.)

]]>What kind of information is lost and in which examples is it essential to keep that information?

As this is about the reconstruction of commutative schemes, then you are asking me about the commutative geometry. I do not have much knowledge on commutative algebraic geometry, and what I got from others is that this in the most classical case of finite type/varieties some finite information which is not very essential, even in the sense of classical Italian school. In noncommutative situations one looses much more. For example, the noncommutative projective space in the sense of Kontsevich and Rosenberg, which is really huge and locally looks almost like a free noncommutative algebra, has the same derived category of coherent sheaves as the commutative projective space. The first has as subvarieties various guys like quantum projective spaces, while the latter does not have them. This is really cruel.

By the way, Urs, I have in 28 given a link where should the discussion on noncommutativity continue. My feeling is that nc Gelfand-Neimark also looses some info even in operator algebraic framework which is by its nature more commutative anyway.

Which would mean that the topos-theoretic description remembers precisely the right aspect of the non-commutativity.

Again you are in operator algebras, what is really not that noncommutative. I agree that for operator algebras the derived picture looses very little (in part because of the existence of involution and of boundedness), unlike in algebraic situation. I am really interested in true noncommutative algebraic spaces, and I do not find any satisfactory treatment from the derived camp yet. I mean even for the *main* and simplest examples of my interest.

It’s maybe interesting for our conversation that in that context, of the internal GN theorem, too, passing from a non-commutative algebra to its poset of commutative subalgebras loses some of the information: under some conditions only the Jordan algebra of the noncommutative algebra is remembered. (As discussed here)

It would be interesting to better understand what that means. It seems that for the non-commutativity of an algebra of observables in quantum theory, in fact the Jordan algebra structure is all that really matters. Maybe. Which would mean that the topos-theoretic description remembers precisely the right aspect of the non-commutativity.

Just an observation. I need to better understand all this.

]]>I need to be looking a bit into the noncommutative Gelfand-Naimark theorem in the topos of copresheaves over the poset of commutative subalgebras of a noncommutative $C *$-algebra. I still need to make up my mind on how to best think of this sizuation and the duality result and the grand hopes that some people have about this and how it all fits into all the rest. Not sure yet.

]]>Hi Zoran,

By Gabriel-Rosenberg a commutative scheme (with some very mild restriction) can be reconstructed from the abelian category of qcoh sheaves. This is not true from the derived category except in extremely special cases

Right, that’s what you said. what I would like to know is concrete examples where one does care about the information that is not seen by the derived or $\infty$-category of quasicohrent sheaves. What kind of information is lost and in which examples is it essential to keep that information?

]]>The continuation of noncommutative bundle discussion could continue better here, and this track left for philosophy books…

]]>Could you point me to some result that would illustrate the phenomenon that you mentioned, that there is a loss of important information when passing to the oo-categories of quasicoherent sheaves on th non commutatiev base space on which one is interested in quantum vector bundles?

By Gabriel-Rosenberg a commutative scheme (with some very mild restriction) can be reconstructed from the abelian category of qcoh sheaves. This is not true from the derived category except in extremely special cases treated e.g. by Bondal-Orlov theorem for projective varieties when the canonical or anticanonical line bundle is ample. There is no essential difference for the reconstruction if one takes stable $(\infinity,1)$-version of derived category – Orlov and Lunts have shown a couple of years ago for smooth quasiprojective varieties that such an enhancement is essentially unique, hence there is no difference in information weather I take old-fashioned derived category or stable quasicategory or A-infty. The derived reconstruction is somewhat better if one keeps the tensor structure, what one does not have in noncommutative case in general.

]]>but what is the real distinguishing feature that means they are not topologies

The category of set valued sheaves is not left exact reflective subcategory of the category of presheaves. Left exact must be dropped in most examples. On the other hand, for abelian sheaves one has the problem with pullbacks of covers. One relevant paper is Rosenberg’s Noncommutative schemes paper (cf. entry noncommutative scheme). Look also for works of van Oystaeyen and his school including his book on nc geometry for associative algebras, and my article on equivariant aspects and my earlier survey on nc localization in nc geometry. Quantum groupoids aka Hopf algebroids are just more general group/like objects which can take place of a structure group. I think Maszczyk has many new examples in that direction. However nothing essentially new in the sense that already for usual Hopf algebras one has all the difficulties with local trivializations.

]]>But I guess I am missing important references (only had a few minutes).

More strongly: the references you put are more of a distraction than use for the central questions of noncommutative topology we discuss above (I apperaciate those authors for other aspects though). Zhang is working in C-star algebraic setup, not algebraic, so not good for the study of quasicoherence. One of his papers has errors in the treatment of involutions as far as I recall. Coquereaux etc. is not very relevant – it just defines, in the affine case, examples for the global sections of what would be an associated bundle, not the associated bundle per se. This is avoiding the main problem, as I did in some early references.

I have written ages ago more relevanty entry noncommutative principal bundle.

In these articles that I found it seems that only quantum vector bundles over the spec of a single nc algebra are considered.

I was telling you above and many times before about my own work which in 1997 went beyond that, e.g. the work on quantum group coherent states. Some useful background is also at noncommutative scheme and gluing categories from localizations (zoranskoda).

]]>Classify “vector bundles” for example with structure Hopf algebra or if you like Hopf-algebrtaic torsors.

Okay, thanks. I looked for some references and have collected what i saw now at quantum vector bundle. But I guess I am missing important references (only had a few minutes).

In these articles that I found it seems that only quantum vector bundles over the spec of a single nc algebra are considered.

Could you point me to some result that would illustrate the phenomenon that you mentioned, that there is a loss of important information when passing to the oo-categories of quasicoherent sheaves on th non commutatiev base space on which one is interested in quantum vector bundles?

]]>@Zoran,

I have vaguely read what you have written about these not-quite-Grothendieck-(co)topologies, but what is the real distinguishing feature that means they are *not* topologies. I know you said that a pullback of a cover (or cocover, depending on your perspective) is not a cover, but I’m just checking that what you have is not a coverage.

Are there links to quantum categories of Street et al? Are there some really basic examples of these not-topologies (like for the category of non-commutative rings)? Do the properties of the (co)covers make sense in any monoidal category?

I’m sorry for asking such basic questions, but I’m intrigued. One thing which I might look into at some point is redoing my work on anafunctors but with quantum categories instead of plain internal categories (which are quantum categories for the case of a cartesian monoidal category). Maybe it won’t work at all, maybe it will. But I’m sure there’s something interesting to get out of it. In particular, what sort of localisations of (2-?)categories of Hopf algebroids will this give rise to?

]]>To add to 20 (discussion with Urs), there are spectra of noncommutative spaces which are real topological spaces with structure stacks; however such constructions, if faithful (determining the space) are not functorial for all types of morphisms (usually they are functorial with respect to some subclass of morphism). This points to the fact that the topological/topos/locale plus sheaf/stack picture needs some refining to absorb noncommutative spaces in nonderived version. Of course, there is a basic belief in the noncommutative community that once, in 23rd century, we will have some sort of genuine spaces behind noncommutative geometry of any kind, with all the flexibility like in the commutative case…but it needs a nontrivial work in nonderived case. Which is interesting, I mean the generalizations of cohomology and sheaf theory to things which are not quite Grothendieck topologies is very interesting problematics.

]]>- Ralf Krömer,
*Tool and object: A history and philosophy of category theory*, Birkhäuser 2007

added to philosophy books. Somebody has some comments on its value ?

]]>Topology is used in algebraic examples to pass between local and global. In defining and calculating with quantum group coherent states I used in loc. cit. together local formulas and global theory (characters, Haar measure etc.). Locality is in the sense of noncommutative localization. Now I found the method very useful. In fact this was the only method I could find after few yars of thinking which lead to a unique (up to overall normalization) invariant coherent state measure which disagrees with the eralier papers which used ad hoc deformations without theory (hence loose invariance both of the measure and of the integrand). I want to continue using such covers in systematic way to study quasicoherent sheaves in noncommutative setup. Loosing this for some ideology of infinity categories is a sacrifice I can not afford, as a practioner from quantum group theory.

Of course, we have here two different kinds of theory, I mean the big site and the small site; I was talking about the small “site” here, but similar problems are in the big site picture, which I understand much less.

]]>Classify “vector bundles” for example with structure Hopf algebra or if you like Hopf-algebrtaic torsors. E.g. the bundles with quantum groups as structure groups. There are geometric methods in representation theory, involving bundles and sheaves and their nonabelian cohomologies over noncommutative spaces. Try for example to represent the derived functors for induction functor for representations of quantum groups as Čech cohomology of some sheaves on quantum flag varieties. There is a problem with refinements here, very noncommutative phenomenon (I mean limiting constructions in Čech cohomology).

]]>