Timeline for Review of Tim Maudlin's New Foundations for Physical Geometry
Current License: CC BY-SA 3.0
18 events
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| May 26, 2017 at 22:25 | comment | added | user100272 | @ScottAaronson Categorical approaches the foundations of quantum mechanics would be an example, as in the work of Abramsky, et al. For the semantics of some programming languages, categorical models are the only known models. Even the étale cohomology, at some point you have to use the category of sheaves as a first class mathematical object and not as a metatheoretical simplification of language. | |
| Jun 18, 2014 at 20:22 | comment | added | Qiaochu Yuan | @Nick: I certainly can't claim the credit - I don't know who the observation is originally due to but I first learned about it from reading Dan Piponi's answer to this MO question: mathoverflow.net/questions/19152/… | |
| Jun 18, 2014 at 19:55 | comment | added | Daniel Litt | (cont.) fiercely original in their own right. While you're right in thinking that the basic definitions of, say, etale cohomology are most conveniently stated in the language of categories, the actual development of the theory uses serious geometry. Grothendieck's "rising tide" quote has, I think, led to the false impression that his work did not involve some serious technical chops. Grothendieck was a master of organization, but I would argue that this was a very small part of the reason for his many successes. | |
| Jun 18, 2014 at 19:55 | comment | added | Nick Gill | @QiaochuYuan, I clicked on the link in your comment and really liked what I read - that's an approach to topology that I've never come across before and I found it most enlightening... I'd give you more than a +1 if I could. | |
| Jun 18, 2014 at 19:55 | comment | added | Daniel Litt | @ScottAaronson: Re: "much of the most admired math in the last century...was all about reorganizing concepts." I think that this is not right. It is true that Grothendieck and company were excellent writers with a beautiful, original vision, and reading EGA one is given the impression that they have simply organized the subject in a natural way, so that everything seem easy. But their less polished stuff (SGA, FGA, etc.) reveals that this is more a byproduct of their writing than of the work itself. Theorems like formal GAGA, Grothendieck duality, etc. are not reorganizations, but are... | |
| Jun 18, 2014 at 14:14 | comment | added | Scott Aaronson | @Felipe: Would such a high-school student have any opinion, positive or negative, about someone who tried to reorganize algebra on a different basis? Anyway, if you read my blog, maybe you'll appreciate the irony. People constantly criticize me for doing nothing but criticizing others, from D-Wave to Lubos Motl to Joy Christian. So then, what happens when I say I liked something, or found it possibly worth looking into further? This thread provides the answer! :) | |
| Jun 18, 2014 at 13:57 | comment | added | Michael Greinecker | @FelipeVoloch My impression from looking at the book is that Maudlin does not want to get rid of general topology, he thinks that it is not be a good base for physical geometry. This says nothing about uses outside geometry. | |
| Jun 18, 2014 at 13:47 | comment | added | Felipe Voloch | I shouldn't really comment on Maudlin as I only had a cursory look at the google books link. My comment was more directed at you, Scott. It read to me as akin to a high school student who complains to his math teacher that he's never going to use algebra. As a long time reader of your blog, I was disappointed. On your comment to Fedja, you can't do some of Grothendieck's stuff (e.g. Étale cohomology) without categories. | |
| Jun 18, 2014 at 13:39 | comment | added | Scott Aaronson | @fedja: That seems like a slightly ironic argument for a mathematician to make. What, for example, can be done with category theory that literally can't be done without it? Am I mistaken that much of the most admired math in the last century (e.g., much of what Grothendieck did) was all about reorganizing concepts to make structure more apparent, rather than just creating whatever tools are needed to solve specific problems? | |
| Jun 18, 2014 at 13:32 | comment | added | Scott Aaronson | @Felipe: For one thing, Maudlin says he wants definitions that work even for discrete collections of points (since, for all we know, physical spacetime is in some sense discrete at the Planck scale). But more broadly, is it that strange to want something more abstract and general than submanifolds of Euclidean space, but less abstract and general than semidecidable properties? | |
| Jun 18, 2014 at 12:51 | comment | added | fedja | The "directed linear structures" look "having something to do with the intuitive concept of a line" only as long as you do not realize the full power of their definition. Maudlin's idea looks neither wrong, nor stupid, nor unoriginal to me (though I am not in a good position to judge the last aspect). However, what one needs to convince others to replace one set of tools with another is not that the new tools look nicer or are more convenient (this is highly subjective and you can argue over it forever) but that one can do something with the new tools that cannot be done with the old ones. | |
| Jun 18, 2014 at 2:36 | comment | added | Felipe Voloch | @ScottAaronson I haven't really read Maudlin but my gut feeling is that he will recover the notion of manifold and to develop the notion of manifold you don't need general topology. If you really want to, you can just talk about submanifolds of Euclidean space and all you need is Calculus. Why bother with anything else if that's what you want? | |
| Jun 18, 2014 at 2:20 | history | edited | Scott Aaronson | CC BY-SA 3.0 |
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| Jun 18, 2014 at 2:18 | comment | added | Scott Aaronson | OK, but suppose your interest wasn't in general semidecidable properties, but specifically in the topological properties of "geometric" spaces (roughly, those intended to model physical space). Could you explain why Maudlin's proposal, of taking (topological) lines rather than open sets as the basic primitive for that purpose, is either wrong or stupid or unoriginal? If so, I'd gladly change my mind about this. | |
| Jun 18, 2014 at 1:39 | comment | added | Felipe Voloch | I think you got things backwards. The abstract notion of topological spaces and open sets are not just to define "space" (for physics or geometry) but as a useful abstraction that allows you to carry over arguments to situations which are not necessarily geometric, e.g. the profinite topology on Galois groups or function spaces ($L^p$, etc). Also, in addition to to Quiaochu's comment, there are other alternative approaches to defining limits and continuity (the basic task of topology), e.g. filters. | |
| Jun 18, 2014 at 1:38 | comment | added | R Hahn | This questioning of the primacy of open sets was discussed at length in this (now closed) MO question: mathoverflow.net/questions/19152/…. | |
| Jun 18, 2014 at 1:20 | comment | added | Qiaochu Yuan | Open sets don't look intuitive as long as you think they're supposed to be describing topology, but in fact they describe something much more general, namely the notion of a semidecidable property: see, for example, qr.ae/GrSBA. The question of whether one can build topology on some other basis was considered e.g. by Grothendieck. | |
| Jun 18, 2014 at 1:00 | history | answered | Scott Aaronson | CC BY-SA 3.0 |