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Tim Maudlin, a philosopher of science at NYU, has a book out called: New Foundations for Physical Geometry: The Theory of Linear Structures.

The section on about the book says the following:

Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time... The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line.

The last line in the quote above caught my eye and a cursory reading of one of the chapters (7: Metrical Structures) on Google Books set off a few alarms. But, I am far from a mathematical physicist and my searches of reviews were fruitless, so my question is:

Is there a link to a review of this book or what is the considered opinion about it among mathematical physicist?

edit 1 I'll understand if this is closed. I suddenly realised that I am effectively indulging in what our friends in the sociology department love to call 'policing the boundaries' of our science.

edit 2 Looking at how things are, I'd vote to close this question too if I could (without intending any offence to those who have participated in the discussion). However, the discussion below made me peek superficially into the history of things. Evidently, the foundations of pont-set topology as we understand it now was established by the 1920s, born from considerations in analysis, it began with Frechet's 1904 thesis, where he based an abstraction of the euclidean space on the concept of limits. It is interesting to note that Ricci and Levi-Civita's Methods de calcul differential absolu et leurs applications was published in 1901 for work done in the previous decade.

user53046
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    You ask for reviews by mathematical physicists, but perhaps many of us would be more interested in reviews specifically by mathematicians. – Joel David Hamkins Jun 16 '14 at 21:14
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    Here are the slides of a talk by Maudlin on the topic. They seem to give a good overview of the basic idea. – Michael Greinecker Jun 16 '14 at 21:21
  • @JoelDavidHamkins Frankly, my reading of that one chapter did not inspire much confidence in the mathematical content of the book. I ask for the opinion of mathematical physicists simply because doing physics is the ultimate aim of the work and I know embarrasingly little about that subject. – user53046 Jun 16 '14 at 21:37
  • @user53046 Apparently, the physics will only come in the second volume. – Michael Greinecker Jun 16 '14 at 21:41
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    the prevalent point of view among physicists is that space nor time are fundamental, but "emergent" (like the concept of a "temperature" is not fundamental); after reading the introduction to this book, it seems very much out of touch with how physicists would think about these issues. – Carlo Beenakker Jun 16 '14 at 21:51
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    You can read a fair amount of it on Google Books; probably enough to draw your own conclusions: http://books.google.com/books?id=10XbAgAAQBAJ – Benjamin Dickman Jun 16 '14 at 23:16
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    @Benjamin Dickman: what conclusions did you draw? I see a philosopher who seems to think he has a better way of formalizing physics, writing about math in about the way you would expect a philosopher to write. My best guess is that that his contribution is technically valid but not terribly interesting from a mathematical standpoint --- more or less undergraduate level stuff --- but that is just a guess based on the small part I read. – Nik Weaver Jun 17 '14 at 00:39
  • @NikWeaver I'm not a mathematical physicist, so I don't think my conclusions would serve much for the purpose of the OP. That said, the very first section gives an argument that $(0,1]$ is not open (in the sense of the standard topology on $\mathbb{R}$) with a foot-note attributing the argument to the Wikipedia page on Topology from 2005. That about summarizes my own personal feelings on the book preview... – Benjamin Dickman Jun 17 '14 at 01:03
  • @Benjamin Dickman: Ouch! I didn't see that. Still, it seems consistent with my first impression. – Nik Weaver Jun 17 '14 at 01:47
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    @CarloBeenakker, I would say instead that the vast majority of physicists have given the nature of space-time very little thought, beyond their course in special relativity, where it's introduced as $\mathbb{R}^4$. Those who actually hold your "prevalent" view, in my experience, have little to base it on besides shared aesthetics. One would expect the reason behind a statement like "physicists believe that" to be based on empirical evidence, but that is certainly not the case here. – Igor Khavkine Jun 17 '14 at 07:46
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    From what I could see, my impression is that it is a wild concoction of some correct elementary mathematics (which I would even call rather ingenious in places, though I have no idea what particular problem the author is trying to solve with all this) and some philosophical gibberish about what something "really is" and what is "fundamental". Unfortunately, the chapters that I would really like to take a look at before passing any judgement are behind the paywall and the book is too recent to appear on ... (well, you know). – fedja Jun 18 '14 at 00:32
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    I'm going to buck the trend here by casting the first close vote. It's sort of an anti-stone soup situation, in which the question was phrased legitimately as a reference request, but it's clearly attracting discussion-y answers.

    But while we're discussing it, I'll just say that I have no idea how this is supposed to be useful to physicists when it doesn't appear to discuss the treatment of infinitesimals at all (and given that it's based on bog-standard classical logic and set theory, I doubt it will have anything interesting to say).

     – Evan Jenkins Jun 18 '14 at 05:33
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    @fedja, it's not (too recent), actually. – Igor Khavkine Jun 18 '14 at 08:13
  • @Igor Khavkine Great. I just didn't try really hard when searching. I'll take a look then, but later. :-) – fedja Jun 18 '14 at 11:47

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I have to confess that I hadn't heard about the book or its author until now, but as far as I can tell from what's available on Google, this particular volume is a book about mathematics, so I hope it gets reviewed as such. I did notice that Mathscinet has listed this book with a review pending.

(Although it would best to wait for a proper review by someone who has access to the whole book, I have to say that I find some claims in the parts that I have looked at, such as "Neither Decartes nor Newton would have recognized the existence of irrational or negative numbers…", or "There may have been loose talk about irrational or negative numbers, but no rigorous arithmetical foundation for them existed. The challenge was taken in 1872 by Richard Dedekind." in the introduction a little dubious.)

Donu Arapura
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  • Are you saying the history is dubious, i.e. you think there were serious attempts to construct the reals out of some more basic objects before? – Monroe Eskew Jun 18 '14 at 03:22
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    Newton certainly recognized the existence of negative numbers, and also irrational numbers, which he called "surd" numbers. See page 3 of his Universal Arithmetick. – John Stillwell Jun 18 '14 at 05:32
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    Monroe, yes I was saying that the history was dubious, although not exactly for the reason you said. The purpose of Dedekind's construction was not to produce negative numbers as seemed to be implied by the quoted sentence. – Donu Arapura Jun 18 '14 at 10:18
  • (I may have been guilty of increasing the noise level on this site. If people want to close the question, I'm fine with that.) – Donu Arapura Jun 18 '14 at 14:05
  • You're right, even Euclid countenanced "incommensurable magnitudes." – Monroe Eskew Jun 18 '14 at 19:25
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I'm not a mathematical physicist---I work in quantum computing theory, which maybe is sort of close if you squint? FWIW, I read the first few chapters of Maudlin's new book and liked them a lot. I remember taking topology as an undergrad and thinking, "why is everything based around 'open sets,' which can be chosen totally arbitrarily except that they have to be closed under unions and finite intersections?" I mean, yes, you can build up a theory on that basis and it works very well. But the notion of open set never impressed itself on me as intuitively central, the way most other basic mathematical notions did---especially given that one can easily define "open sets" (for example, in finite spaces) that have nothing whatsoever to do with the intuitive concept of "openness" that supposedly motivated the definition in the first place. So I wondered: would it be possible to build up topology on some completely different basis? This is the main question that Maudlin sets out to answer (affirmatively) in this book. And it's a big undertaking, and one that many people will probably regard as quixotic and unnecessary even if it succeeds---which might be why no one tried it before (or maybe they did; I can't say for certain about that). In the preface, Maudlin compares his situation to that of someone who realizes that the Empire State Building would've been better if it had been built a few feet to the left: even if that's true, it's far from obvious that it's worth the effort now to move the thing! But I, for one, am happy to see someone probe the foundations of topology in this way---especially someone who writes as clearly as Maudlin, so that I can actually understand where he's going and why.

Physics won't be covered until the second volume. I honestly don't know yet whether there are any real applications to physics, but if there are, one could regard them as just icing.

Scott Aaronson
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    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, http://qr.ae/GrSBA. The question of whether one can build topology on some other basis was considered e.g. by Grothendieck. – Qiaochu Yuan Jun 18 '14 at 01:20
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    This questioning of the primacy of open sets was discussed at length in this (now closed) MO question: http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets. – R Hahn Jun 18 '14 at 01:38
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    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. – Felipe Voloch Jun 18 '14 at 01:39
  • 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. – Scott Aaronson Jun 18 '14 at 02:18
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    @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? – Felipe Voloch Jun 18 '14 at 02:36
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    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. – fedja Jun 18 '14 at 12:51
  • @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? – Scott Aaronson Jun 18 '14 at 13:32
  • @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? – Scott Aaronson Jun 18 '14 at 13:39
  • 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. – Felipe Voloch Jun 18 '14 at 13:47
  • @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. – Michael Greinecker Jun 18 '14 at 13:57
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    @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! :) – Scott Aaronson Jun 18 '14 at 14:14
  • @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... – Daniel Litt Jun 18 '14 at 19:55
  • @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. – Nick Gill Jun 18 '14 at 19:55
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    (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. – Daniel Litt Jun 18 '14 at 19:55
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    @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: http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets – Qiaochu Yuan Jun 18 '14 at 20:22
  • @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. –  May 26 '17 at 22:25