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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.

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closed as primarily opinion-based by Evan Jenkins, Lucia, Stefan Kohl, Felipe Voloch, S. Carnahan Jun 19 at 2:29

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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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 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 at 21:21
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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 at 21:51
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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 at 7:46
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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 at 0:32

2 Answers 2

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.)

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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 at 3:22
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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 at 5:32
    
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 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 at 14:05
    
You're right, even Euclid countenanced "incommensurable magnitudes." –  Monroe Eskew Jun 18 at 19:25

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.

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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, 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 at 1:20
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This questioning of the primacy of open sets was discussed at length in this (now closed) MO question: mathoverflow.net/questions/19152/…. –  R Hahn Jun 18 at 1:38
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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 at 1:39
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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 at 2:36
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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 at 12:51

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