Review of Tim Maudlin's New Foundations for Physical Geometry 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. 
 A: 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.)
A: 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.
