New Foundations for Physical Geometry: The Theory of Linear Structures

by Tim Maudlin

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. This is the initial volume in a two-volume set, the firstof which develops the mathematical structure and the second of which applies it to classical and Relativistic physics. The book begins with a brief historical review of the development of mathematics as it relates to geometry, and an overview of standard topology. The new theory, the Theory of Linear Structures, is presented and compared to standard topology. The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line. Axioms for the Theory of Linear Structures are laid down, and definitions of other geometrical notions developed inthose terms. Various novel geometrical properties, such as a space being intrinsically directed, are defined using these resources. Applications of the theory to discrete spaces (where the standard theory of open sets gets little purchase) are particularly noted. The mathematics is developed upthrough homotopy theory and compactness, along with ways to represent both affine (straight line) and metrical structure.

Genres: Philosophy, Science, Physics

374 pages, ebook

First published January 1, 2014

Reviews

[Ege]

https://ndpr.nd.edu/reviews/new-found...

[Joseph · Rating 4 out of 5]

I am a mathematician -- specifically a complex analyst. As mathematics, I thoroughly enjoyed this book. I don't think necessarily that topology needs to be supplanted as much as Maudlin seems to think, but I find this idea of defining things in terms of lines to be a fascinating thing. I also appreciate his discussion of how there are aspects of submetrical structure that open sets do not catch. The Theory of Linear Structures seems here to give a wider array of tools for analyzing different spaces.

On a philosophical level, I do find certain aspects unsatisfactory. For instance, I think that as much as possible, mathematics should not be grounded in set-language or set-theoretic foundations, since sets do not seem to have ontology in the same way that numbers or lines do. However, this is not really an argument against Maudlin's system since point-set topology is (if possible) even more grounded in sets than the theory of linear structures. I also think I would distance myself from his seeming skepticism of relative numeric structures.

However, the mathematics was interesting mathematics, and I would love to see if modern analysis could be developed in terms of linear structures and if it would change anything in it at all -- up to and including providing new insights that might help solve problems that until now seemed intractable.