Space Time And The Categories
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Categories, Bundles and Spacetime Topology
Author: C.T. Dodson
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-03-09
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown'The point of a Pin'. van Gulik's TheChinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging SUbdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
Space, Time, and Spacetime
Author: Lawrence Sklar
language: en
Publisher: Univ of California Press
Release Date: 2023-07-28
In this book, Lawrence Sklar demonstrates the interdependence of science and philosophy by examining a number of crucial problems on the nature of space and time—problems that require for their resolution the resources of philosophy and of physics. The overall issues explored are our knowledge of the geometry of the world, the existence of spacetime as an entity over and above the material objects of the world, the relation between temporal order and causal order, and the problem of the direction of time. Without neglecting the most subtle philosophical points or the most advanced contributions of contemporary physics, the author has taken pains to make his explorations intelligible to the reader with no advanced training in physics, mathematics, or philosophy. The arguments are set forth step-by-step, beginning from first principles; and the philosophical discussions are supplemented in detail by nontechnical expositions of crucial features of physical theories. In this book, Lawrence Sklar demonstrates the interdependence of science and philosophy by examining a number of crucial problems on the nature of space and time—problems that require for their resolution the resources of philosophy and of physics. “/DIV
Space, Geometry, and Kant's Transcendental Deduction of the Categories
Thomas C. Vinci aims to reveal and assess the structure of Kant's argument in the Critique of Pure Reason called the "Transcendental Deduction of the Categories." At the end of the first part of the Deduction in the B-edition Kant states that his purpose is achieved: to show that all intuitions in general are subject to the categories. On the standard reading, this means that all of our mental representations, including those originating in sense-experience, are structured by conceptualization. But this reading encounters an exegetical problem: Kant states in the second part of the Deduction that a major part of what remains to be shown is that empirical intuitions are subject to the categories. How can this be if it has already been shown that intuitions in general are subject to the categories? Vinci calls this the Triviality Problem, and he argues that solving it requires denying the standard reading. In its place he proposes that intuitions in general and empirical intuitions constitute disjoint classes and that, while all intuitions for Kant are unified, there are two kinds of unification: logical unification vs. aesthetic unification. Only the former is due to the categories. A second major theme of the book is that Kant's Idealism comes in two versions-for laws of nature and for objects of empirical intuition-and that demonstrating these versions is the ultimate goal of the Deduction of the Categories and the similarly structured Deduction of the Concepts of Space, respectively. Vinci shows that the Deductions have the argument structure of an inference to the best explanation for correlated domains of explananda, each arrived at by independent applications of Kantian epistemic and geometrical methods.