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Continuous Mathematics

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Boundaries of a Complex World

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

Besides its essential role in the development of calculus, analysis, geometry, and algebraic geometry, topology has major direct applications and contributions in the study of image reconstruction and recognition, modeling, graphs, and networks, fluid mechanics, protein folding, robotics, and fundamental physics. Topology is the part of mathematics that investigates space from a qualitative point of view, roughly speaking without using the concept of distance. It may seem counter-intuitive to do geometry in a metric-free world, but the habitual dualities small/large, or near/far, etc., are more complicated than they appear because they need a comparison relation with a unit of measurement, and the order relation in the set of positive real numbers. Topological intuition does not need these concepts. It is to geometry what logic is to algebra. In his book on topology, Pavel Alexandrov says [149]: “The specific attraction and in large part the significance of topology lies in the fact that its most important questions and theorems have an immediate intuitive content and thus teach us in a direct way about space which appears as the place in which continuous processes occur.” An example is given in Fig. 4.1, where a mug is smoothly deformed into a zero volume surface, the Klein bottle.

A boundary is that which is an extremity of anything

Euclid

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Authors and Affiliations

  1. Embry-Riddle Aeronautical University, Department of Mathematics, Daytona Beach, FL, USA

    Andrei Ludu

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  1. Andrei Ludu
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Ludu, A. (2016). Continuous Mathematics. In: Boundaries of a Complex World. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49078-5_4

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  • DOI: https://doi.org/10.1007/978-3-662-49078-5_4

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  • Print ISBN: 978-3-662-49076-1

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