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A Geometry of ApproximationPre-Topological and Topological Approximation Operators

A Geometry of Approximation: Pre-Topological and Topological Approximation Operators Chapter 3 Pre-Topological and Topological Approximation Operators 3.1 Information, Concepts and Formal Operators So far we have listed a number of well-defined mathematical instru- ments that act on either sides of abstraction we are dealing with, that is, points and properties. But what is the informational interpretation of the above machinery? What is a plausible philosophical interpreta- tion of all these mathematical results? Are they able to provide points (i.e. noumena) with a proper informational and conceptual structure based on their manifested properties (i.e. phenomena)? First of all, from Corollary 2.3.1 we can say that the operators A, C and IT S give the intensional (or formal) images of the extensional structures that are defined by int, cl and est on G and, symmetrically, through int, cl and est we obtain extensional (or concrete) images of the formal structures that A, C and IT S define on M . However, this imaging is not a mere mirroring, since intensional structures are not definable without the extensional structures, and the other way around. This is a plausible point of view under any non-mechanical approach to cognitive acts. Thus we maintain that P-systems equipped with the basic construc- tors and the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Geometry of ApproximationPre-Topological and Topological Approximation Operators

Part of the Trends in Logic Book Series (volume 27)
Editors: Pagliani, Piero; Chakraborty, Mihir
A Geometry of Approximation — Jan 1, 2008
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Publisher
Springer Netherlands
Copyright
© Springer Netherlands 2008
ISBN
978-1-4020-8621-2
Pages
73 –105
DOI
10.1007/978-1-4020-8622-9_3
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 3 Pre-Topological and Topological Approximation Operators 3.1 Information, Concepts and Formal Operators So far we have listed a number of well-defined mathematical instru- ments that act on either sides of abstraction we are dealing with, that is, points and properties. But what is the informational interpretation of the above machinery? What is a plausible philosophical interpreta- tion of all these mathematical results? Are they able to provide points (i.e. noumena) with a proper informational and conceptual structure based on their manifested properties (i.e. phenomena)? First of all, from Corollary 2.3.1 we can say that the operators A, C and IT S give the intensional (or formal) images of the extensional structures that are defined by int, cl and est on G and, symmetrically, through int, cl and est we obtain extensional (or concrete) images of the formal structures that A, C and IT S define on M . However, this imaging is not a mere mirroring, since intensional structures are not definable without the extensional structures, and the other way around. This is a plausible point of view under any non-mechanical approach to cognitive acts. Thus we maintain that P-systems equipped with the basic construc- tors and the

Published: Jan 1, 2008

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