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A Geometry of ApproximationFrames (Part I)

A Geometry of Approximation: Frames (Part I) Chapter 4 4.1 Frame – Approximation Any complete account for an history of the concept of an approximation is trivially non-feasible. Indeed, approximation is a fundamental concept in a number of fields and it is not likely to try and list even a small amount of them. However, because of its historical role in Mathematics we recall Archi- medes’ exhaustion method (ca. 287–212 BC). This method is based on the observation that we can compute the area of, for instance, a circle, by means of a series of more and more refined approximations from below and from above. Figure 4.1: Archimedes’ exhaustion method 107 108 4 Frames (Part I) More precisely, Archimedes’ main idea was to try and approximate the circle using inscribed (lower approximating) and circumscribed (upper approximating) regular polygons, as shown in Figure 4.1. However, this method may be traced back to Eudoxus of Cnidos (ca. 400–347 BC) and to the well-known paradoxes by Zeno of Elea (ca. 495–435 BC) accounted by Plato in “Parmenides”, which can be considered a first kind of (never ending) approximation reasoning (see Kline [1972]). 4.2 Frame – Classification An approximation process is a means for classifying objects. Indeed “concrete” items are http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Geometry of ApproximationFrames (Part I)

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
107 –166
DOI
10.1007/978-1-4020-8622-9_4
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 4 4.1 Frame – Approximation Any complete account for an history of the concept of an approximation is trivially non-feasible. Indeed, approximation is a fundamental concept in a number of fields and it is not likely to try and list even a small amount of them. However, because of its historical role in Mathematics we recall Archi- medes’ exhaustion method (ca. 287–212 BC). This method is based on the observation that we can compute the area of, for instance, a circle, by means of a series of more and more refined approximations from below and from above. Figure 4.1: Archimedes’ exhaustion method 107 108 4 Frames (Part I) More precisely, Archimedes’ main idea was to try and approximate the circle using inscribed (lower approximating) and circumscribed (upper approximating) regular polygons, as shown in Figure 4.1. However, this method may be traced back to Eudoxus of Cnidos (ca. 400–347 BC) and to the well-known paradoxes by Zeno of Elea (ca. 495–435 BC) accounted by Plato in “Parmenides”, which can be considered a first kind of (never ending) approximation reasoning (see Kline [1972]). 4.2 Frame – Classification An approximation process is a means for classifying objects. Indeed “concrete” items are

Published: Jan 1, 2008

Keywords: Topological Space; Distributive Lattice; Formal Concept; Concept Lattice; Intuitionistic Logic

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