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Ways of Advancing Knowledge. A Lesson from Knot Theory and Topology

  • Emiliano IppolitiEmail author
Chapter
Part of the Studies in Applied Philosophy, Epistemology and Rational Ethics book series (SAPERE, volume 25)

Abstract

The examination of the construction of several approaches put forward to solve problems in topology and knot theory will enable us to shed light on the rational ways of advancing knowledge. In particular I will consider two problems: the classification of knots and the classification of 3-manifolds. The first attempts to tell mathematical knots apart, searching for a complete invariant for them. In particular I will examine the approaches based respectively on colors, graphs, numbers, and braids, and the heuristic moves employed in them. The second attempts to tell 3-manifolds apart, again searching for a complete invariant for them. I will focus on a specific solution to it, namely the algebraic approach and the construction of the fundamental group, and the heuristic moves used in it. This examination will lead us to specify some key features of the ampliation of knowledge, such as the role of representation, theorem-proving and analogy, and will clear up some aspects of the very nature of mathematical objects.

Keywords

Topological Space Fundamental Group Algebraic Structure Mathematical Object Reidemeister Move 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I would like to thank Justin Roberts (Dept. Mathematics of University of California, San Diego) for his help with knot theory and topology.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Sapienza University of RomeRomeItaly

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