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5. Reflections on the Axiomatic Approach to Continuity

  • 2022
  • OriginalPaper
  • Chapter
Published in:
Author
John L. Bell
Published in
Axiomatic Thinking II
Publisher
Springer International Publishing

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Abstract

The chapter delves into Hilbert's axiomatic treatment of continuity, highlighting its significance in both physics and mathematics. It discusses the Archimedean axiom and its role in physics, leading to the formulation of the Physical Continuity Axiom (PCA). The text also explores Brouwer's Continuity Principle (BCP) and its implications for intuitionistic mathematics, contrasting it with classical mathematics. Additionally, it introduces Synthetic Differential Geometry (SDG) as an axiomatic framework for continuum mechanics, emphasizing its use of infinitesimals and the Principle of Infinitesimal Linearity. The chapter concludes with historical observations on the development of infinitesimal calculus and its philosophical implications.
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previous chapter Well-Ordering Principles in Proof Theory and Reverse Mathematics
next chapter Abstract Generality, Simplicity, Forgetting, and Discovery
Title
Reflections on the Axiomatic Approach to Continuity
Author
John L. Bell
Copyright Year
2022
Book
Axiomatic Thinking II

Print ISBN: 978-3-030-77798-2

Electronic ISBN: 978-3-030-77799-9

Copyright Year: 2022

DOI
https://doi.org/10.1007/978-3-030-77799-9_5
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