In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert’s axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere.

The title takes its name from David Hilbert’s seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert’s return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come.

The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.

Chapter 8 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

mathematical logic

foundations of mathematics

history of mathematics and logic

philosophy of mathematics and logic

general philosophy of science

• Introduction to Axiomatic Thinking

• Logic and Reasoning

• Propositional Logic

• Predicate Logic

• Set Theory Foundations

• Axiomatic Systems

• Proof Techniques

• Mathematical Structures

• Number Systems and Axioms

• Consistency, Completeness, and Independence

• Applications of Axiomatic Methods

• Historical Development of Axiomatic Systems

• Limitations of Axiomatic Systems

• Advanced Topics in Axiomatic Thinking

• Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal

  Fernando Ferreira

• Carl Friedrich von Weizsäcker-Zentrum, Universität Tübingen, Tübingen, Germany

  Reinhard Kahle

• Department of Mathematics, ETH Zurich, Zurich, Switzerland

  Giovanni Sommaruga

Book Title
Axiomatic Thinking I

Editors
Fernando Ferreira, Reinhard Kahle, Giovanni Sommaruga

DOI
https://doi.org/10.1007/978-3-030-77657-2

Publisher
Springer Cham

eBook Packages
Mathematics and Statistics, Mathematics and Statistics (R0)

Copyright Information
Springer Nature Switzerland AG 2022

Hardcover ISBN
978-3-030-77656-5
Published: 14 October 2022

Softcover ISBN
978-3-030-77659-6
Published: 15 October 2023

eBook ISBN
978-3-030-77657-2
Published: 13 October 2022

Edition Number
1

Number of Pages
XIV, 205

Number of Illustrations
5 b/w illustrations, 4 illustrations in colour

Topics
Mathematical Logic and Foundations, Philosophy of Mathematics, History of Mathematical Sciences