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Contemporary Mathematical Thinking

Part of the Synthese Library Book Series (volume 474)

DOI: 10.1007/978-3-031-27548-7

ISBN: 978-3-031-27547-0

Publisher: Springer International Publishing - Springer Journals

Feb 16, 2023

1 - 10 of 10 Chapters

LitStream

Chapter 1Introduction

Patras, Frédéric

Pages: 1 –5

[The second half of twentieth century mathematics was profoundly marked by the rise, domination, and decline of mathematical structuralism. It is impossible to understand the current state and the future of mathematics without understanding the corresponding endeavour and going back over its...
LitStream

Chapter 2Style in Mathematics

Patras, Frédéric

Pages: 7 –22

[The convictions and beliefs related to paradigms and the aesthetic notions associated with mathematical practice can be grouped under the term of mathematical style of thought. The existence of styles is not an accidental phenomenon, but rather the manifestation of general properties of...
LitStream

Chapter 3From Plato to Husserl

Patras, Frédéric

Pages: 23 –36

[The facade platonism displayed by the mathematical community is unsatisfactory. Husserl’s transcendental idealism seems to be the way today to reconcile mathematical thinking with its intuitive origins.]
LitStream

Chapter 4The Origins of Modern Mathematics

Patras, Frédéric

Pages: 37 –51

[The creation of modern mathematics standards for definitions and proofs, as well as for the organisation of the corpus, took a long time. The analysis of preliminary stages leads to highlight the gradual emergence of a new understanding of mathematics in the work of Gauss, Galois, Dedekind and...
LitStream

Chapter 5Axioms and Intuitions

Patras, Frédéric

Pages: 53 –66

[Originally thought of as a way to build foundations, the axiomatic method was progressively used more and more to organise and expand theories, their scope and boundaries. Despite their formal content and nature, the role of axioms in mathematics cannot be understood without taking into account...
LitStream

Chapter 6The Structuralist Current

Patras, Frédéric

Pages: 67 –83

[The advent and the reign, for a moment unconditional, of the idea of structure, together with the key role played by Bourbaki in the mathematical structuralism endeavour, are central to 20th century mathematics. However, the very idea of structure is less precise than it would seem at first...
LitStream

Chapter 7Structures and Categories

Patras, Frédéric

Pages: 85 –98

[Bourbaki would have finally stumbled on epistemological difficulties: his initial programmatic conceptions proved unsatisfactory in view of developments in contemporary mathematics, which would require new ideas and new ambitions. This chapter looks further into Bourbaki’s mathematical...
LitStream

Chapter 8The Homes of Thought

Patras, Frédéric

Pages: 99 –110

[The work of Alexander Grothendieck has opened new horizons to the theory of knowledge. Grothendieck expressed his views on science in texts that are more literary than directly philosophical—if one accepts that philosophical thought must be characterised by systematic organisation and anchoring...
LitStream

Chapter 9To the Meeting of Reality

Patras, Frédéric

Pages: 111 –121

[Contemporary mathematics is convinced of the decisive role played by the physical, life and social sciences. At the same time, the quarrels between pure and applied mathematics are brought to lose their importance.]
LitStream

Chapter 10Conclusion

Patras, Frédéric

Pages: 123 –125

[Since the establishment of the structuralist method, the most eminent mathematicians have often supported the idea of the independence of mathematical contents from their philosophical implications or presuppositions. We argue that modernity must go in another direction and renew the dialogue....

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