How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

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Princeton University Press, 02.05.2010 - 424 Seiten

To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results.


Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure.


The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory?


Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

 

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LibraryThing Review

Nutzerbericht  - kukulaj - LibraryThing

Ach, I really wanted to give this five stars. Byers does a great job of showing how ambiguity and paradox are at the core of what mathematics is about. Of course it is also a paradox that mathematics ... Vollständige Rezension lesen

LibraryThing Review

Nutzerbericht  - SpaceyAcey - LibraryThing

Uses more words than necessary to explain his ideas. I kind of understand what he's trying to say but not really. I'm sure there is a more eloquent way to convey his ideas. Vollständige Rezension lesen

Inhalt

INTRODUCTION
1
CHAPTER 1
25
CHAPTER 2
66
The Contradictory in Mathematics
80
CHAPTER 3
110
CHAPTER 4
136
CHAPTER 5
193
CHAPTER 9
200
Ideas Logic and Paradox
253
CHAPTER 7
284
CHAPTER 8
327
Is Mathematics Algorithmic
368
Notes
389
Index
407
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Über den Autor (2010)

William Byers is professor of mathematics at Concordia University in Montreal. He has published widely in mathematics journals.

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