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Experiencing Mathematics: What Do We Do, When We Do Mathematics? [Pehme köide]

  • Formaat: Paperback / softback, 291 pages, kaal: 542 g
  • Sari: Monograph Books
  • Ilmumisaeg: 30-Jan-2014
  • Kirjastus: American Mathematical Society
  • ISBN-10: 082189420X
  • ISBN-13: 9780821894200
Teised raamatud teemal:
  • Formaat: Paperback / softback, 291 pages, kaal: 542 g
  • Sari: Monograph Books
  • Ilmumisaeg: 30-Jan-2014
  • Kirjastus: American Mathematical Society
  • ISBN-10: 082189420X
  • ISBN-13: 9780821894200
Teised raamatud teemal:
Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really do when they are doing mathematics, Reuben Hersh offers an escape from this trap. This book of selected articles and essays provides an honest, coherent, and clearly understandable account of mathematicians' proof as it really is, and of the existence and reality of mathematical entities. It follows in the footsteps of Poincare, Hadamard, and Polya. The pragmatism of John Dewey is a better fit for mathematical practice than the dominant ""analytic philosophy''. Dialogue, satire, and fantasy enliven the philosophical and methodological analysis.

Reuben Hersh has written extensively on mathematics, often from the point of view of a philosopher of science. His book with Philip Davis, The Mathematical Experience, won the National Book Award in science.

Arvustused

I view Hersh as a hero: not a perfect idol, but more a real-life hero who through stubborn hard work and prolific writing has made a difference in the types of conversations we are having now about ourselves and the ways we relate to the world. ... [ A]s I dove into the book, I found myself fascinated by its riches, and surprised that [ it] worked so well. ... I think most readers would agree that the sequencing of the articles actually worked ... many will enjoy getting a complete overview of Hersh's argument. ... I remain enthralled by Hersh's ideas and impressed by his persistent defense of his controversial but significant perspective. I believe the American Mathematical Society has done a service to the mathematical community by putting together this collection. ... Reuben Hersh's collection is full of provocative ideas, offering perspectives on our profession that may help us understand better ourselves and our craft and even to teach our students better. This volume will remain on my easy-to-reach shelf for a long time to come." - MAA Reviews

"... I found the author's arguments powerful and compelling, and conveyed with great clarity and concision. ... It is refreshing to occasionally step back and talk about mathematics rather than doing it, and this book provides solid rhetorical ammunition." - LMS Newsletter

Preface xi
Credits xiii
Acknowledgments xvii
Overture 1(4)
The Ideal Mathematician 5(8)
Philip J. Davis
Manifesto 13(4)
Self-introduction 17(4)
Chronology 21(14)
Mathematics Has a Front and a Back 35(6)
Part 1 "Mostly for the right hand"
41(132)
Introduction
43(4)
True Facts About Imaginary Objects
47(4)
Mathematical Intuition (Poincare, Polya, Dewey)
51(22)
Summary
51(8)
Mathematical Intuition
59(2)
Polya
61(2)
Mental Models
63(3)
Mental Models Subject to Social Control
66(2)
Dewey and Pragmatism
68(4)
Acknowledgments
72(1)
To Establish New Mathematics, We Use Our Mental Models And Build On Established Mathematics
73(16)
Introduction
73(2)
Established mathematics
75(2)
Mathematicians' proof vs. axiomatic proof
77(1)
Mathematicians' proof is semantic, not syntactic
78(2)
Established mathematics is fallible
80(2)
Published vs. private, rigorous vs. plausible
82(1)
Established mathematics is not controversial
83(2)
Acknowledgments
85(4)
How Mathematicians Convince Each Other or "The Kingdom of Math is Within You"
89(26)
A quote from Hardy
90(1)
What some mathematicians say they are doing
91(4)
The Materialist versus the Platonist: Changeux and Connes
95(6)
What, then, is a mathematicians' proof?
101(1)
Relation between formal proof and mathematicians' proof
102(2)
Aristotle, Kant, and Locke
104(1)
Is this mere Platonism?
104(1)
Heron's area theorem
105(3)
Conclusions
108(1)
Acknowledgments
108(1)
Appendix
108(7)
On the interdisciplinary study of mathematical practice, with a real live case study
115(10)
Wings, not foundations!
125(6)
1 What foundations?
125(3)
2 Lived experience as "foundation"
128(3)
Inner Vision, Outer Truth
131(6)
Mathematical Practice as a Scientific Problem
137(10)
Atiyah's pleasant surprise
137(1)
Does "existence" matter?
137(2)
For a multi-disciplined study of mathematical practice
139(1)
The basic problem
140(2)
Timely or timeless?
142(2)
Conclusion
144(1)
Educational implications
144(3)
Proving is Convincing and Explaining
147(10)
I What is proof?
147(1)
II Proof among professional mathematicians
147(1)
III Three meanings of "proof"
148(1)
IV Variation in proof standards
149(1)
V The four-color theorem
150(3)
VI Proof in our classrooms
153(2)
VII Coda
155(2)
Fresh Breezes in the Philosophy of Mathematics
157(6)
Foundations lost
157(1)
Phil / m and phil / sci
158(2)
Taking the test
160(3)
Definition of mathematics
163(4)
Introduction to "18 Unconventional Essays on the Nature of Mathematics"
167(6)
Part 2 "Mostly for the left hand"
173(60)
Introduction
175(2)
Rhetoric and Mathematics (with Philip J. Davis)
177(14)
Part 1 Mathematics as Rhetoric
178(4)
Part 2 Rhetoric in Mathematics
182(6)
Closure
188(3)
Math Lingo vs. Plain English: Double Entendre
191(4)
Independent Thinking
195(4)
The "Origin" of Geometry
199(6)
The Wedding
205(2)
Mathematics and Ethics
207(6)
Ethics for Mathematicians
213(4)
Under-represented Then Over-represented: A Memoir of Jews in American Mathematics
217(10)
Paul Cohen and Forcing in 1963
227(6)
Part 3 Selected book reviews
233(30)
Introduction
235(2)
Review of Not Exactly...In Praise of Vagueness by Kees van Deemter
237(4)
Review of How Mathematicians Think by William Byers
241(6)
Review of The Mathematician's Brain by David Ruelle
247(4)
Review of Perfect Rigor by Masha Gessen
251(4)
Review of Letters to a Young Mathematician by Ian Stewart
255(2)
Review of Number and Numbers by Alain Badiou
257(6)
Part 4 About the Author
263(8)
An amusing elementary example
265(2)
Annotated research bibliography
267(2)
Poems
269(2)
Curriculum Vitae 271(2)
List of articles 273(6)
Index 279
Reuben Hersh is emeritus professor of mathematics at the University of New Mexico, USA.