2010;
317 pp;
Hardcover

MSC: Primary 00; 97;

Print ISBN: 978-0-8218-4761-9

Product Code: MBK/71

List Price: $62.00

AMS Member Price: $49.60

MAA Member Price: $55.80

**Electronic ISBN: 978-1-4704-1604-1
Product Code: MBK/71.E**

List Price: $62.00

AMS Member Price: $49.60

MAA Member Price: $55.80

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#### Supplemental Materials

# Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice

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*Alexandre V. Borovik*

This is an unusual and unusually fascinating book.

Readers who never thought about mathematics after their school years will be amazed to discover how many habits of mind, ideas, and even material objects that are inherently mathematical serve as building blocks of our civilization and everyday life.

A professional mathematician, reluctantly breaking the daily routine, or pondering on some resisting problem, will open this book and enjoy a sudden return to his or her young days when mathematics was fresh, exciting, and holding all promises.

And do not take the word “microscope” in the title too literally: in fact, the author looks around, in time and space, focusing in turn on a tremendous variety of motives, from mathematical “memes” (genes of culture) to an unusual life of a Hollywood star.

—Yuri I. Manin, Max-Planck Institute of
Mathematics, Bonn, and Northwestern University

It is an unusual book that casts new and paradoxical light on the nature
of mathematics.

This book will be interesting—perhaps for different reasons—to
school teachers of mathematics, to math majors at universities, to graduate
students in mathematics and computer science, to research mathematicians
and computer scientists, to philosophers and historians of mathematics,
and to psychologists and neurophysiologists.

The author's goal is to start a dialogue between mathematicians and
cognitive scientists. He discusses, from a working mathematician's point
of view, the mystery of mathematical intuition: why are certain
mathematical concepts more intuitive than others? To what extent does the
"small scale" structure of mathematical concepts and algorithms reflect
the workings of the human brain? What are the "elementary particles" of
mathematics that build up the mathematical universe?

One of the principal points of the book is the essential vertical unity of
mathematics, the natural integration of its simplest objects and concepts
into the complex hierarchy of mathematics as a whole. The same ideas and
patterns of thinking can be found in elementary school arithmetic and in
cutting-edge mathematical theories. There are no boundaries between
"recreational", "elementary", "undergraduate", and "research" mathematics;
the book freely moves throughout the whole range. Nevertheless, the author
takes great care in keeping the book as non-technical as possible.

The book is saturated with amusing examples from a wide range of
disciplines—from turbulence to error-correcting codes to logic—as
well as with just puzzles and brainteasers. Despite the very serious subject
matter, the author's approach is lighthearted and entertaining.

#### Readership

Undergraduate students, mathematicians, and the general scientific audience interested in cognitive aspects of mathematics.

#### Reviews & Endorsements

...an ambitious book that covers a great deal of material...[The book contains] many gems, amusing observations, thought-provoking links between mathematics and cognition, and unconventional ideas that every math teacher should be forced to wrestle with...Recommended.

-- J.F. Kolacinski, Choice

#### Table of Contents

# Table of Contents

## Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice

- Cover Cover11 free
- Title page iii4 free
- Contents vi7 free
- Illustration x11 free
- Preface xi12 free
- List of figures xxiii24 free
- List of photographs xxv26 free
- Part I. Simple things: How structures of human cognition reveal themselves in mathematics 128 free
- A taste of things to come 330
- What you see is what you get 2350
- The wing of the hummingbird 4370
- Simple things 6188
- Infinity and beyond 89116
- Encapsulation of actual infinity 117144
- Part II. Mathematical reasoning 133160
- What is it that makes a mathematician? 135162
- “Kolmogorov’s logic” and heuristic reasoning 169196
- Recovery vs. discovery 187214
- The line of sight 197224
- Part III. History and philosophy 215242
- The ultimate replicating machines 217244
- The vivisection of the Cheshire Cat 247274
- References 281308
- Index 307334 free
- Back Cover Back Cover1345