Hardcover ISBN:  9780821847619 
Product Code:  MBK/71 
317 pp 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $52.80 
Electronic ISBN:  9781470416041 
Product Code:  MBK/71.E 
317 pp 
List Price:  $62.00 
MAA Member Price:  $55.80 
AMS Member Price:  $49.60 

Book Details2010MSC: Primary 00; 97;
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, MaxPlanck 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 cuttingedge 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 nontechnical as possible.
The book is saturated with amusing examples from a wide range of disciplines—from turbulence to errorcorrecting 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.ReadershipUndergraduate students, mathematicians, and the general scientific audience interested in cognitive aspects of mathematics.

Table of Contents

Part I. Simple things: How structures of human cognition reveal themselves in mathematics

1. A taste of things to come

2. What you see is what you get

3. The wing of the hummingbird

4. Simple things

5. Infinity and beyond

6. Encapsulation of actual infinity

Part II. Mathematical reasoning

7. What is it that makes a mathematician?

8. “Kolmogorov’s logic” and heuristic reasoning

9. Recovery vs. discovery

10. The line of sight

Part III. History and philosophy

11. The ultimate replicating machines

12. The vivisection of the Cheshire Cat


Additional Material

Reviews

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


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 Get Permissions
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, MaxPlanck 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 cuttingedge 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 nontechnical as possible.
The book is saturated with amusing examples from a wide range of disciplines—from turbulence to errorcorrecting 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.
Undergraduate students, mathematicians, and the general scientific audience interested in cognitive aspects of mathematics.

Part I. Simple things: How structures of human cognition reveal themselves in mathematics

1. A taste of things to come

2. What you see is what you get

3. The wing of the hummingbird

4. Simple things

5. Infinity and beyond

6. Encapsulation of actual infinity

Part II. Mathematical reasoning

7. What is it that makes a mathematician?

8. “Kolmogorov’s logic” and heuristic reasoning

9. Recovery vs. discovery

10. The line of sight

Part III. History and philosophy

11. The ultimate replicating machines

12. The vivisection of the Cheshire Cat

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