### It is an unusual book

which casts new and paradoxical light on the nature of
mathematics.

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

The author attempts 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 are more intuitive than the others? To what extent the "small scale" structure of mathematical concepts and algorithms reflects the workings of the human brain? What are the "elementary particles'' of mathematics which 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 the 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 of 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 just puzzles and brainteasers. Despite the very serious subject matter, the author's approach is lighthearted and entertaining.

2000 Mathematics Subjects Classification 00A30, 00A35

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

The author attempts 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 are more intuitive than the others? To what extent the "small scale" structure of mathematical concepts and algorithms reflects the workings of the human brain? What are the "elementary particles'' of mathematics which 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 the 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 of 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 just puzzles and brainteasers. Despite the very serious subject matter, the author's approach is lighthearted and entertaining.

2000 Mathematics Subjects Classification 00A30, 00A35