Preface Skvoz volxebny i pribor Levenguka . . . Nikola i Zabolockii The portrayal of human thought has rarely been more powerful and convincing than in Vermeer’s Astronomer. The painting cre- ates the illusion of seeing the movement of thought itself—as an embodied action, as a physical process taking place in real space and time. I use the Astronomer as a visual metaphor for the principal aim of the present book. I attempt to write about mathematical think- ing as an objective, real-world process, something which is actually moving and happening in our brains when we do mathematics. Of course, it is a challenging task inevitably, I have to concentrate on the simplest, atomic activities involved in mathematical practice— hence “the microscope” in the title. Among other things, • I look at simple, minute activities, like placing brackets in the sum a + b + c + d + e. • I analyze everyday observations so routine and self-evident that their mathematical nature usually remains unnoticed: for ex- ample, when you fold a sheet of paper, the crease for some rea- son happens to be a perfectly straight line. • I use palindromes, like MADAM, I’M ADAM, to illustrate how mathematics deals with words composed of symbols—and how it relates the word symmetry of palindromes to the geometric symmetry of solid bodies. • I even discuss the problem of dividing 10 apples among 5 people! xi

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