1

A Taste of Things to Come

This is the opening chapter of the book, and I use it to set the tone of

my narrative: I start with some simple mathematical observations

and briefly discuss what they possibly say about the inner workings

of our minds. Surprisingly, this discussion very naturally involves

some non-trivial ideas and results from the frontier of mathemati-

cal research. But it is better to see it for yourself.

1.1 Simplest possible example

Simplicity, simplicity, simplicity!

I say, let your affairs be as two or three,

and not a hundred or a thousand;

instead of a million count half a dozen,

and keep your accounts on your thumb-nail.

Henry David Thoreau, Walden

In my account, I am not afraid to be very personal, almost sen-

timental, and have decided to start the discussion of the “simple

things” of mathematics by turning to my memories from my school

years.

Always test a mathematical theory on

the simplest possible example—and ex-

plore the example to its utmost limits.

I had my most formative mathe-

matical experiences at the tender age

of thirteen, when I still lived in my

home village on the shores of Lake

Baikal in Siberia. I learned elemen-

tary calculus from two thin booklets

sent to me from a mathematics corre-

spondence school: The Method of Co-

ordinates [266] and Functions and Graphs [267]. Much later in my

life I met one of the authors of the books, the famous mathemati-

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http://dx.doi.org/10.1090/mbk/071/01