xviii Preface
questions which have puzzled me throughout my professional ca-
reer in mathematics and which continue to puzzle me.
Perhaps, my questions are naive. However, I worked on the book
for several years and kept the text on the Web, returning to it from
time to time to add some extra polish or to correct the errors. So
far, the changes in the book were limited to expanding and refining
the list of questions, not inserting answers—I cannot find any in
the existing literature. This is one of the reasons why I believe that
perhaps at least some of my questions deserve a thorough discus-
sion in the mathematical, educational, and cognitive science com-
My last apology concerns the use of terminology. Some terms
and expressions which attained a specialized meaning in certain
mathematics-related disciplines are used in this book in their (orig-
inal) wider and vaguer sense and therefore are more friendly to the
readers. To fend off a potential criticism from nit picking special-
ists, I quote a fable which I heard from one of the great mathemati-
cians of our time, Israel Gelfand:
A student corrected an old professor in his lecture by
pointing out that a formula on the blackboard should con-
tain cotangent instead of tangent. The professor thanked
the student, corrected the formula and then added:
“Young man, I am old and no longer see much difference
between tangent and cotangent—and I advise that you do
so as well.”
Indeed, when mathematicians informally discuss their work,
they tend to use a very flexible language—exactly because the prin-
cipal technical language of their profession is exceptionally precise.
I follow this practice in my book; I hope it allows me to be friendly
towards all my readers and not only my fellow mathematicians.
Acknowledgements: Inspiration and Help
The gods have imposed upon my writing
the yoke of a foreign tongue
that was not sung at my cradle.
Hermann Weyl
I thank my children, Sergey and Maria, who read a much ear-
lier version of the book and corrected my English (further errors
introduced by me are not their responsibility) and who introduced
me to the philosophical writings of Terry Pratchett. I am grateful
to my wife, Anna, the harshest critic of my book; this book would
never have appeared without her. She also provided a number of
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