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- munities. 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 illustrations.
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