Eight Lessons on Infinity

You are probably not in the target audience for Haim Shapira's Eight Lessons on Infinity: A Mathematical Adventure.  If you are an MAA member, reading this review on the MAA web site, the chances are good that you are familiar with the basic material covered in this book -- the proof there are an infinite number of primes, Zeno's paradox, the Grand Hotel Infinity, and of course Cantor's diagonalization argument.   So you probably don't want to get this book to read for your own edification.  The question is whether to buy it for an inquisitive young person in your life.  Is this book a good non-technical introduction to those topics?  My answer is a qualified "Yes.''

 

I've never met Haim Shapira, but I know a lot of people like him.  He's the sort of person that stops you at a social occasion and asks, ``Hey, do you want to see something cool?''  Then he sits down and starts doodling some interesting problem on a cocktail napkin.  His excitement for mathematics jumps off of every page of this book.  He loves little puzzles and problems, and he can't help but throw them at you, even when sometimes they don't exactly fit his topic.  

 

That brings me to my first qualification.  The inquisitive young person you give this book to has to be a little bit patient.  The book opens with a "Warm-up,'' titled ``A short introduction to thinking.'' This is a place where Shapira crams in a bunch of problems that he just thinks are neat, even though they don't have much to do with his topic.  Your young person may easily spend a lot of time in this section, but they may also be dismayed that there are so many problems here that don't have anything to do with infinity.  Then even when Shapira gets on to the main topic, he sometimes goes off on tangents, just because an interesting puzzle or proof comes to his mind.  So be sure to warn your young person that the book isn't quite as focused as it could be.

 

Shapira's style is breezy and fresh.  I don't know how many different presentations of the Grand Hotel Infinity I've read in my life, but Shapira's is one of the best.  The problems build in the usual way.  The hotel is full, and a single guest shows up, then a finite number of guests, then a countably infinite number of guests.  But the story is well-told, and the solution is explained clearly.   Shapira even manages to introduce a group of guests corresponding to the points of the unit interval, and his explanation for why they can't be accommodated is his first introduction to Cantor's diagonalization argument.  It's all done in a style that is likely to captivate and challenge your inquisitive young person, without burying them in technicalities.

 

My other qualification has to do with the pacing of the lessons.    Lessons 1 through 7 work at a reasonable pace through Number Theory, prime numbers, Zeno's paradox, set theory, and the Grand Hotel Infinity.  Then Lesson 8, ``Cardinals and the taming of the infinite,'' tackles Cantor head on, and it goes on for almost 60 pages.  It could easily have been broken up into two chapters for easier consumption.

 

But these are really quibbles.  On the whole, I think Shapira's book is an original and interesting new entry in the category of math books for the general audience.  It will make a great gift for the inquisitive young person in your life, or perhaps a prize at your next high school math contest.

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Benjamin V.C. Collins stayed in the Grand Hotel Infinity once but was annoyed when he kept having to get up in the middle of the night to move to a different room to make space for new guests.