One, Two, Three: Absolutely Elementary Mathematics by David Berlinski

Cover of One, Two, Three (Pantheon)

David Berlinski is an interesting guy for a lot of reasons. He seems to be driven to explain fairly complex things in a way that mortals can understand. One, Two Three is an excellent example of this.

Absolutely, Elementary, Mathematics

How the natural numbers are conceived of and constructed should be part of the basic training of every elementary and secondary school mathematics teacher.  After all it is absolutely elementary mathematics (AEM) and what could be more fundamental and important than that? It sure sounds important. Sadly, I think such a deep probing of what on the surface seems like it ought to be brain-dead simple, or (gasp) even obvious, is usually brushed off as being too abstract and impractical. It is abstract, but then again all of mathematics starting with 1 + 1 = 2 is entirely abstract.

Maybe you don't need to know much about the foundations of the counting numbers and why they work the way they do to be able to teach someone to add, subtract, multiply and divide. But, what happens when asked by the curious mind that most dreadful of questions—why? As in, why does that work? Why that way and not some other way? You had better be prepared for that day, and this book will help you get there.

Introduction to Mathematical Philosophy—for Dummies

Well, not quite. But for those of us that have slogged through Bertrand Russell's classic 1919 work covering the same material that this one mostly does, it is easy to see why Berlinski felt this book was needed. I've read Russell's book and understood a fair bit of it. Didn't quite latch on to a fair other bit (mostly in the second half).

After reading One, Two, Three I don't think I understand the second half of Introduction to Mathematical Philosophy any better (I admit that I simply read it, and did not really work at it—big difference), but I most certainly do understand the first half even better than I did before.

Russell's name pops up now and again, so it is pretty clear what (and who) Berlinski was thinking about when he tackled this subject. It is pretty much guaranteed that the following will be be better understood after reading:

• Why and how we count
• Why the counting numbers exist
• How one number can be obtained from another
• The true relationship between addition and subtraction
• The true relationship between multiplication and addition

Probably a few other things too. Some readers may be put off in places by Berlinski's rather flippant style and frequent attempts (not always successful) at humor and lightening of the natural heaviness of the material. But, that is part-and-parcel of how he goes about making things plain. Over all, he succeeds rather well.

Hey Math Teacher, Read This

Anyone who is motivated to teach mathematics for any reason beyond, "Hey, it pays the bills," ought to be interested in the fundamentals of the natural numbers. Next time you see your child's math teacher ask if they have read Introduction to Mathematical Philosophy. If they say no, then ask if they have read One, Two, Three. If the answer is still no, ask then if they know what the successor function is. If they do not know you have every right to ask—why not? Then, after they wriggle out of it, buy them a copy of One, Two, Three.

One, Two, Three. David Berlinski. 2011. Pantheon Books: New York, NY.