Although many characters in the book seem to know about and have interest in mathematics, it is only her grandparents who are really mathematicians. They were both talented math students at the start of WWII. Her grandmother wound up working at Bletchley Park and has fond memories of Alan Turing. However, because of his opposition to the war, her grandfather was not allowed to work there. As a result, he has a bit of a "chip on his shoulder" about this. While her grandmother works on proving the Riemann Hypothesis, her grandfather writes a regular column on mathematical puzzles in a science magazine. (The book describes Martin Gardner as an American version of Alice's grandfather...though I suppose it is really the other way around.) But, what most interests the grandfather is breaking famous codes, since he would like to prove to the world that he was good enough to have worked at BP even though he wasn't there. For instance, he works on the Voynich Manuscript (which he of course does not decode), but also on the "Stevenson/Heath Manuscript". This famous manuscript (well, famous in the book anyway) is supposed to give the location of a treasure. When Alice is a little girl, her grandfather tells her that he has solved the mystery and knows where the treasure is located. But, he refuses to tell anyone. Instead, he encodes his answer in the form of a necklace with the symbol "alef" (the first letter of the Hebrew alphabet and Cantor's symbol for the various levels of infinity) and the number "2.14488156Ex48" in a locket. He promises that when she is old enough, Alice will be able to decode it and decide what to do with the treasure. Along with a subplot about large corporations and their manipulation of consumers for profit, and an introspective glimpse into the inner-workings of the mind of a teenage girl, Alice's attempt to rediscover her grandfather's solution is a main focus of the novel.

There is certainly quite a lot of mathematics in this entertaining book. Perhaps most impressive is how well Thomas is able to weave the mathematics into the story. Having read many works of mathematical fiction, I feel qualified to say that the author of PopCo is unusually talented in her ability to include many, complicated mathematical ideas in a story in a way that does not seem either forced or overly didactic. She also successfully avoids making use of the mathematical stereotypes that are nearly ubiquitous in the works listed on this site. Among the mathematical themes that readers will encounter in this book are: transfinite cardinals (that is, the different "sizes" of infinity), trapdoor codes (RSA encryption, the role of number theory in cryptography), the Monty Hall problem (a simple but subtle example of probability theory), the Riemann Hypothesis, Pythagoras' numerical analysis of pleasing musical tones, Gödel's incompleteness theorem, logical paradoxes, Conway's "Game of Life", and a table of the first 1000 prime numbers. We also read about mathematicians: Turing, Erdös, Hardy, etc.

Although Thomas seems to have a very broad knowledge of mathematics, and works all of it very convincingly into the story, she does make a few mistakes. For instance, although she does seem to understand the idea of logical paradoxes in set theory and that it is somehow related to the idea that sets can contain themselves, she mistakenly believes that "the set of all sets" is obviously paradoxical. Although it is true that the Zermelo-Fraenkel Axioms of set theory exclude such a set, there is no immediate paradox associated to the idea that a set could contain itself. Rather, the problem arises when one encounters sets that only contain themselves when they don't, such as "the set of all sets which do not contain themselves".

Similarly, the author becomes a bit confused about Gödel's incompleteness theorem and its use of self-referential statements which are only true if they are false. Thomas does an excellent job of discussing the "coding scheme" that Gödel uses to transform statements into numbers. However, her description of what was done with it is not quite accurate:

We do have the statement "1+1=2" as a truth in arithmetic. Thomas's description (or is it Alice's description?) suggests we also know that "if 1+1=2 then 1+1(not=)2". This would mean that we know mathematics does contain contradictions. That would be pretty awful! In fact, as she suggests at the end of the paragraph (but either doesn't understand or doesn't explain well) is that it is actually a situation in which there are two possibilities: either inconsistency (such as 1+1 being 2 and not being 2 at the same time) or incompleteness (that there are true statements we simply can't ever prove). However, to be able to see that, Thomas/Alice needs to add another level to her description. She needs to be able to encode not just statements of what you get when adding numbers together, but statments about whether something is provable or not. Moreover, it is not really mathematics as a whole which suffers from these limitations, but rather only a certain type of axiomatic system.

Let me explain: Instead of the example she gave, it would have been more accurate to say that Gödel encoded in his system the statement "This statement cannot be proved" as an arithmetical statement. Then, as I said above, there are two possibilities: either the statement can be proved (in which case there is a contradiction, since it said it could not be proved) or it cannot be proved (in which case the statement is 100% accurate, but unprovable)! Since he encoded all of this into simple arithmetic statements, we then know the same is true of any first order, constructible axiomatization of arithmetic. Any attempt to completely capture arithmetic using these particular types of axioms is doomed to either leave some questions undecidable or to be internally contradictory.

She also does a poor job of explaining the Riemann Hypothesis, although in that case I am not certain whether she fails to understand the statement of the conjecture or was just satisfied with a rather loose and poetic restatement of it. Fortunately, these few relatively insignificant mischaracterizations of advanced mathematics do not come close to spoiling the book. For one thing, even if she misses a few small details, the author understands the main point of the mathematics she is discussing well enough to use it sensibly in the story.

Moreover, the book successfully creates a fictional universe in which mathematics is everywhere. And, as a math professor, I have to appreciate this bit of "pro-math" progaganda. Even though she works at a toy company as a "creative", the mathematics that Alice learned from her grandparents is useful in her everyday life. It is also a fictional universe in which giant corporations use media to manipulate the buying public into giving them money. Come to think of it, it is a fictional universe that sounds very much like the real one in which I live.

Mathematically inclined readers may also be interested in the self-referential aspects of the book, as towards the end Alice begins discussing her ideas for writing a book about the company she works for. (She decides not to use the company's real name, to avoid legal problems, and picks the name "PopCo" as a good pseudonym.)

The author once had a blog at "bookgirl.org" which contained more information about the book. It seems to have disappeared. At the moment, however, it is still possible to read her old postings using the Wayback Machine. For instance, here you can read an interesting passage about Cantorian set theory ("sizes" of infinity) which was cut from the book and here is another "deleted scene" about computability.

See also my review of PopCo which was published in the AMS Noticed (Feb. 2006).