In fact, three related stories are intertwined in this novel, and each of them has a significant mathematical component. Primarily, the protagonist is Alexandra Jacobi-Semeu, a math professor in communist Romania in the 1970s, whose life falls apart when her brother defects to Canada. Eventually she makes the difficult decision to join him, leaving behind her daughter, Ada, whom she loves. This is possible because she has an international reputation not only as a good but as an amazing research mathematician. (In particular, it is because of pressure from other mathematicians around the world that she is allowed to attend a conference abroad, allowing her escape, and she is welcomed at the University de Montreal and later the Institute for Advanced Studies in Princeton where her research can really flourish. All the time, however, she thinks of her daughter and wonders whether she has made the right choice.

Nearly the opposite choice was made by her thesis advisor, Asuero Aroso. Although he appears in the book as an elderly man caring for his bedridden wife, we hear stories of his youth that he shares with Alexandra. For instance, we learn that he was once a student of David Hilbert's at Gottingen, and that Hilbert considered him a better mathematician than John von Neumann! [See note below from author.] In a development that contrasts Alexandra's plight nicely, he explains that he was offered a job at Gottingen by Richard Courant [Note: I had earlier written another name here, but was corrected by the author], but that he turned down that offer so that he could be with the woman he loved in Romania. As it turns out, this plan does not work out as he had expected. He is forced out of his job because he is Jewish, his relationship with his wife is frequently troubled, and because he delays publication of his proof of the independence of the continuum hypothesis, credit for the proof goes instead to Kurt Gödel. Therefore, it seems that he speaks from experience when he advises Alexandra to follow her brother to Canada.

The third storyline in Continuums is that of Alexandra's brother, Leonard. Like his sister, he trained to be a research mathematician. However, believing that he does not have his sister's raw talent, he decides instead to work for the government in a position that will provide him with money, but that he will pursue his dream of writing about mathematics even if he can not contribute to mathematics itself. When he gets to Canada, he writes a monograph about Olinde Rodrigues, an under-appreciated mathematician whom many consider to have priority in the discovery of the quaternions. After this, he writes a book about Aroso and finally, despite the critical failure of the previous two books, he also plans to write about Hermann Grassmann. At one point, a girlfriend points out to Leonard that he has a problem with commitment, not only to a romantic relationship but to his dreams. She argues that he does not dedicate himself to being a mathematician, to the jobs he has taken instead, to his writing or to his wife and child. So, presumably, in this regard he represents the problem of not making a decision, though his decision to leave Romania clearly had consequences for himself and others.

In many ways, the author Carr resembles Leonard Jacobi. He too was a young man who fled communist Romania for Canada. The brief biography on the cover indicates that he worked as an engineer in the aerospace industry, but the book demonstrates a familiarity and understanding of pure math research that goes beyond what one might expect of an engineer. And, like Leonard, he now has written about mathematics, specifically about the under-appreciated mathematicians Grassmann and Rodrigues as well as (the fictional) Aroso. I wonder if he, too, is a person who decided that he could not go into math research because he was not a "real genius". Personally, I believe that such arguments are misguided. I really am not certain that there are any geniuses...but even if there are, there are certainly also many mathematicians who are just intelligent and dedicated. And these "ordinary" mathematicians can also have successful careers and make contributions to the future of mathematics.

Mathematics is discussed frequently throughout the book, and considering that Carr is not a professional research mathematician, he does a remarkably good job of presenting it. With one notable exception (where he describes a couple with no common interest as "two sets without a union" rather than "two sets without an intersection"), it sounds entirely believable. The Continuum Hypothesis is discussed frequently (and once appears in dialogue written in terms of Cantor's notation with alef's, as if it were possible to typeset spoken language in TeX!), as is analytic number theory and some Lie theory. It might be possible to read the book without knowing what these things are in advance, but they are not really explained and many appear in literary metaphor, something would be lost. For instance, an analogy is made between the independence of the continuum hypothesis and the freedom of choice that people have.

From my perspective, this book deserves a lot of credit for presenting such a realistic and undiluted representation of the world of research mathematics. More importantly, from a literary perspective, it does a fantastic job of setting up a scenario in which the reader can explore some deep philosophical questions about the human condition. Its greatest weakness, however, is the lack of "poetry" in Carr's prose. His sentences are blunt and to the point, but individually lack the beauty that we see in math and in even a sad life when looking at the novel as a whole.

One very interesting thing to me is how similar this book is to Uncle Petros and Goldbach's Conjecture. Consider:

Aroso is a prodigy from Turkey who studies with Hilbert at Gottingen.	Petros is a prodigy from Greece who works with Hardy and Littlewood in Cambridge
After he is "scooped" in publishing a famous theorem from set theory, Aroso becomes a mathematical recluse and tells his story to his young student.	After he is "scooped"in publishing a famous theorem from number theory, Petros becomes a mathematical recluse and tells his story to his young nephew.
Alexandra's brother decides that he should not be a research mathematician like her because he could only be a good mathematician, never a great one.	Petros' nephew decides that he should not be a research mathematician like him because he could only be a good mathematician, never a great one.

Another similarity is that both books discuss Georg Cantor and his mental illness, although UPGC suggests that it was Cantor's mathematics that drove him crazy while Continuums (more sensibly in my opinion) tells the less dramatic but more believable story that he was always a victim of bipolar disorder and would have been like that even if he had not done any math.

Still, despite these similarities and the fact that UPGC is much older (now practically a classic of mathematical fiction), I strongly recommend Continuums. It is much more realistic than UPGC and presents a more nuanced, less simplistic view of math and the human condition.