Diary of a Bad Year is an unusual book in which the top of every page is a portion of an essay written by the protagonist, Señor C, for his German publisher, the middle of every page tells of his relationship with a young woman from his point of view, and the bottom of every page is told from the point of view of Anya, the young woman. Since Señor C is (by his own admission), too old for a sexual relationship, this makes for a strange sort of love story. But, this structured way of presenting the two viewpoints and contrasting them with thoughts on politics, science, ethics and economics is quite interesting. Perhaps the most difficult thing about reading this book is that it is difficult to accept it as fiction when Señor C is so clearly a representative of the author (an older, successful writer, born in South Africa but living in Australia) and the opinions expressed in the essays presumably are his own. It leaves one feeling that the author really wants to have a chance to present you with his thoughts in essay form, but realizing that they are not quite convincing or interesting enough on their own, he has dressed them up with a thin story.

Most of those opinions concern politics, where Coetzee presents an anarchist philosophy: governments are nothing but gangs of bandits that have seized control and any method of selecting a leader which is objective (whether it is an election or the coronation of the previous leader's first born son) is equally good. This cynical view is useful for criticizing the totalitarian excesses of the George W. Bush administration and the pandering of Tony Blair, but perhaps not as effective when he attempts to use it to promote communism or defend the origins of Apartheid. Other opinions expressed are typical of older writers of any age (e.g. the youth today have lost their appreciation for good music, society has become less civil, etc.)

In any case, for obvious reasons, we will focus here on the mathematical implications of the book rather than the political ones. In two chapters towards the middle, the essays focus on philosophical questions about mathematics and probability. Then, in the following three chapters, the bottom of the page story involves a discussion between Anya and her greedy investor boyfriend, Alan, about these same questions. Later, the essay tries to apply mathematical concepts to questions of ethics.

The essay presents a nice description of the natural numbers (1, 2, 3, 4, etc.) and the way that they are more structured than, for example, just a list of words describing certain abstract mathematical objects. Then, it poses the interesting question of whether these mathematical objects have a real existence in some sense, or whether we have created them as the consequences of the rules of arithmetic that we similarly made up. Of course, this question is not new. (See, for example, this Wikipedia entry on the philosophy of mathematics.) And, unsurprisingly, Coetzee is not able to resolve this question on which philosophers and mathematicians are themselves split. I suspect that Coetzee, like me, tends to believe that the numbers as mathematical objects are a human creation. In particular, he makes the interesting point that if one believes the numbers have an existence outside of human thought, then one might have to be worried that the rules we have created for figuring out what the next number after a given number (as we know that 19872 is the number after 19871) might be wrong in that instance because (in fact) it happens to be something else.

When Anya inquires about Alan's opinion on this subject, he takes a more practical view that might be closer to Platonism...but really sounds more like he is not interested in the philosophical question at all. To him, numbers are just the things we use to answer questions like how much money we will make from a particular investment.

I am afraid I cannot agree with Señor C's essay when it goes on to suggest that mathematics is nothing more than these rules for the arithmetic of the natural numbers. Although it is perhaps true that this piece of mathematics is richer than many people would expect, or that many of the interesting philosophical questions about mathematics already apply even to this simplest portion of the whole, I think it would greatly diminish mathematics to limit it to only those portions that are essentially equivalent to the mathematics of whole numbers. Geometry, calculus, abstract algebra, etc. are in some senses much bigger than the mathematics of whole numbers. I also cannot say that I understand his point about Zeno's paradox, where he seems to present this paradox and its traditional resolution through convergent series, as being somehow relevant to the philosophical question under consideration.

In a later portion of the essay, the focus is placed on probability. In particular, the author ponders both what is meant by a statement such as ``overweight men have an increased probability of suffering a heart attack" and whether the universe is deterministic (as was certainly believed by those who thought that Newtonian mechanics was an entirely accurate description of reality) or merely probabilistically determined by laws (as many people believe to be the case with quantum mechanics). Again, these are both interesting questions, and so the reader who has not previously thought about it may benefit from encountering them here. But neither the essays nor Alan's reactions seem to add anything interesting to the debate. Alan accepts the current scientific dogma that reality involves inherent randomness in the form of the collapse of the quantum wave function, and Señor C remains cynical and argues that probabilistic statements (even epidemiological ones such as "smoking cigarettes increases your chances of getting lung cancer") are essentially meaningless.

When it comes to the application of mathematics to ethics, Coetzee limits himself to the comparison of two acts to determine which one is worse. He points out that in math there is a difference between a wholly ordered set (in which any distinct pair of elements can be placed around a ``less than'' symbol in a unique way to form a true statement about which one is ``larger'') and a partial ordering (in which the question of which is larger does not apply to some elements). His point is that there is no reasonable way to compare the ethical weight of some acts.

Whether you enjoy reading this book may depend to a large extent on how you will react to Coetzee's opinions on various subjects. If you already agree with his viewpoints, you may feel cheered by seeing them repeated here. If you can be open-minded and consider them, you may learn something even if you do not come to agree. However, if you have strong opinions that differ from his, this book may simply come across as being self-righteous and annoying. For me, the most interesting thing about the book was its structure. I found it almost impossible to read each page from top to bottom since that would disrupt the train of thought in the separate portions. On the other hand, since the different portions of the page are thematically and chronologically related, it does not make sense to read them as three separate books either. As a result, I was jumping around in an interesting way, and so had some more control over the way it was read than I normally do in reading a novel.

By all accounts, Coetzee has written better works of fiction (although I still do consider the structure of this book to be creative and interesting). I now wonder whether any of those contain references to mathematics. If you can answer this question, please write to let me know.