There are, roughly, two sorts of people who might consider reading this very short book: those who know or work with a fairly large amount of mathematics, and those who don’t. There are different things that should be said about the book to each group. Let’s take the latter group first.

One dictionary definition of “apology” is “a formal spoken or written defense of some idea, religion, philosophy, etc.” People who’ve had little exposure to mathematics beyond the basics of ordinary arithmetic, solving simple equations, and perhaps a little plane geometry may hope that the apology in this book could help them understand what sorts of things are in more advanced kinds of mathematics. They may wonder why anyone who doesn’t have to use anything beyond the basics might be interested in learning more about such things. Even someone exposed to basic calculus in college might have the same hope. For them the book may be somewhat disappointing, as it offers only two simple examples, and some general remarks on the characteristics of the “best” kind of mathematics.

These two examples are the fact that the square root of two isn’t a rational number – i. e. the ratio of two integers – and the fact that there are prime numbers larger than any given number one could choose. The explanations give a taste of what typical (but very simple) mathematical proofs are like. There are some who may find that enlightening if they have never been exposed to such things. But it’s pretty thin gruel. It’s like a foreigner dropping into a good roadside diner in Iowa and trying to understand from that what the U. S. as a whole is like.

What may possibly be somewhat more informative is Hardy’s discussion of his claim that “A mathematician, like a painter or a poet, is a maker of patterns.” (p.84) This may be of interest to readers who are curious about how professional mathematicians think about their craft. It is a little bit complicated. First off, the patterns must have both “beauty” and “seriousness”. Each of those requires further explanation. Of beauty, Hardy is pretty vague – something that’s typical of discussions of almost any kind of “beauty”. What he says is that the ideas which make up the pattern must “like the colours [of a painting] or the words [of a poem] must fit together in a harmonious way.” (p. 85) But that just shifts the issue to what “harmonious” means. Hardy gives up by saying “It may be hard to *define* mathematical beauty, but that is just as true of beauty of any kind.” (p. 85) One just “knows” it when one sees it.

As for “seriousness”, that’s a more convoluted matter. Hardy says “The seriousness of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the *significance* of the mathematical ideas which it connects.” And “a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.” (p. 89) The two examples Hardy gives of “significant” theorems do illuminate “significance” to some extent – if one happens to be aware of the vast complex of connected ideas (having to do with the deeper nature of numbers and geometrical objects).

Hardy goes on to discuss what “significance” involves. “There are two things at any rate which seem essential, a certain *generality* and a certain *depth*; but neither quality is easy to define at all precisely.” (p. 103) He makes a stab at it anyhow. “Generality” is sort of related to abstractness, and “depth” has “*something* to do with *difficulty*.” (p. 109).

It turns out that it is not only the ultimate theorems themselves that must have “beauty”, but the proofs should also have that as well. And in this case there should be “a very high degree of *unexpectedness*, combined with *inevitability* and *economy*.” (p. 113) Both of Hardy’s examples show this pretty well. Those are very elementary theorems, but even very advanced theorems can exhibit this beauty if they follow in this way from slightly less advanced ones. Professional mathematicians are generally not very satisfied with even (seemingly) simple theorems, like the Four Color Theorem, if their proofs are quite messy (which the proof of the FCT is to an extreme).

So, will a reader who has little exposure to mathematics beyond the basics, but is sincerely curious to understand more, be satisfied with Hardy’s narrative? I’d say that such readers would have to answer that for themselves. Probably what Hardy has to say will be most appreciated if and when readers go on to actually study more advanced mathematics. Then, and only then, will it be really worthwhile to use what Hardy’s written in order to properly appreciate what is learned in the more advanced study. So this gives us an answer to what readers with much more exposure to mathematics may find valuable in Hardy’s *Apology*: It may help clarify what they enjoy about mathematics, namely the aesthetic enjoyment of mathematics that is especially “elegant”. However, they’ve probably come to similar conclusions on their own.

Hardy does deal with another issue that is considerably more controversial: that is, whether it’s important for the “best” mathematics to have – or not have – practical utility and applicability. Hardy, a confirmed pacifist, insisted that “Real mathematics has no effect on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.” (p. 140) Although counterexamples have appeared since that was written, they are quite insignificant in comparison to the beauty and (can I say?) grandeur of both theories. (“Grandeur” is a property a related collection of theorems on the same topic may possess. Hardy doesn’t mention it.)

It’s a waste of time to argue over the relative merits of “pure” vs. “applied” mathematics. If a particular professional mathematician prefers one over the other, it’s just a matter of individual taste. If a given body of mathematics happens to have important applications, that fact needn’t either enhance nor detract from whether the mathematics has significant value to those who work on it. And so neither case requires more or less “apology” (in the sense of an excuse or justification) that needs to be given for working on it.

Hardy offers very little of a biographical nature in his *Apology*, although there are a few pages along such lines at the end. This is rather unfortunate. One would have hoped that such an eminent mathematician as Hardy might have offered a rather more detailed account of his professional working habits and how he arrived at his most noteworthy accomplishments. He wrote some great textbooks (alone or in collaboration), but his best theoretical work was in a legendary collaboration with J. E. Littlewood.

The few biographical remarks that Hardy offers are somewhat downbeat, as the *Apology* was written late in his life, when he felt most of his creative energy had faded. Too, he seemed to have had for much of his life somewhat of an inferiority complex. Hardy was eight years older than Littlewood, so he may have always been at a disadvantage to the vigor of his collaborator. Littlewood also published one notable book for a general audience (Littlewood’s Miscellany), which is quite different from the *Apology*, and has its own idiosyncratic strengths and weaknesses. Unfortunately, neither man had almost anything to say about their unique collaboration.

The reprinted (1967) edition of the *Apology*, which came out 20 years after Hardy’s death, has a long foreword by Hardy’s friend C. P. Snow. As Snow wasn’t a mathematician, it has little to say about mathematics, but contains some very useful biographical details, including much about Hardy’s other great collaboration, with Srinivasa Ramanujan. Professional mathematicians may find this foreword the most interesting part of the book.