Littlewood’s Miscellany is a good choice to read along with G. H. Hardy’s A Mathematician’s Apology (which I reviewed here). That’s not because it says anything more than Hardy’s book about the celebrated collaboration. It doesn’t. But it does give a reasonable snapshot of the world of mathematics in the period, roughly, from 1910 to 1950.
But first, let’s get a few negative observations out of the way. Littlewood begins almost immediately with this: “Anyone open to the idea of looking through a popular book on mathematics should be able to get on with this one.” I think that’s overly optimistic. In terms of the interest of people only casually curious about the nature of mathematics, the Miscellany does not compare favorably with Hardy’s Apology. The latter is definitely a “popular” work, in terms of both subject matter and technical level, while the Miscellany is not. Except for some catty gossip about academic life in England in the early decades of the 20th century, and the people (e. g. Bertrand Russell) associated with it, there’s not much in the book to interest the general reader. In all fairness, I think that only people with a strong interest in mathematics will find much in the Miscellany to hold their attention. And even that will be somewhat limited since contemporary mathematics is quite different from that of Littlewood’s heyday.
Another negative is that Littlewood is very candid in stating that a miscellany “is a collection without a natural ordering relation.” In other words, there’s no common thread or theme running through the book, not even some natural progression from one topic to the next. Some chapters seem to be merely disparate comments that Littlewood may have jotted down on a scrap of paper or a diary page at random times. Other chapters deal with nitty-gritty details of mathematical topics that are probably only of minor interest to contemporary math lovers – such as ballistics (guns and stuff), notations for extremely large numbers, and an examination in excruciating detail of the astronomical data that led to the discovery of Neptune. Notably lacking is any insight into the nature of Littlewood’s collaboration with Hardy. Littlewood has no more to say about that than the latter did.
For better or for worse, contemporary mathematics is vastly different from the mathematics of Littlewood and his era. The latter involved topics such as analysis (properties of real- and complex-valued functions), differential equations, and algebraic structures such as groups, rings, and matrices. Active areas of modern mathematics are things like “category theory”, abstract algebraic geometry, and higher-dimensional geometric objects. Littlewood’s and Hardy’s expertise was primarily in the mathematical analysis of their day, and with its application to physics and number theory. Littlewood can hardly be faulted for writing about what he knew best. But readers must understand that his topics are not as prominent in contemporary mathematics as they once were.
All that said, the Miscellany is still very much worth reading for the seriously mathematically curious. First of all, that’s because of the historical insight it gives to a certain period in the long evolution of mathematics. Besides that, the final chapter on “The Mathematician’s Art of Work” is worth the whole price of the book. In just 12 pages it provides numerous gems of insight into how working mathematicians actually go about doing what they do.
Littlewood builds on what other leading mathematicians, such as Poincaré and Hadamard, have written about the habits that are essential for mathematical creativity. When one is in the “preparation” stage of dealing with a difficult problem, one should experiment diligently with a variety of approaches. Then, after the “incubation” stage, when one has finally obtained a key insight, concentrated effort is generally required to verify the insight. Regarding these periods of conscious effort, Littlewood has this to say:
Either work all out or rest completely. It is too easy, when rather tired, to fritter a whole day away with the intention of working but never getting properly down to it. This is pure waste, nothing is done, and you have had no rest or relaxation. I said ‘Work all out’: speed of associative thought is, I believe, important in creative work.
That sounds like excellent advice to me. It applies not only to doing mathematics, but many other creative activities, such as novel writing and computer programming. But it all depends on whether you have a clear idea of what you need to do, or whether instead you need to spend more time “incubating” the next step of the work. Trying to be productive when you don’t really know what needs to be done – the next lemma to prove or the next plot twist to imagine – is likely to be wasted time.