An Applied Mathematician's Apology

by Lloyd N. Trefethen

In 1940 G. H. Hardy published A Mathematician's Apology, a meditation on mathematics by a leading pure mathematician. Eighty-two years later, An Applied Mathematician's Apology is a meditation and also a personal memoir by a philosophically inclined numerical analyst, one who has found great joy in his work but is puzzled by its relationship to the rest of mathematics.

Genres: Mathematics, Nonfiction

Unknown Binding

First published July 30, 2022

Reviews

[Shyan · Rating 3 out of 5]

Has some nice insights and reflections. Perhaps too much bragging on the author's part.

[Jeff · Rating 5 out of 5]

This is a fantastic book. If I had known this type of mathematics existed when I was in school my career would likely have been quite different. Lots of interesting anecdotes about well known numerical analysts. Goes into enough mathematical depth that you can look up the stuff you're interested in to get into the technical details while remaining light enough to read casually.

One thing I appreciate is how Trefethen doesn't shy away from the human element of mathematics: concerns with prestige, whether work is important or just filling in details, etc. This is a perspective that mathematicians and scientists rarely write about despite virtually all of them being influenced by it.

[Gavin · Rating 3 out of 5]

Plain, brief, unprepossessing. Maybe deep for these reasons.

Now I'm no god, but remember, I am a leading figure in one of mathematics' big subdisciplines and the senior among 15 statutory chairs in one of the world's top mathematics departments. So let's ask, how many works by these Fields medalists have I read?

The answer is, exactly one. This is the inspiring textbook Complex Analysis by Ahlfors that I studied in Math 213a, sophomore year at Harvard. Beyond that, I've read some pages by Serre, Hörmander, Milnor, Atiyah, Smale, Lions, and Tao. But Ahlfors, Fields medalist 86 years ago, is the only one of the 60 who wrote something I have come anywhere close to reading in full.

All areas of intellectual activity have proliferated in specializations with the years, as more and more people make contributions, but still, this situation is extreme. Can you imagine a novelist who's never read a book by Mann, Hemingway, Márquez, Lessing, or Morrison? An economist who's never read anything by Samuelson, Arrow, Friedman, Kahneman, or Krugman?

...Whoever they are influencing so greatly, it is not the Professor of Numerical Analysis at Oxford. And, of course, they have been equally little influenced by me...

In my early years I took it for granted that the more mainstream mathematicians, the leaders in each specialized field, understood what was important in their areas. It troubled me, therefore, to notice that my own work wasn't building on theirs. I would investigate a problem and make a good contribution, often a genuine discovery, without ever mastering or in the end even attempting to master the results of the nonnumerical experts in the area... privately I interpreted the situation as a deficiency on my part. I knew I was doing good work, but I supposed it would be even better if I had the strength of character to absorb the papers of Adamjan, Arov, and Krein in support of my Carathéodory-Fejér approximation, to immerse myself in the theories of the great Louis Nirenberg while I was working on PDEs at the Courant Institute, or to digest Dunford and Schwartz when I was writing the book on pseudospectra... If ignoring the masters were truly an error, then I would have found not infrequently in my career that my contributions later turned out to have been anticipated, or invalidated, by the work of others. This has not happened. Everything I've done has remained valid and original, some things more important than others, of course, but almost never mistaken or redundant... What in the world is going on with mathematics if careful attention to the works of the leaders of approximation theory, complex analysis, real analysis/PDEs, functional analysis, and stochastic analysis need not be on the path to making contributions in these fields?

[Emilio Minichiello · Rating 3 out of 5]

(3 stars) This book is supposed to be a kind of response to Hardys book A mathematicians apology, but for applied mathematicians. But I found the book lacking all of the qualities that made hardys book enjoyable and interesting. Trefethen gives a little bit of an autobiography, which was interesting, but only went up to his schooling days. I would have liked to know more about his rise to being an oxford professor, with some anecdotes and interesting details. Instead we get a flat and lengthy recounting of how the chebfun software that he worked on works. Hardys book is about how math is an art and how it should be regarded as such. Trefethens book is about himself and how annoyed he is that there is a gap between applied and pure mathematics, with hardly any deep analysis or suggestions on how to fix the situation.