Prof. Ian Stewart FRS is clever and well-regarded. For a long time, his book on Galois theory was on my to-read list. This book was a major disappointment. It started off, in prospect, as a possible five stars, but it rapidly slid down to two.
What are the problems? Too many to list, but here are some.
First, there is actually precious little mathematics here, esp. in the first hundred pages or so. Then the text is littered with statements that were almost literally painful to read. At one point, he observes that the number of bits required to encode the human genome is approximately the same as the capacity of a CD - thus 'we are roughly as complex as Seargent Peppers Lonely Hearts Club Band'. This is a _completely_ content free remark, for reasons that I am sure Prof. Stewart is aware of, when he is making any effort at all. He implies that we didn't 'really' know that a reef-knot cannot be untied, until topologists managed to prove it in this century. This is a serious confusion of models and reality. It is more accurate to say that we have known, _with absolute certainty_ that you cannot untie a reef-knot with fixed ends, we juat haven't bothered to shoe-horn that knowledge into the language of algebraic topology. Presumably we didn't know until this century either (because mathematics tells us that you can) that you could take a sphere the size of a football apart, and put it together as a sphere the size of the sun? This chapter ends up in a discussion of protein structure that I expected to build to some interesting mathematical theory for solving the protein folding problem (more a statistial physics problem than a mathematics problem, per se, I would have thought), but that ends up by saying nothing more than that there is surely some metric under which there is a continuously descending path in the potential space for a protein, because otherwise it wouldn't reliably fold (you don't say) but we don't know what that metric looks like - in the meantime some people have made the problem into a video game, and it turns out that there are people who are good at solving the problem. Cool! (I don't really think so, more sunday supplement cute, actually. What has that got to do with mathematics, or science, really?
He has a discussion of the structure of viruses that tells us (and not a lot of people know this) that some viruses have a structure that is found in 3d cross-sections of 4d lattices - I expected him at this point to provide a causal explanation of why this might be so, but the chapter just stops at this point.
There are various remarks about data analytics (e.g. clustering, classification tree construction) that avoid all concrete detail and look to be just plain misleading about computational complexity.
I could go on(really? you ask), but this review is already too long.
In 1968, J. Maynard Smith published a perfect little classic of a book, 'Mathematical Ideas in Biology' (CUP, 130pp in my copy). It is long out of print, but surely easy to track down. Stewart isn't in competition.