This book was sent to me by the publisher as a review copy.

It is not easy to write a review for an anthology of writings, but I think that in such cases what is best discussed is the choice of writing and its range, both topically and in terms of level. In this case we have some 30 short essays, covering a huge range of topics, as well as a real breadth of complexity. I will highlight some of my particular favourites, though I should say from the outset that I really enjoyed reading just about everything in this book. There were perhaps two or three posts which didn’t resonate with me, but out of 30, that is pretty good, given my personal tastes.

The collection starts with a lovely essay discussing the interplay between the teaching, and the practice of mathematics, and in particular the role of rigour, formality and proof in these two somewhat separate directions. There are some lovely examples in this chapter about methods for guiding mathematical exploration in the classroom, and these are certainly things that I can take on for my own teaching.

Two chapters follow on from this, also talking to a large extent about rigour and abstraction on the life and works of G H Hardy, and of course Ramanujan. These are nicely contrasting, talking on the one case about Hardy’s desire to be as pure as possible, and in the second case, talking about the work for which Hardy was perhaps most impactful – that of a relatively simple mathematical conjecture in the world of genetics.

The book then starts to delve more into the world of specific mathematical objects and ideas: Stacking wine bottles (with some wonderful geometrical constructions, showing some very surprising results), billiard trajectories, Moonshine and its relation to string theory, and beyond, the abc conjecture and the mystical world of Mochizuki, cellular automota and fractals.

There are then some posts on the relationship between art and mathematics, though for me, the post on maths at the Metropolitan Museum in New York feels a little contrived.

From here the book goes more into the world of mathematics education with some very interesting essays on the meaning of truly understanding maths. This again, for a lecturer is extremely interesting, as what may seem useful – the ability for a student to explain their mathematical reasoning, may actually come from a place of rote learning, rather than true understanding, and that sometimes a student with real fluency will have an intuitive feeling of what to do, without necessarily being able to explain why. This isn’t necessarily a stage that should be criticised or penalised, just because someone can’t articulate why they need to do something, for which they have a strong feeling. It also penalises those who may not have the verbal skills, but are nevertheless proficient at some mathematical technique.

The rest of the book is somewhat of a miscellany, including discussions about probabilistic programming, spotting errors in data, the meaning of depth in mathematical proofs, and finally, how to write a mathematics book.

This was overall a really enjoyable collection to read, and I am going to go back and find some of the collections put together by Pitici from past years. This book could be read by anybody with an interest in mathematics, though perhaps some of the chapters would be a little inaccessible to anybody below a keen undergraduate level.

How clear is this post?