Ian Stewart's 'Letters to a Young Mathematician' is a spiritual successor to G. M. Hardy's 'A Mathematician's Apology', although Ian views the subject more positively and presents it as a series of letters. If you can suspend your disbelief that Ian writes in the same style as he ages 30 years, while you rise from a secondary schooler to an "emeritus guru", like Ian himself, then Ian seems to play the part of a mentor quite well.

The book opens with a preface explaining how the mathematicians of yore lived a seemingly much more fulfilling life, both academic and personal, than the mathematicians of today. "A typical day for Hardy consisted of a maximum of four hours of intensive thought about research problems; the rest of the day was then occupied watching the game of cricket... A typical day for the modern academic is ten or twelve hours long, with teaching commitments, research grants to pursue, research to be carried out, and liberal doses of pointless bureaucracy to get in the way of anything creative." I never want to get into academia because it seems like I'd be playing office power games with other autists, putting two bad things together.

Ian then begins the book by basically trying to convince a secondary schooler to get into mathematics. I'm too far deep to back out now, even if I wanted to, but it was still enjoyable hearing about all the exciting prospects which await me. Ian's examples of mathematics in the real world include basically all of physics and biology, including plenty of "dynamics"es, rainbows, and how birds space themselves out evenly when sitting on a one-dimensional plane. Ian acknowledges that experiencing all of this would require a mathematician to go outside, and so he argues that mathematics is elegant and beautiful in its own right. Unfortunately, he primarily picked geometrical examples, like how it's possible to construct a regular 17-gon but not a regular heptagon. I'm sure I'd find the proof elegant if I understood it.

Ian explains how he got into mathematics, giving credit to two good teachers. When reading this chapter, I immediately noticed a parallel to my own experience. The first of Ian's two fostered a very competitive learning environment, which worked to the advantage of most students. Part of me wishes my class's test scores were publicised in secondary school too, but maybe I'd have ranked so low in the class I'd've just given up. I definitely have a teacher to thank for keeping me in the top set even though my maths grade was subpar in year 9. The latter of Ian's two star teachers, being excited to have a number of capable sixth formers, introduced Ian and his friends to the world of "proper mathematics". This teacher even facilitated Ian doing an S-Level in year 12, which seemed to lift him into the University of Cambridge. I agree with Ian that mathematics is a field which is easier to fall out of than fall into. Unlike fields like law or marketing, everyone gets "a chance" with mathematics, but the subject is unforgiving and will try its hardest to throw you off. Furthermore, the learning process is always active, and even with the best instructors, you have to be self-sufficient with it.

Ian spends a few pages talking about all the interesting topics in mathematics, advancing from discussing applications as he'd done before. He then advances to discussing mathematics philosophically, which is fun for about five minutes. Mathematics is an infinite set which is provably impossible to completely, undeniably, rigorously define. This is such that an alien species might have correct mathematics, just as we do, but only understand completely different parts of it, and of course, in completely different notation. There are some nice spiritual ideas about what mathematics is. The Platonists think ideal mathematical objects like spheres exist in some abstract universe, and like Erdős' idea that God has a book of the most elegant proofs.

One of my favourite quotes in the book is about university. "For the first six months you'll wonder why the school ever let you in. (After that you'll wonder how some of the others were let in.)" I have not been to university, so I can't comment on its accuracy yet, but I might offer a verification, or a disproof by counterexample, next year.

I'd say the most interesting chapter is chapter 6: 'How Mathematicians Think'. Ian lays out how Jacques Hadamard and Henri Poincaré presented mathematical thought processes in a few stages.

1. Consciously battle the problem by picking off simple examples. Lay out the possible approaches, and prepare visual images or other aids to help you spot patterns.
2. Just as the problem appears to be impossible, incubate it in your subconscious mind for however long is necessary, until you have a sudden revelation and discover the idea which makes the problem appear "trivial".
3. Verify that your idea works, and use it to finish solving the problem.
4. Write a fully symbolic and rigorous proof, or whatever is necessary for your solution. Unfortunately, in this process, the illumination of the subconscious is obscured, ironically making it difficult for readers to understand how you came to the solution.

At my level, I don't spend as much time in the subconscious stage as researchers. Even at university, most problems are either almost entirely methodical, or will reveal their weaknesses at some point in stage 1. This is only fair, as examinations have a time limit, and you can't expect your subconscious to do too much when it's given just five minutes and a trip to the toilet, although I've experienced those revelations sometimes. In any case, you know when the magical revelation happens, and it's a glorious feeling. Truly, "everything is either impossible or trivial".

Proofs are the core of proper mathematics for at least two reasons. The practical-minded argument is that proofs are needed to ensure we aren't wasting our time on "incorrect" mathematics, and so engineers' bridges won't split in half. The proper reason is that the logic of a proof can be very interesting and elegant, and it'd be a shame to let a conjecture go unjustified, even if the computer can say it's true for all 'n' up to 10^80. The most elegant proofs can be found in Erdős' God book, and mirror great verbal stories. I find computer-assisted proofs interesting, and having a programming language for proof writing, checking every line is valid, is the most fitting thing ever. Unfortunately, the homogeneity of computer-assisted proofs means some of the creativity, story, and personal touch of the writer are lost. The more brute-forcey a computer-assisted proof gets, the longer and harder it is to read, and the more elegance is lost.

By chapter 14, the person Ian is hypothetically writing to is starting their PhD programme. It's unlikely that I'll ever pursue a PhD myself, so the remainder of the book isn't directly relevant to me, but it was informative to read a brief overview of how mathematics in modern academia works. The one big question which is relevant to me is "Pure or Applied?", and assuming I'm not going into pure maths research, it will have to be (primarily) applied. Of course, "pure maths" comes up everywhere in "applied maths", and the two sets are both loosely defined and probably *shouldn't* be defined, but they are for convenience. Anyway, I don't like how Ian approaches this topic because he spends too much time talking about mechanics and the woes of the people who do it. Ian inadvertently makes the "Pure Vs. Applied" debate seem like a "Proper Maths Vs. Mechanical Engineering and Some Physics" debate. In reality, I'm probably more of an applied (statistics, computer science...) person, but in the debate Ian describes, I'd probably be a dogmatic purist.

The last few chapters also show another problem of the book, that it's primarily written for an American audience. This isn't a serious problem, but it does make the last few chapters even less relevant to me. Ian presents a few amusing stories from mathematicians, including himself. For instance, Ian claims there was "a prominent professor of number theory who had the habit of turning up at the start of a visiting speaker's seminar, falling asleep within minutes of the talk starting, snoring loudly the whole way through, and then asking penetrating questions by the end when the audience's applause woke him up". Ian goes on to give some good advice about working with other people. Don't trust them to make the right decision, even when you've told them the correct information and what decision to make, and they agree with you. There's a time in the process when people are actually paying attention, and a debate can be swayed either way. Make your point too early, and people will forget about it. Make it too late, and nobody will care. I don't mind the prospect of playing these social games in general, but it seems ridiculous to have to do it as an academic researcher.

In all, the book provides some valuable guidance, but I'm not sold on the format. Ian could've written a more detailed and focused book for either secondary schoolers, undergraduate students, or postgrads in half the words each. Of course, Ian expects the reader to read through the book over the span of some days or weeks, not a lifetime, so he makes each section approachable for everyone, but the format's flaws can't be entirely ameliorated. Around the start of the book, Ian provides lots of good ideas of what you can do with mathematics, and what careers you can go into, but as expected, he assumes the person he's writing to takes the academic route every step of the way. The insight Ian gives into academia is surface-level, but valuable nonetheless, but also enough for me to decide I want no part in it. Unsurprisingly, I personally found the book hit its stride in the chapters which are theoretically targeted towards undergrads and those entering university. Fortunately, in this section, Ian references a plethora of intriguing material which should be more focused than his book, and I'll be sure to read some of the books he mentions. Ian clearly planned ahead to hear the complaint I made so I can't blame him much.