I'd say those first few chapters are far and above the best example I've seen of a mathematician getting across the essence of pure maths and why it appeals to them. Unfortunately, though, from then on the book gets bogged down in the problem that almost always arises, that what delights and fascinates mathematicians tends to raise a big 'So what?' in the outside world.
Neale interlaces attempts getting closer and closer to the conjecture, working down from a proof of primes several millions apart to under 600, adding in other, related mathematical work, for example on building numbers from squares and combinations of primes, but increasingly it's a frustrating read, partially due to necessary over-simplification. Time and again we're told about something, but effectively that it's too complicated for us to understand (or we'll come back to it in a later chapter), and this doesn't help make the subject approachable. I understand that a particular mathematical technique may be too complicated to grasp, but if so, I'm not sure there's any point telling us about it.
Part of the trouble is, most of us can only really get excited about maths if it has an application - and very little of what's described here does as yet. I'm not saying that pure mathematics is a waste of time. Not at all. Like all pure research, you never know when it will prove valuable. Obscure sounding maths such as symmetry groups, imaginary numbers and n-dimensional space have all proved extremely valuable to physics. It's just that while the topic remain abstract, it can be difficult to work up much enthusiasm for it.
At the beginning of the book, Neale draws a parallel with rock climbing, and that we are to the mathematicians scaling the heights like someone enjoying a stroll below and admiring their skill. And, in a way, this analogy works too well. We can certainly be impressed by that ability - but a lot of us also see rock climbing as a waste of time and consider it as interesting if you aren't actually doing it as watching paint dry.
It's not impossible to make obscure mathematics interesting - Simon Singh proved this with Fermat's Last Theorem. But that was achieved with writing skill by spending most of the book away from the obscure aspects. I'm beginning to suspect that making high level mathematics approachable is even more difficult than doing that maths in the first place.
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