In my teenage years, I read Hofstadter's Goedel, Escher, Bach. It blew me away. I looked for another book in its class. Penrose's The Emperor's New Mind looked like it might have been that, and... it wasn't. It was reasonably disappointing, but hey, it was a difficult act to follow.
Rather more recently, I found The Road to Reality in a local charity shop. It's a huge doorstop. I flicked through, thinking 'Hmmm, not another Penrose', but the contents surprised me, very pleasantly. It's a non-rigourous explanation of modern physics, and the maths required for it. It's perhaps a pop science book, but that's really pushing it. The required level is more 'a numerate degree' than even A-level maths!
Anyway, as it's a long book (1000 pages), roughly divided into 'maths required for modern physics' and 'physics', I thought I'd put in a milestone review at the end of the maths section (about 400 pages). The starting point seems simple - Pythagoras' theorem, then complex numbers, a little real number calculus... then quickly we're into complex analysis, Fourier transforms, manifolds, symmetry groups, and before you know it you're on gauge connections! The recap of complex analysis I found fun, Riemann surfaces and their classification was fascinating, and by the time we were going full-on with the tensors I must admit I was getting mostly lost!
It's a breathtaking introduction to so much fascinating maths. The Pythagoras' theorem I mentioned above included an explanation of hyperbolic geometry, and many other topics were presented in ways that are slightly askew from the normal presentation, which gave great insights. Seeing Fourier series as Laurent series when the repeating pattern is mapped onto the unit circle was incredibly helpful to me, and his explanation of 'spin half' particles struck a real cord with me.
Overall, he seems to have a great geometric view of the world, which coincides marvellously with my preferences, which meant that many of his explanations worked really well for me, when they might have fallen flat for others. Having said that, he does seem quite willing to quickly switch to algebra too, and shows plenty of dexterity there. His way of putting physical mathematics into his own style shows a real mastery of the subjects.
For now, though, I think I'll put the reading on hold. I'll let the ideas, in their relatively intuitive form, sink in slowly, perhaps re-read other mathematicians' treatments of these areas (time to break out the books on complex analysis, differential geometry and topology!), and then come back to the last 600 pages a little fresher and more prepared... but I am looking forward to it.