This is a book in the 'compact collection of science/maths essays' category beloved of publishers because they can charge hardback prices for fewer than 80 pages of text - but don't let that put you off, because it's excellent. Terence Tao introduces six areas of mathematics: numbers, algebra, geometry, probability, analysis and dynamics. Each of these is descriptive rather than practical (there are only a couple of school-level equations in sight), done in a way that gets to the heart of 'what's the point of this subject' with precision and a good level of storytelling. It's a shame in a way that analysis and dynamics form the final third of the book (apart from a short epilogue), because they don't work as well as the other sections. For the first four, practically everyone will have a vague idea of what they are and what they do - but Tao is then able to open our eyes to their wonders and power. Because Tao has to stay away from any detail of what...
A quote on the back cover of this book describes it as monumental - it certainly feels big enough to use as part of a structure with 664 pages plus a further 112 pages of end notes. Colin Gillespie's aim is to 'bring together centuries of ancient thought and modern insights to unearth what is real and what is not'. That certainly sounds like it would need a lot of pages, if it is to be achieved at all. A starting point is to establish what 'real' means. Gillespie gives us the guidance that 'to accept something as truly real, it must be an accurately described objective aspect of the universe.' That may seem obvious, but as he points out 'little in philosophy or physics even aims to pass this test.' Gillespie builds a new conception of physics based entirely on space. As he puts it 'space isn't something; it is literally everything.' In this worldview, the universe began with a single 'space quantum' - a Planck-sized six-dimensiona...