Sir Roger Penrose, distinguished Mathematician and Physicist, prolific Author, and perhaps somewhat confused philosopher of consciousness has written a quite remarkable book.

The Road to Reality : A Complete Guide to the Laws of the Universe attempts to cover general relativity, quantum mechanics and the modern theories that attempt to unify them both (i.e. String theory, Loop Quantum Gravity & etc.) -- starting from elementary mathematics! His course in mathematics is the first third of the book, but note that he manages to finish-off complex analysis in thirty pages. The Amazon review calls the book "thrillingly difficult". I doubt if I'm going to do more than skip around in it, but I've enjoyed the first 200 pages so far.

Sir Roger's major purpose, I think, is to explore the remarkable coincidence that mathematics is so useful to describe the physical world; he leaves no brick in his edifice unexamined. At the outset, for example, when introducing the real numbers (!) and after commenting that the Pythagoreans assassinated those who admitted to the existence of irrational numbers, he examines the question of whether the Reals are needed to describe the physical universe. If spacetime is quantized, then (for instance) the diagonal of any square is some (exact!) number of planck length -- and the side times the square root of two is only an approximation anyway.

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This was an argument of R. Buckminster Fuller. He coined the term Scheherezade Numbers as the very large integers made up of products of primes such that whaever maths you needed would result in (usually) an integer result. I don't remember all of it (and didn't understand all of it when I read it), but he also argued against other mathematical conceits such as lack of dimensionality (when we draw a triangle, it still has depth, the molecules of graphite or ink making it up, and is really a 3D spiral, not a 2D triangle).

All drawn from my hazy recollections of his book, Synergetics.

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