last posted on it, the latest of which is on string theory.
But I came across a 1000 page book by Roger Penrose that has distracted me: The Road to Reality. Penrose is a 80+ year old Oxford physicist who worked in the area of cosmology, not least as a mentor to Stephen Hawking. What attracts me about the book is the way that he blends the expansion of mathematical understanding with the expansion of our knowledge of physics.
1. The first chapter is called, "The roots of science." I won't go into much detail but he basically is making an argument that mathematics is objectively true. Math is not a matter of opinion. Fermat's Last Theorem is not true because Andrew Wiles came up with a proof that was pleasing before someone came up with a disproof that was pleasing. Fermat's Last Theorem was true before Fermat came across it, and it would be true even if no one had yet produced a mathematical proof.
The way Penrose expresses this idea is by affirming Plato. Plato believed that there was an independent reality to ideas apart from the concrete instantiations of them in the world. Penrose isn't wanting to make a big deal of this. He wants us basically to see this as a way of saying that math is objectively true. In the real world, we have approximations, but mathematical ideas are real in a way.
I would personally rather go Aristotle on him. Math is an abstraction of the real world. Math is objectively true but as an abstraction of the concrete world. One corresponds to one thing. Two to two things. Multiplication is multiple addition. Division is multiple subtraction. Exponents are a particular kind of multiplication.
We use a base ten coincidentally because we happen to have ten fingers, but this is just a way of talking about reality. "Natural math" is probably more based in e or pi. These last two paragraphs are my thoughts rather than those of Penrose.
2. The last part of this first chapter presents his sense that the whole of our mental world comes from the physical world, and the whole of math comes from our mental world, and the whole of the physical world comes from the mathematical world. He leaves room for the possibility that there may be left overs. Some of our ideas may be distinct from the physical world. Some of the physical world may be apart from math. Some of math may be independent of our mental world.
Interesting, although not why I bought the book yet.