Notes on Chapter 1, The Road to Reality, by Roger Penrose.
(Introduction)Thales of Miletus:
Many sites on the web attribute all kinds of achievements to Thales — for example, that he considered the earth to be a sphere — but this great article combs primary sources, and notes:

There is a difficulty in writing about Thales and others from a similar period. Although there are numerous references to Thales which would enable us to reconstruct quite a number of details, the sources must be treated with care since it was the habit of the time to credit famous men with discoveries they did not make. Partly this was as a result of the legendary status that men like Thales achieved, and partly it was the result of scientists with relatively little history behind their subjects trying to increase the status of their topic with giving it an historical background.

Roger Penrose doesn’t quite say that Thales invented the mathematical proof, so he’s off the hook. Thales is the first recorded person to replace existing supernatural explanations for events with natural ones, though: He proposed earthquakes happen because the earth floats on water, rather than because there exist angry gods. Thus began the long retreat of mysticism and religion as authorities for exposition. (A shorter biography of Thales.)

Pythagoras of Samos:
Another great article by J J O’Connor and E F Robertson, this time about Pythagoras. They quote Aristotle:

The Pythagorean … having been brought up in the study of mathematics, thought that things are numbers … and that the whole cosmos is a scale and a number.

Now that theoretical approaches which see the universe as a giant quantum calculating machine are in the ascendant, it may turn out that humanity’s very first naturalist guess was a remarkably good one.

Mathematical Platonism:
Penrose has argued previously, notably in Shadows of the Mind, in favor of mathematical platonism — the view that mathematical notions (numbers, primeness, the Mandelbrot set) have an existence that is independent both of the physical world and of our mental world. This is not a controversial view for mathematicians; almost all are mathematical platonists, first and foremost intuitively. But what does mathematical platonism mean to Penrose?

It may be helpful if I put the case for the actual existence of the platonic world in a different form. What I mean by this ‘existence’ is really just the objectivity of mathematical truth.

Penrose illustrates all this with an earlier image from Shadows: He sees each of these three worlds as emanating from an other, to produce an Escher-like paradox. In it, our mental states are represented by a subset of states in the physical world, the physical world is controlled by a subset of mathematical notions, and we use a subset of our mental world to grasp these mathematical conceptsFor Road, he further modifies his illustration to address the possibility that 1) not all of the mind is represented by a physical state, leaving room for a soul, 2) there exist true mathematical notions that are not accessible to our reason, and 3) that there are physical processes that lie outside the realm of mathematical control..

It’s a snug diagram. My main reservation is that the link drawn from the mental world to the mathematical notions it can grasp is not “generative,” like the other two links. We can perceive mathematical notions, but this perception does not generate them in a way that is analogous to how physical states can generate mental processes — mathematical notions are not dependent upon our perception for their existence, at least not if you are a mathematical platonist.

My own prejudices tend towards the physical world merely being a “concretized” rendition of the mathematical world, rather than separate from it. I suspect that at the most basic level, physical reality is exactly defined by fundamental mathematical processes, and the universe really is constructed from such “platonic shapes”. Penrose seems to be saying something similar, though he also appears embarrassed by this early outbreak of metaphysical supposing in his first chapter, and so heads off into 360 pages of mathematics, starting with chapter two.