It looked like a deal that I couldn't pass up when I found it in a quaint Viennese bookshop: Roger Penrose's 1099 page The Road to Reality: The Complete Guide to the Laws of the Universe, an encyclopedic account of state-of-the-art physical science, for the bargain price of 30 Euros. (One might have called it, following Gamov: 1, 2, 3, ....1099.)
I greedily snatched it up as, I thought, the analogue today of Aristotle's Complete Works (or at least of the physical treatises thereof). With Aristotle on one end of the shelf, and Penrose's tome on the other, I'd be set; I'd own both the earliest and most recent authoritative surveys of physical science.
(I was doubly wrong about the book, hwoever. It wasn't a bargain: unbelievably, one can buy it new on Amazon for just over $17. And it proves to be rather eccentric in many places, and not comprehensive.)
Naturally at some point I scanned the book for its treatment of philosophers, especially of the ancient variety. Frege, Russell, Leibniz are mentioned in passing. Aristotle is dealt with briefly (but surely wrongly) in an elegant discussion of "Aristotelian" space-time--that is, absolute Euclidean space-time. Pythagoras is praised over a couple of pages as probably having originated the notion of a demonstration, and, of course, for his sense that reality is deeply mathematical.
But to Plato, shockingly, the first 383 pages of the book are dedicated. At least, that's how Penrose describes his preliminary discussion of the mathematical notions required for modern physics: it is, he says, an examination of 'Plato's world':
Let us now take a glimpse into Plato's world--at least into a relatively small but important part of that world, of particular relevance to the nature of physical reality (23).This is 'Plato's world' because:
Plato made it clear that mathematical propositions--the things that could be regarded as unassailably true--referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world (11).Penrose gives Fermat's last theory as an example of the reality and objectivity of this 'world', and then also the Mandelbrot set:
The point that I wish to make is that no one, not even Benoit Mandelbrot himself when he first caught sight of the incredible complications in the fine details of the set, had any preconception of the set's extraordinary richness. The Mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. If it has meaning to assign an actual existence to the Mandelbrot set, then that existence is not within our minds, for no one can fully comprehend the set's endless variety of unlimited complication. Nor can its existence lie within the multitude of computer printouts that begin to capture some of its incredible sophistication and detail, for at best those printouts capture but a shadow of an approximation to the set itself. Yet it has a robustness that is beyond any doubt; for the same structure is revealed--in all its perceivable details, to greater and greater fineness the more closely it is examined--independently of the mathematician or computer that examines it. Its existence can only be within the Platonic world of mathematical forms.This is an idea that does not easily go away. Penrose is no dummy; and I've heard many mathematicians (well familiar with set theory, logicism and intuitionism) say the same thing.