Platonic-Physical-Mental worlds - Penrose

Roger Penrose has written an astounding book: The Road to Reality. Not content with asking the reader, and apparently he expects non-scientists to read this book, to learn about quantum mechanics and general relativity (and all of their modern descendants, worthy or not), he spends the first 350 pages introducing enough mathematics to get you most of a way through a degree in the subject. He wants to show that descriptions of science can amaze, but science itself is an awe-inspiring, humbling, and humanising endeavour.

His discussions of everyday numbers and geometry show that little of what we take as being “real” can be considered as such. But his task is to provide a road to reality (the “road” is to use Platonic objects and logic), so after dismantling our universe he reconstructs it with sounder building blocks (Platonic objects) and cement (logic).

Here, for example, is how he treats numbers. I don’t think that there is anything new in the following, but I’ve never seen it explained so well in a single place. Obviously what follows is a summary of his work, and I recommend that you borrow or buy a copy of the book and read it for yourself.

Since we cannot be sure that physical objects around us have any true identity or permanence, we need another way to construct the natural numbers 0,1,2,3, . . . with which we count. Cantor showed how to do this using Platonic objects. First consider an empty set. That’s right; a collection with nothing in it. This is usually called the null set and denoted by the Greek letter phi, Φ. Any set can be defined by listing its objects, and the usual notation is to do so between braces: Φ = { }. The null set has no objects inside it: we use it to define 0. Now consider this set: {Φ}. This is the set with one object inside, the null set. This defines the number 1. Next, we look at {Φ, {Φ}}. This set has two objects inside and defines the number 2. We can go on like this forever, defining all the natural (counting) numbers without ever referring to anything not in the Platonic world.

Hooray. Once we have the natural numbers, it’s easy to extend to the integers, namely all the whole numbers positive and negative. All we have to do is define a “sign” and then how to manipulate things with either a positive or negative sign.

Huzzah. The ancient Greeks knew how then to deal with rational numbers, those numbers formed from a ratio of two integers. We think of the concept of a fraction as obvious, as being one number divided by another, but it is not at all obvious, nor is it immediately clear what this means in Platonic terms (see the preface of Penrose’s book, for example). But it was Eudoxus who first wrote down the rules on how to order, add and otherwise manipulate the rationals. Especially important, in the context of our next paragraph, is the ordering of the rationals so that we know of any two rationals which is larger than the other.

Spiffing. The next leap is brilliant, and can only be hinted at in vague terms here. Dedekind gave a formal procedure for defining (although not constructing) the irrational numbers. The irrational numbers can be talked about in physical-world terms as those numbers whose decimal expansion never repeats or terminates. You cannot write such a number as the ratio of two integers (hence the name). His procedure can be visualised as first laying out all of the rationals in a long line from negative numbers (as big as you like) on the left to positive numbers (as big as you like) on the right, with 0 in the “middle” (this is why ordering is so important). Now take an infinitely sharp knife and cut the line anywhere you like. If it makes a cut on a rational number, fine, try again. If not, then you have fallen between two rational numbers, and the set of rationals to the left of the cut has no largest member and those to the right no smallest. In fact, you have defined an irrational number. Although I have talked about laying out the numbers and slicing up the line, the procedure is purely formal: it exists entirely in the Platonic world.

These types of numbers - the integers, rationals, and irrationals - taken together form the real numbers. Everything is purely Platonic up to now. But here is the deep mystery: reals exist in the physical world too. For example, the calculus, describing changes of quantities in the real world, rests upon the existence of reals. So the mind-blowing thing is that the purely Platonic contains within it the Physical, which contains within it the Mental world, which somehow contains within it the Platonic. If that doesn’t make you fall out of your chair, read it again until it does!