Why “Platonic”?

Penrose calls one of the three worlds in the theory of three worlds as Platonic. Why?

Plato

The rich Athenian citizen Plato was a follower of the philosopher Socrates. He set up a school of philosophy in the 4th century B.C., which was fundamental for European philosophy and has crucially shaped philosophical discussions until the present day. If Roger Penrose thus calls one of the three worlds “Platonic”, he refers to Plato and specifically to one particular question and the discourse about it, which is still of great significance today. This question is, “Are ideas real?”

Plato’s realism of ideas

Subsequent philosophers often presented the issue as a conflict between Plato and his disciple Aristotle. Plato is ascribed the attitude that ideas were not only real but even constitute actual reality, while what we describe as reality were mere shadows of the original ideas. Penrose calls the abstract world of mathematics “Platonic”, thus referring to Plato’s thought that abstract ideas had the quality of reality.

The world of ideas as one of the three worlds

Of course the Platonic reality of ideas was also disputed, and Europe’s history of philosophy is full of pros and cons about it, which under the headings of realism, nominalism and the problem of universals have shaped the philosophers’ discourse for many centuries and exert an influence in the background even now. Like Plato, Penrose’s theory attributes a reality to the abstract Platonic world, but not an exclusive reality, as would an uncompromising Platonic realism, but as one of the three real worlds, which interact with each other. Thus the theory of the three worlds is not about which world is the real and true one – which was discussed at length as the problem of universals – but about how the interaction takes place between them.

But let’s return to the Platonic world. What distinguishes it from the other two worlds?

Characteristics of the Platonic world

 Nonlocality

Where is the number “3”? Can you point your finger at it somewhere in your environment?

Of course you can point at three apples, at three pencilled lines or at three coffee cups, but this is not the number three; rather, they are apples, pencilled lines and coffee cups. The number three remains abstract. No one can point at it.

Naturally, you can also point at the word “three” or at the “3” in this text, but these are only symbols of the number and not the number itself. The number itself remains abstract; it simultaneously exists everywhere and nowhere.

Symbols are always in a certain place, they are thus localised. The number itself, however, is non-local, i.e. there is no place in the universe wh  ere the number is found; rather, it can be found everywhere. It exists on earth, on the moon and equally in Andromeda. This nonlocality is a very elementary property of the objects of the Platonic world; in particular, it distinguishes them from the objects in the physical world, in which objects are locally defined, i.e. localised.

Timelessness

 The Platonic world’s relationship with time is analogous to locality:

1 plus 2 is 3 – this is true now, was true yesterday and will be true tomorrow and for ever thereafter. In this sense, we can describe the Platonic world as a place of eternal truths, in stark contrast to the physical world, which is subject to constant change. If it rains today, the sun may shine tomorrow; 1 plus 2 is 3 every day and any day. This timelessness applies to all mathematical statements, but also to their objects, again in contrast to the objects of the physical world: the number 3 is timeless, whereas 3 apples are not.

This is a text in the series about the theory of the three worlds