God is always doing geometry. What Plato meant by this is no clearer now than it was in the first century A.D., when the dinner conversation quoted above (which I'm about to paraphrase) took place. Diogenianus, one of the guests, recalls that it is Plato's birthday and proposes to “bring him into the conversation” by debating the meaning of this nugget—“if indeed the statement is Plato's.” Plutarch, who knew a lot about such things, replies that although the quote does not appear explicitly in any of Plato's books, it is well enough attested, and it is in character.

There follow the opinions of the three remaining guests, and finally that of Plutarch. Drastically abridged, they are:

• • Geometry focuses the mind on the abstract rather than on the sensorial.

• • Geometry is divine in that “geometric proportion … befits a moderate oligarchy or a lawful monarchy … it distributes to each according to his worth,” whereas arithmetic proportion is egalitarian.

• • The essence of geometry is boundaries, so God does geometry when he bounds matter to create.

• • Geometry intervenes when proportion and measure and number are used to order chaotic nature.

Although encompassing, these are not entirely persuasive explanations; make of them what you will. I bring them up largely for fun, but also to illustrate the breadth of interpretation that the notion of geometry has traditionally enjoyed, and so to defend my choice of title for this volume. One might consider also Klein's definition in his Erlangen program.