Grid Worlds

What is this?

An invitation to explore Humean Supervenience.

Humean Supervenience?

Take a look at Small World above.

Small World is a 15 by 15 grid of cells. Each cell is either on or off. The distance between two cells is the minimum number of steps between them, traveling up, down, left, or right. Time is discrete: t=0, t=1, t=2, ... Each frame shows the state of the cells at a point in time. The frames together show Small World's mosaic, from its Big Bang to its Big Crunch.

Humean Supervenience about Small World is the thesis that every question about Small World is settled by its mosaic.

But that's obviously true.

Is it?

Here are some questions about Small World:

• Is the central cell on at t=7?
• Are all the cells ever on?
• Is any cell always on?

These questions are settled by the mosaic. If these were all the questions about Small World, then Humean Supervenience about Small World would be obviously true.

But there are more. For example:

• If the top-right cell had been on at t=0, would the bottom-left cell have been on at t=1?
• Is it a law of Small World that if a cell is on at t then it's on at t+1?
• Is Small World deterministic?
• Why was the top-right cell on at t=7?

Humean Supervenience about Small World says that these questions too — in fact, all questions — are settled by Small World's mosaic.

But that's obviously false.

Is it?

Enter David Lewis. He tries to show how every question about Small World is settled by its mosaic. So he defends Humean Supervenience about Small World.

Except not really. What Lewis actually does is defend Humean Supervenience about our world.

[Humean Supervenience about our world] says that in a world like ours, the fundamental relations are exactly the spatiotemporal relations: distance relations, both spacelike and timelike, and perhaps also occupancy relations between point-sized things and spacetime points. And it says that in a world like ours, the fundamental properties are local qualities: perfectly natural intrinsic properties of points, or of point-sized occupants of points. Therefore it says that all else supervenes on the spatiotemporal arrangement of local qualities throughout all of history, past and present and future. — Humean Supervenience Debugged, Mind, 103: 473–90

Every question about our world, says Lewis, is settled by the spatiotemporal arrangement of local qualities, aka our world's mosaic.

How does he argue for this?

At length. (See David Lewis's Metaphysics for a survey and references.) But with too few examples.

Thus this tool?

Yes. '"What is the use of a book," thought Alice "without pictures or conversation?"'. And what is the use of a theory without examples? Maybe Grid Worlds can provide them.

Humean Supervenience about our world is a more interesting claim than Humean Supervenience about Small World, because there are more, and more interesting, questions about our world than about Small World. (It's a small world after all.) Still, there are interesting questions about Small World, like those above. And showing how questions about Small World are settled by Small World's mosaic may illuminate how questions about our world are settled by our world's mosaic — according to Lewis. Small World, and other such grid worlds, are small enough to work with, but big enough to make the work worthwhile.

So let's be resolutely concrete. No discussion of Humean Supervenience without a grid world to illustrate!