"Let the whole outside world consist of a long paper tape…"
John von Neumann 1948
Thought experiments are the metaphysician's primary tool. Limited only by logic we can test metaphysical hypotheses at any possible world we can imagine. But imagination has its limits.
Can we imagine two possible worlds, each entirely empty except for a single object; the difference being only that one object is stationary whereas its counterpart moves?
In possible worlds containing only two particles perpetually moving relative to each other but never colliding, must there be a determinate answer to what would happened if they did collide?
If the actual world were changed at the moment of the Big Bang just enough so that Romney would have won in 2012, would his opponent still have been Obama?
Much can hang on how we answer questions like these but answers are not easy. The possible worlds in question seem either too poorly furnished or too cluttered with bewildering detail for imagination or argument to get a grip.
In this post I want to introduce a new device for thinking about metaphysical problems; they are a kind of philosophical workbench on which we can test competing metaphysical theories. I call them "Turing Worlds".
In this post I am not going to use Turing Worlds to advance any very radical metaphysical thesis. Instead I'm going to show how some well know problems play out in these worlds. Think of it as an exercise of calibrating a new tool against known quantities.
Roughly: a Turing World is a possible world whose workings are entirely comprehensible as the operation of a Turing Machine.
I will assume all readers have got the gist of Turing machines. Here is a particularly nice example.
The video shows an actual machine. That is, it shows a machine embedded in the actual world and governed by actual physics. This requires there be a lot of hidden electronics behind the scenes that we would have to understand to really understand how it works. It is only a superficially simple machine with complex inner workings embedded in our complex world.
To build a Turing world we want to abstract away as much of the complexity as we can. So, to a first approximation, we can think of a Turing world -- as Von Neuman instructs-- as a tape; infinite in length; divided into cells; each cell occupied by a zero or one.
At any given moment there will be one cell whose contents are directly relevant to what happens next. Intuitively, this is the cell that is under the scanner but we can do without the apparatus of the scanner itself: in my illustration I have substituted a blue glow.
The scanner position changes over time; moving left or right relative to the tape but never moving more than one cell at a time. Before the scanner moves it prints either a "1" or "0" on the selected cell. Sometimes this changes the contents of that cell, sometimes not. Sometimes the scanner prints a symbol but does not move.
Figure 1 depicts the sort of world Von Neuman described but — with all due respect to that great man— there must be more to a computable world than what we see here. In a Turing machine what happens next doesn't just depend on what symbol is under the scanner; it also depends upon the machine's computational state. Computational states are features of the world that vary over time in systematic ways that can result in something different happening even when the same symbol is at the scanner.
To build Turing worlds we will have to give them computational states. It is easy to imagine what the computational state of a Turing world might be-- think of the kind of hidden machinery that drives the machine in the video. It is far less obvious what a computational state metaphysically must be. For the purposes of this post I am going to assume that the computational state of a world at any moment must at the very least correspond to some intrinsic property of something in that world at that time; some property that changes over time.
Staying as neutral as I can about what those properties might be I'll build this into my illustrations of Turing worlds with a simple label as in figure 2, illustrating worlds in states A and B.
With states in the picture we can describe what happens in a Turing world will in terms of the transition rules of that world. Here is a table of transition rules for a very simple Turing world.
The state diagram on the right describes the same transitions as the table on the left. The machine/world depicted here has two states, 'A' and 'B'. The table tells us, for example, that if in state 'A' when a '1' is under the scanner cell, the machine will do '0→B': that is, it will print a '0', move the tape to the right and move to state B. The '↓' symbol means the tape does not move.
The transition table for an actual Turing machine like the one in the video gives an abstract description of its workings but the reason the actual machine instantiates one table rather than another— indeed, the reason it works at all—depend on its underlying machinery. Tinkering with that machinery can change the table that describes it.
Which brings us to the signal difference between a Turing World and a world that merely has a Turing machine in it. I stipulate as a matter of definition:
The transition table that describes a Turing World must be entailed by the physical laws of that world.
One way of framing this requirement would be to say that the transition rules of a Turing world are it's laws. I myself am happy with this way of putting things, but this formulation might be contentious.
There are metaphysical theories that require laws, properly so called, to be rooted in particular kinds of intrinsic properties of things, or of universals, or in relations between universals. Whether or not that is so is a question that I think Turing Worlds might help us figure out. But that is a good reason not to beg such questions from the outset. Better, instead, to say, more cautiously, that the transition table of a Turing machine describes certain regularities and stipulate— as (D1) does — that a Turing World corresponding to a particular transition table will be a world where those regularities are nomological. That leaves us free to argue later on about precisely what 'nomological' might mean.
While (D1) is neutral on the question of whether Turing transition rules could be the basic laws of any world, I will continue to depict the ontology of Turing worlds in the austere manner of Figure 2 and talk as if their transition tables exhaust their physics. If this leads to disagreement down the line we will have learned something about the upshots of our different assumptions about the laws.
I take it as uncontroversial that there can be and in fact are physical processes which are a) describable in computational terms b) proceed as they do as a matter of physical law. The encoding of proteins by DNA is an obvious example.
With these caveats in place we should all be able to agree that Turing Worlds are metaphysically possible. They are real possible worlds in whatever sense possibilities are "real". That means that generalizations over possible worlds of the sorts that metaphysicians are wont to give ought to apply to Turing Worlds and we can test metaphysical hypothesis by seeing if we can construct Turing Worlds that refute them.