In my article Where to apply Wolfram’s rules? I introduced a radical idea.
When we’re applying a rule to a graph in Wolfram Physics, there are generally many possible places in the graph we could apply the rule.
So when we’re applying this rule:
to this graph:
we can apply it in 2 different ways in any of 46 different places:
So that’s a total 92 different ways to apply the rule to the graph, each resulting in a different next state of the universe.
Here’s the radical idea: rather than choose one of those 92 possible universes, we choose not to choose. We have no reason to prefer any one of those 92 possible universes over any other, so we don’t prefer any one of them over any other. Instead, we keep each of them in mind.
The trouble is, when we apply the rule to the graph again, we’re going to have to apply it to all 92 possible universes. In each case, we’ll be able to apply the rule in another 92 or so different ways, yielding another 92 or so possible universes for each of the 92 universes we’ve been holding in our mind. So now we’re keeping in mind as many as 92 × 92 = 8,464 different universes.
And when we apply the rule to the graph again, we’re going to have to apply it to all 8,464 possible universes. In each case, we’ll be able to apply the rule in another 92 or so different ways, yielding another 92 or so possible universes for each of the 8,464 universes we’ve been holding in our mind. So now we’re keeping in mind as many as 92 × 8,464 = 778,688 different universes.
The number of possible universes we have to keep in mind gets extremely large extremely quickly.
To help us visualize all these possible universes, we’re going to need the multiway graph.
Who’s who
I’m not sure there’s room on this screen to draw a multiway graph of 778,688 different universes.
So let’s start small.
To apply that rule I’ve been using, we find two edges from the same node, delete one of the edges, then create three new edges from the three existing nodes to a new node:
Here’s the simplest universe that matches the rule:
You can see that I’ve labelled the nodes A, B and C. How we identify nodes is going to prove surprisingly important.
There are two ways we can apply the rule to this simple universe.
The node I’ve labelled A – the one the two edges come from – is the only possible match for node 1 in the rule. But there are two different ways we can match nodes 2 and 3.
We could match node B to node 2 and node C to node 3:
or we could match node C to node 2 and node B to node 3:
OK, I know what you’re thinking, what does it matter whether we match node B to node 2 or node 3? what does it matter whether we match node C to node 3 or node 2?
I mean, seriously, how pedantic do we want to be here?
Well, let’s see what happens with these two ever-so-slightly different matches.
Remember, the rule stipulates that we delete the edge from node 1 to node 2, and keep the edge from node 1 to node 3.
So if we match node B to node 2 and node C to node 3, we end up deleting the edge from node A to node B:
But if we match node C to node 2 and node B to node 3, we end up deleting the edge from node A to node C:
Here’s where we end up once we’ve created the three new edges from the three existing nodes, A, B and C, to a new node, D.
If we matched node B to node 2 and node C to node 3, we end up with this graph:
You can see that the edge from node A to node B has been deleted, but the edge from node A to node C remains.
But if we matched node C to node 2 and node B to node 3, we end up with this graph:
In this case, you can see that the edge from node A to node C has been deleted, but the edge from node A to node B remains.
I know, I know. You’re probably thinking: how different are these two graphs, really?
That’s a crucial question. Are these two different universes? Or are they the same universe?
Topologically, they’re the same. I mean, they both have three edges going to a central node, and one more edge between two of the three peripheral nodes. No difference.
Except... well, remember my mentioning that how we identify nodes is going to prove surprisingly important? My labelling the nodes A, B, C and D makes these two universes different. In one universe, node A is connected to node B; in the other, node A is connected to node C:
Think of it from the point of view of the nodes. Imagine a universe in which Adam is married to Brittany:
Now imagine a universe in which Adam is married to Claire:
Are these the same universes? Well, not from the point of view of Adam, Brittany and Claire, they’re not.
People are not interchangeable. Brittany and Claire are not the same person. Being married to Brittany is different from being married to Claire.
And nodes? Are nodes interchangeable?
Here’s one way of thinking about it. The nodes and edges of the graph are space. If nodes were interchangeable, that’d be like different regions of space being interchangeable. On a large scale, at least, that simply isn’t the case in our universe: Bangkok and Chicago are two different cities; Brazil and Cambodia are two different countries; Betelgeuse and Canopus are two different stars. One region of space is not the same as another region of space.
In Wolfram Physics, nodes are not interchangeable. Node B and node C are two different nodes.
Which means that the two universes we arrived at after the first application of the rule are different:
Many worlds
So here’s our first multiway graph:
Here’s what this multiway graph is telling us: if we apply our rule to the universe at the top, we’ll end up with one of the two universes at the bottom, depending on which way we apply the rule.
And here’s what the multiway graph is not telling us.
It’s not telling us which of the two universes we’ll end up with.
Indeed, we’re not going to choose between the two universes. Remember our radical idea: rather than choose one of these two possible universes, we’re going to keep both of them in mind.
And here’s something else the multiway graph is not telling us.
It’s not telling us that the universe has bifurcated into two separate universes. This is not like the many-worlds interpretation of quantum mechanics, which does suggest that at every decision point, the universe splits into two or more parallel universes.
As we’ll see, different paths through the multiway graph can come back together. This makes it quite unlike the many-worlds interpretation of quantum mechanics, in which the parallel universes remain forever disconnected.
All the multiway graph is telling us is that if we apply our rule to the universe at the top, we’ll end up with one of the two universes at the bottom, depending on which way we apply the rule.
Endless possibilities
So let’s apply the rule a couple more times.
Since we’re keeping two possible universes in mind, we’re going to apply the rule to both of them:
For each of these two possible universes, there are, again, two ways we can apply the rule:
giving us four different next states of the universe:
As before, we can count four different universes only if we assume that nodes are not interchangeable.
The middle two graphs above, for example, are topologically the same, but the nodes are in different places. In the graph on the left, it’s node B that’s out there on its own, whereas in the graph on the right, it’s node C that’s out there on its own:
Interestingly, though, we do now have topological differences between our four universes. The first two graphs above, for example, are topologically different. In the graph on the left, the node that’s out there on its own, node B, is connected to a node, node D, with two other edges (the edge from node C and the edge to node E), whereas in the graph on the right, the node that’s out there on its own, again node B, is connected to a node, again node D, with three other edges (the edges from nodes A and C and the edge to node E). So the two graphs are topologically different:
Adding the four possible next states of the universe gives us a three-level multiway graph:
Let’s go one level deeper.
For each of the four possible universes we’re now keeping in mind, there are four ways we can apply the rule:
giving us sixteen different next states of the universe:
Adding these sixteen possible next states gives us a four-level multiway graph:
Collapsing consciousness
What is the multiway graph?
It’s a way of visualizing all the possible universes we have to keep in mind if we choose not to choose between them.
It might seem perverse not to choose. Even with the smallest of universes, the number of possibilities we have to keep in mind gets extremely large extremely quickly.
Wouldn’t it be simpler to choose a single path through the multiway graph? In other words, wouldn’t it be simpler, at every step, to choose one of the possible ways to apply a rule rather than consider them all?
Well, yes, it would be simpler... but that might not be the way our universe works.
By choosing not to choose, we’ll be able to derive aspects of quantum mechanics from Wolfram Physics.
And we’ll arrive at a concept of the observer that promises to resolve issues related to the collapse of the wavefunction that have plagued quantum mechanics ever since Schrödinger put his metaphorical cat into a metaphorical cage.
And maybe, just maybe, by choosing not to choose, we’ll arrive at a model of consciousness itself.