The multiway graph shows every possible evolution of the universe.
So, if we can compute every possible reality, does that mean that there’s no single objective reality?
Well, the causal graph, it turns out, collapses every possible reality into a single objective reality in a way that’s so unexpected that you’ll be left wondering: how did that just happen?
Abstracter and abstracter
So far, in my exploration of Wolfram Physics, I’ve introduced you to three ever more abstract graphs.
I’ve shown you the hypergraph. It is space... and everything in space.
And I’ve shown you the multiway graph. It’s every possible evolution of the hypergraph.
And I’ve shown you the causal graph. It captures the causal connections between every possible event in the evolution of the hypergraph.
Here’s the bad news if you struggle with abstraction.
For the next few articles, I’m going to be focusing the causal graph, the most abstract of the three.
I’ll do my best, whenever I talk about the causal graph, to relate what I say back to the multiway graph and the hypergraph.
Still, you might be thinking: the hypergraph seems concrete, I can imagine the universe as a hypergraph; even the multiway graph doesn’t seem too abstract, I can imagine the universe branching and branching and branching; but why can’t we stop there? why do we have to get still more abstract? why do we have to go to the causal graph?
One answer is that good things will come from it: mass/energy, momentum, special relativity, general relativity and quantum mechanics all emerge from the causal graph.
But there’s a deeper answer.
The hypergraph represents one reality, one universe.
The multiway graph represents every possible reality, every possible universe.
But the causal graph, it turns out, represents a reality that’s the same regardless of which of those possible universes you find yourself in.
In other words, the causal graph represents a single objective reality.
One level deeper
The causal graph I’ve shown you before is no longer going to hack it.
If we’re going to get a glimpse of a single objective reality, we’re going to have to go one level deeper.
I’m going to use the same rule as before:
but I’m going to apply it to every possible state of the universe in every possible way one more time than before.
That gives us a multiway graph that’s one level deeper:
and a causal graph that’s one level deeper:
As you can see, thing get pretty busy pretty quickly when you’re applying a rule to every possible state of the universe in every possible way.
So let’s remind ourselves what we’re looking at here.
One way or another
Let’s zoom in on one region of the multiway graph.
Each of those white hypergraphs is a possible state of the universe.
And each of those red arrows is an application of the rule, taking us from one possible state of the universe to another.
Remember, the multiway graph isn’t just showing one possible evolution of the universe, it’s showing every possible evolution.
OK, so let’s focus on one of those possible evolutions.
Let’s imagine that, from the initial state of the universe, of the two possible applications of the rule, the one that actually happens, for us, is the one on the left:
which takes us to this state of the universe, from which, of the two possible applications of the rule, the one that actually happens, for us, is again the one on the left:
which takes us to this state of the universe, from which, of the four possible applications of the rule, the one that actually happens, for us, is again the one on the left:
which takes us to this state of the universe, from which, of the six possible applications of the rule, the one that actually happens, for us, is once again the one on the left:
That’s one possible path through the multiway graph, one possible evolution of the universe.
Here’s another:
And another:
What we’re doing here is reducing the multiway graph from every possible evolution of the universe to one possible evolution of the universe.
After all, it’s hard to imagine every possible history of every possible universe. It’s much easier to think of ourselves as existing in a particular universe with a particular history.
So what does this reduction of every possible evolution of the universe to one possible evolution of the universe do to the causal graph?
Reductive
Let’s remind ourselves what’s what in the causal graph.
Each of those white oblongs is an event, an application of the rule, taking us from one possible state of the universe to another.
And each of the yellow arrows is a causal relationship, indicating that one event has to happen before another another event can happen.
Again, the causal graph isn’t just showing the causal relationships between a particular sequence of events in a particular evolution of the universe, it’s showing the causal relationships between every possible event in every possible evolution of the universe.
But just as we did with the multiway graph, we can reduce the causal graph from every possible evolution of the universe to one possible evolution of the universe.
Take that first path through the multiway graph:
Suppose we found ourselves in this particular universe with this particular history:
As we looked back through this history, we wouldn’t be all that interested in the events that could have happened, but didn’t; these events on the far side of the multiway graph, for example:
We’d be more interested in the events that did happen, in our universe; these events:
What are the causal relationships between these events, the ones that actually happened in our universe?
Well, that’s a much simpler causal graph:
Let’s take out the multiway graph, leaving just the reduced causal graph:
Now when it comes to the causal graph, it doesn’t matter where we put the events, what matters is which events are causally connected to which other events.
So we can arrange the events in the reduced causal graph to see the shape of it more clearly:
So that’s what the causal graph looks like when we reduce it from every possible evolution of the universe to one possible evolution of the universe.
Meanwhile, in a slightly different universe...
Let’s try another possible evolution of the universe.
Take that second path through the multiway graph:
Suppose we found ourselves in this particular universe, the one next door to the universe we were in before, with this particular history:
Here are the events that happened in this universe:
And here are the causal relationships between these events:
Again, let’s take out the multiway graph:
and arrange the events so that we can see the shape of the causal graph more clearly:
Oh, that’s a coincidence... it’s the same shape as before.
Meanwhile, in a completely different universe...
OK, so let’s take a completely different evolution of the universe.
Let’s take that third path, the one that zig-zags through the right-hand side of the multiway graph:
Suppose we found ourselves in this particular universe, quite a way from the universes we were in before, with this particular history:
Here are the events that happened in this universe:
And here are the causal relationships between these events:
Once again, let’s take out the multiway graph:
and arrange the events so that we can see the shape of the causal graph more clearly:
Oh, hang on, that... can’t be a coincidence... it’s the same shape again.
Plus ça change
What’s going on here?
No matter which path we take through the multiway graph – this one...
...or this one...
...or this one...
... – no matter which possible universe we find ourselves in – this one...
...or this one...
...or this one...
... – the causal graph is the same shape:
This is not a coincidence.
What’s going on here is causal invariance.
A single objective reality
I’ve said before what causal invariance is.
If a rule is causal invariant, it means that whenever there’s a two-way branch in the multiway graph, there exist paths through the multiway graph along which the two branches come back together.
But there’s another way to define causal invariance.
If a rule is causal invariant, then the causal graph is the same shape regardless of which path you take through the multiway graph.
That’s why it’s called causal invariance: because the causal graph is invariant.
The strange thing about causal invariance – well, one of the many strange things about causal invariance – is that these two definitions are mathematically equivalent.
Even stranger is what this means for how we, as observers, perceive reality.
If a rule is causal invariant, then different obervers, following different paths through the multiway graph, might disagree about the precise order of events.
One observer might think that event A happened, then event B, while another observer might think that event B happened, then event A.
But both observers would agree that events A and B must both happen before event C can happen.
In other words, if a rule is causal invariant, then different observers might follow different paths through the multiway graph, but they see the same causal graph.
Or, to be precise, they see isomorphic causal graphs.
Which means that different observers can disagree about all sorts of things, but they’ll agree about the causal graph.
It’s as if the hypergraph – space and everything in space – is a contingent reality: it depends on which evolution of the universe you find yourself following...
...but the causal graph – which events have to happen before which other events can happen – is objective reality: it’s the same no matter who you are or where you are.
The causal graph collapses every possible reality into a single objective reality.