I don't understand why you feel that there needs to be a space in order to have a collection of nodes and edges that connect the nodes.

It isn't as if there is an actual line connecting two dots across space.

It's just, there's a set of "points" (there doesn't need to be any things about these "points" other than that they are distinct. You could think of them as urelements I guess.), and another set of "edges", which are each a set of two elements from the first set.

A pair of "points" are "connected" by an "edge" iff the set comprised of those two "points" is in the second set.

This doesn't need a notion of space to work, unless you need like, an idea of space to have collections of things. So unless "the collection of odd numbers" needs an idea of space, then graphs don't need space.

This conversation has given me lots to think about and I was ready to let it trail off. But wanted to answer you just out of respect.

My mistake was thinking that he was proposing something about the nature of physical reality. Ie. that physical reality was literally a network of interconnected nodes at its fundamental level. Something along the lines of macroscopic -> atomic -> sub atomic -> wolfram-nodal. A la, string theory, et. al.

From the replies here, I gather this has nothing to do with physical reality.. he's not describing how this nodal network might be implemented in reality... just that it must be, because the predictions come out correctly.

I'm as dumb as they come, so take this with a grain of salt: I think he is describing (a theory about) physical reality. To give you a different perspective: You position his theory of the universe as a network of nodes along with distances/measurements. But if you think for a moment about the void that you have to cross to get from one object to another (like from our star system to another) what is it? If you put it under an electron microscope it doesn't reveal itself.

It's "nothing" as far as we're concerned but despite being intangible, despite being nothing, it has rules and laws that govern matter that traverses it just like every other part of our universe. So what is space itself? What is it composed of? You can't take a quick step outside of the universe to get a look under the hood because you are part of the universe. You can only talk about it in terms of it's rules and behavior–in the abstract.

No, I don't think that's it. He is saying that physical reality is literally a network of interconnected nodes at its fundamental level. I think you're still assuming that these graphs have to be embedded in some space to be real.

If we get fundamental enough, we can go below geometry, into algebra.

For a less abstract example, what's the radius of an electron in modern physics?

These graphs have to be embedded into a computation device to be real and changing in time. To build such a computation device, maybe you need time and space. It could even be that our Universe is of the minimum complexity required to build a computation device capable of computing the graphs that describe our Universe.

> These graphs have to be embedded into a computation device to be real and changing in time.

No, no, no! These graphs are meant to be more fundamental than spacetime. They can change in the sense that there are algebraic rules for stepping them from one state to another, but this step isn't in our time. Our space and time should be generated by operating on these graphs (let's say "stepping" them, as long as we remember that it's not a time-step in our spacetime, but just an algebraic operation).

EDIT> I should say that this whole idea is a model. The question of whether a model is real or not independent of the system being modeled is a philosophical one that we can't really do justice to in an HN comment thread.

This is a classic debate in the philosophy science. Should theories describe the world, or merely make predictions about it? Because no one these days believes that quantum mechanics really describes the world, we live firmly in the realm of theories being things that make predictions.

The existence of the atom, however, is not what is being predicted. A theory should predict the sensor readings of particular instruments during particular experiments. Those sensor readings are what is real; "The Atom" is the abstraction.

The very idea that "things exist" is an abstraction. A thing exists if we can model it, and that model corresponds with reality.

This is a fundamental philosophical question. The same question can be asked about more traditional physical models, e.g. what is Einstein's spacetime composed of, and how did his equations of general relativity come to be? Are they real, or are they just a model that we use to describe reality?

Over time, physicists have come to take models very seriously (thanks in part to Einstein's 1905 papers). That is, when our model says that electrons act like particles and waves, we assume that electrons really are these odd things that are somewhere in between a particle and a wave.

So, if it turns out that Wolfram's ideas have merit, and there really is an underlying spacetime network which can be used to model and derive all of the known equations of physics (and likely some unknown equations), then at that point we would have to take very seriously the existence of such a network. We would be forced to think of a seemingly continuous spacetime as a special facet or feature of this underlying network, in the same way that we now think of space as a special feature of the more general spacetime.

> But a theory is supposed to model something that actually exists, yes?

Yes, but "model" is the key word. If it provides a set of rules which predict the way thing that exist in the world behave, then it models something (or set of things) that exits in the real world.