If we “drill down” to what we’ve called above the “molecular level” of mathematics, what will we find there? There are many technical details (some of which we’ll discuss later) about the historical conventions of mathematics and its presentation. But in broad outline we can think of there as being a kind of “gas” of “mathematical statements”—like 1+1=2 or x+y=y+x—represented in some specified symbolic language. (And, yes, Wolfram Language provides a well-developed example of what that language can be like.)
But how does the “gas of statements” behave? The essential point is that new statements are derived from existing ones by “interactions” that implement laws of inference (like that q can be derived from the statement p and the statement “p implies q”). And if we trace the paths by which one statement can be derived from others, these correspond to proofs. And the whole graph of all these derivations is then a representation of the possible historical development of mathematics—with slices through this graph corresponding to the sets of statements reached at a given stage.
By talking about things like a “gas of statements” we’re making this sound a bit like physics. But while in physics a gas consists of actual, physical molecules, in mathematics our statements are just abstract things. But this is where the discoveries of our Physics Project start to be important. Because in our project we’re “drilling down” beneath for example the usual notions of space and time to an “ultimate machine code” for the physical universe. And we can think of that ultimate machine code as operating on things that are in effect just abstract constructs—very much like in mathematics.
In particular, we imagine that space and everything in it is made up of a giant network (hypergraph) of “atoms of space”—with each “atom of space” just being an abstract element that has certain relations with other elements. The evolution of the universe in time then corresponds to the application of computational rules that (much like laws of inference) take abstract relations and yield new relations—thereby progressively updating the network that represents space and everything in it.
But while the individual rules may be very simple, the whole detailed pattern of behavior to which they lead is normally very complicated—and typically shows computational irreducibility, so that there’s no way to systematically find its outcome except in effect by explicitly tracing each step. But despite all this underlying complexity it turns out—much like in the case of an ordinary gas—that at a coarse-grained level there are much simpler (“bulk”) laws of behavior that one can identify. And the remarkable thing is that these turn out to be exactly general relativity and quantum mechanics (which, yes, end up being the same theory when looked at in terms of an appropriate generalization of the notion of space).
But down at the lowest level, is there some specific computational rule that’s “running the universe”? I don’t think so. Instead, I think that in effect all possible rules are always being applied. And the result is the ruliad: the entangled structure associated with performing all possible computations.
But what then gives us our experience of the universe and of physics? Inevitably we are observers embedded within the ruliad, sampling only certain features of it. But what features we sample are determined by the characteristics of us as observers. And what seem to be critical to have “observers like us” are basically two characteristics. First, that we are computationally bounded. And second, that we somehow persistently maintain our coherence—in the sense that we can consistently identify what constitutes “us” even though the detailed atoms of space involved are continually changing.
But we can think of different “observers like us” as taking different specific samples, corresponding to different reference frames in rulial space, or just different positions in rulial space. These different observers may describe the universe as evolving according to different specific underlying rules. But the crucial point is that the general structure of the ruliad implies that so long as the observers are “like us”, it’s inevitable that their perception of the universe will be that it follows things like general relativity and quantum mechanics.
It’s very much like what happens with a gas of molecules: to an “observer like us” there are the same gas laws and the same laws of fluid dynamics essentially independent of the detailed structure of the individual molecules.
So what does all this mean for mathematics? The crucial and at first surprising point is that the ideas we’re describing in physics can in effect immediately be carried over to mathematics. And the key is that the ruliad represents not only all physics, but also all mathematics—and it shows that these are not just related, but in some sense fundamentally the same.
In the traditional formulation of axiomatic mathematics, one talks about deriving results from particular axiom systems—say Peano Arithmetic, or ZFC set theory, or the axioms of Euclidean geometry. But the ruliad in effect represents the entangled consequences not just of specific axiom systems but of all possible axiom systems (as well as all possible laws of inference).
But from this structure that in a sense corresponds to all possible mathematics, how do we pick out any particular mathematics that we’re interested in? The answer is that just as we are limited observers of the physical universe, so we are also limited observers of the “mathematical universe”.
But what are we like as “mathematical observers”? As I’ll argue in more detail later, we inherit our core characteristics from those we exhibit as “physical observers”. And that means that when we “do mathematics” we’re effectively sampling the ruliad in much the same way as when we “do physics”.
We can operate in different rulial reference frames, or at different locations in rulial space, and these will correspond to picking out different underlying “rules of mathematics”, or essentially using different axiom systems. But now we can make use of the correspondence with physics to say that we can also expect there to be certain “overall laws of mathematics” that are the result of general features of the ruliad as perceived by observers like us.
And indeed we can expect that in some formal sense these overall laws will have exactly the same structure as those in physics—so that in effect in mathematics we’ll have something like the notion of space that we have in physics, as well as formal analogs of things like general relativity and quantum mechanics.
What does this mean? It implies that—just as it’s possible to have coherent “higher-level descriptions” in physics that don’t just operate down at the level of atoms of space, so also this should be possible in mathematics. And this in a sense is why we can expect to consistently do what I described above as “human-level mathematics”, without usually having to drop down to the “molecular level” of specific axiomatic structures (or below).
Say we’re talking about the Pythagorean theorem. Given some particular detailed axiom system for mathematics we can imagine using it to build up a precise—if potentially very long and pedantic—representation of the theorem. But let’s say we change some detail of our axioms, say associated with the way they talk about sets, or real numbers. We’ll almost certainly still be able to build up something we consider to be “the Pythagorean theorem”—even though the details of the representation will be different.
In other words, this thing that we as humans would call “the Pythagorean theorem” is not just a single point in the ruliad, but a whole cloud of points. And now the question is: what happens if we try to derive other results from the Pythagorean theorem? It might be that each particular representation of the theorem—corresponding to each point in the cloud—would lead to quite different results. But it could also be that essentially the whole cloud would coherently lead to the same results.
And the claim from the correspondence with physics is that there should be “general laws of mathematics” that apply to “observers like us” and that ensure that there’ll be coherence between all the different specific representations associated with the cloud that we identify as “the Pythagorean theorem”.
In physics it could have been that we’d always have to separately say what happens to every atom of space. But we know that there’s a coherent higher-level description of space—in which for example we can just imagine that objects can move while somehow maintaining their identity. And we can now expect that it’s the same kind of thing in mathematics: that just as there’s a coherent notion of space in physics where things can for example move without being “shredded”, so also this will happen in mathematics. And this is why it’s possible to do “higher-level mathematics” without always dropping down to the lowest level of axiomatic derivations.
It’s worth pointing out that even in physical space a concept like “pure motion” in which objects can move while maintaining their identity doesn’t always work. For example, close to a spacetime singularity, one can expect to eventually be forced to see through to the discrete structure of space—and for any “object” to inevitably be “shredded”. But most of the time it’s possible for observers like us to maintain the idea that there are coherent large-scale features whose behavior we can study using “bulk” laws of physics.
And we can expect the same kind of thing to happen with mathematics. Later on, we’ll discuss more specific correspondences between phenomena in physics and mathematics—and we’ll see the effects of things like general relativity and quantum mechanics in mathematics, or, more precisely, in metamathematics.
But for now, the key point is that we can think of mathematics as somehow being made of exactly the same stuff as physics: they’re both just features of the ruliad, as sampled by observers like us. And in what follows we’ll see the great power that arises from using this to combine the achievements and intuitions of physics and mathematics—and how this lets us think about new “general laws of mathematics”, and view the ultimate foundations of mathematics in a different light.