How does the “size of mathematics” compare to the size of our physical universe? In the past this might have seemed like an absurd question, that tries to compare something abstract and arbitrary with something real and physical. But with the idea that both mathematics and physics as we experience them emerge from our sampling of the ruliad, it begins to seem less absurd.

At the lowest level the ruliad can be thought of as being made up of atoms of existence that we call emes. As physical observers we interpret these emes as atoms of space, or in effect the ultimate raw material of the physical universe. And as mathematical observers we interpret them as the ultimate elements from which the constructs of mathematics are built.

As the entangled limit of all possible computations, the whole ruliad is infinite. But we as physical or mathematical observers sample only limited parts of it. And that means we can meaningfully ask questions like how the number of emes in these parts compare—or, in effect, how big is physics as we experience it compared to mathematics.

In some ways an eme is like a bit. But the concept of emes is that they’re “actual atoms of existence”—from which “actual stuff” like the physical universe and its history are made—rather than just “static informational representations” of it. As soon as we imagine that everything is ultimately computational we are immediately led to start thinking of representing it in terms of bits. But the ruliad is not just a representation. It’s in some way something lower level. It’s the “actual stuff” that everything is made of. And what defines our particular experience of physics or of mathematics is the particular samples we as observers take of what’s in the ruliad.

So the question is now how many emes there are in those samples. Or, more specifically, how many emes “matter to us” in building up our experience.

Let’s return to an analogy we’ve used several times before: a gas made of molecules. In the volume of a room there might be 10^{27} individual molecules, each on average colliding every 10^{–10} seconds. So that means that our “experience of the room” over the course of a minute or so might sample 10^{39} collisions. Or, in terms closer to our Physics Project, we might say that there are perhaps 10^{39} “collision events” in the causal graph that defines what we experience.

But these “collision events” aren’t something fundamental; they have what amounts to “internal structure” with many associated parameters about location, time, molecular configuration, etc.

Our Physics Project, however, suggests that—far below for example our usual notions of space and time—we can in fact have a truly fundamental definition of what’s happening in the universe, ultimately in terms of emes. We don’t yet know the “physical scale” for this—and in the end we presumably need experiments to determine that. But rather rickety estimates based on a variety of assumptions suggest that the elementary length might be around 10^{–90} meters, with the elementary time being around 10^{–100} seconds.

And with these estimates we might conclude that our “experience of a room for a minute” would involve sampling perhaps 10^{370} update events, that create about this number of atoms of space.

But it’s immediately clear that this is in a sense a gross underestimate of the total number of emes that we’re sampling. And the reason is that we’re not accounting for quantum mechanics, and for the multiway nature of the evolution of the universe. We’ve so far only considered one “thread of time” at one “position in branchial space”. But in fact there are many threads of time, constantly branching and merging. So how many of these do we experience?

In effect that depends on our size in branchial space. In physical space “human scale” is of order a meter—or perhaps 10^{90} elementary lengths. But how big is it in branchial space?

The fact that we’re so large compared to the elementary length is the reason that we consistently experience space as something continuous. And the analog in branchial space is that if we’re big compared to the “elementary branchial distance between branches” then we won’t experience the different individual histories of these branches, but only an aggregate “objective reality” in which we conflate together what happens on all the branches. Or, put another way, being large in branchial space is what makes us experience classical physics rather than quantum mechanics.

Our estimates for branchial space are even more rickety than for physical space. But conceivably there are on the order of 10^{120} “instantaneous parallel threads of time” in the universe, and 10^{20} encompassed by our instantaneous experience—implying that in our minute-long experience we might sample a total of on the order of close to 10^{500} emes.

But even this is a vast underestimate. Yes, it tries to account for our extent in physical space and in branchial space. But then there’s also rulial space—which in effect is what “fills out” the whole ruliad. So how big are we in that space? In essence that’s like asking how many different possible sequences of rules there are that are consistent with our experience.

The total conceivable number of sequences associated with 10^{500} emes is roughly the number of possible hypergraphs with 10^{500} nodes—or around (10^{500})^{10500}. But the actual number consistent with our experience is smaller, in particular as reflected by the fact that we attribute specific laws to our universe. But when we say “specific laws” we have to recognize that there is a finiteness to our efforts at inductive inference which inevitably makes these laws at least somewhat uncertain to us. And in a sense that uncertainty is what represents our “extent in rulial space”.

But if we want to count the emes that we “absorb” as physical observers, it’s still going to be a huge number. Perhaps the base may be lower—say 10^{10}—but there’s still a vast exponent, suggesting that if we include our extent in rulial space, we as physical observers may experience numbers of emes like (10^{10})^{10500}.

But let’s say we go beyond our “everyday human-scale experience”. For example, let’s ask about “experiencing” our whole universe. In physical space, the volume of our current universe is about 10^{78} times larger than “human scale” (while human scale is perhaps 10^{270} times larger than the “scale of the atoms of space”). In branchial space, conceivably our current universe is 10^{100} times larger than “human scale”. But these differences absolutely pale in comparison to the sizes associated with rulial space.

We might try to go beyond “ordinary human experience” and for example measure things using tools from science and technology. And, yes, we could then think about “experiencing” lengths down to 10^{–22} meters, or something close to “single threads” of quantum histories. But in the end, it’s still the rulial size that dominates, and that’s where we can expect most of the vast number of emes that form of our experience of the physical universe to come from.

OK, so what about mathematics? When we think about what we might call human-scale mathematics, and talk about things like the Pythagorean theorem, how many emes are there “underneath”? “Compiling” our theorem down to typical traditional mathematical axioms, we’ve seen that we’ll routinely end up with expressions containing, say, 10^{20} symbolic elements. But what happens if we go “below that”, compiling these symbolic elements—which might include things like variables and operators—into “pure computational elements” that we can think of as emes? We’ve seen a few examples, say with combinators, that suggest that for the traditional axiomatic structures of mathematics, we might need another factor of maybe roughly 10^{10}.

These are incredibly rough estimates, but perhaps there’s a hint that there’s “further to go” to get from human-scale for a physical observer down to atoms of space that correspond to emes, than there is to get from human-scale for a mathematical observer down to emes.

Just like in physics, however, this kind of “static drill-down” isn’t the whole story for mathematics. When we talk about something like the Pythagorean theorem, we’re really referring to a whole cloud of “human-equivalent” points in metamathematical space. The total number of “possible points” is basically the size of the entailment cone that contains something like the Pythagorean theorem. The “height” of the entailment cone is related to typical lengths of proofs—which for current human mathematics might be perhaps hundreds of steps.

And this would lead to overall sizes of entailment cones of very roughly 10^{100} theorems. But within this “how big” is the cloud of variants corresponding to particular “human-recognized” theorems? Empirical metamathematics could provide additional data on this question. But if we very roughly imagine that half of every proof is “flexible”, we’d end up with things like 10^{50} variants. So if we asked how many emes correspond to the “experience” of the Pythagorean theorem, it might be, say, 10^{80}.

To give an analogy of “everyday physical experience” we might consider a mathematician thinking about mathematical concepts, and maybe in effect pondering a few tens of theorems per minute—implying according to our extremely rough and speculative estimates that while typical “specific human-scale physics experience” might involve 10^{500} emes, specific human-scale mathematics experience might involve 10^{80} emes (a number comparable, for example, to the number of physical atoms in our universe).

What if instead of considering “everyday mathematical experience” we consider all humanly explored mathematics? On the scales we’re describing, the factors are not large. In the history of human mathematics, only a few million theorems have been published. If we think about all the computations that have been done in the service of mathematics, it’s a somewhat larger factor. I suspect Mathematica is the dominant contributor here—and we can estimate that the total number of Wolfram Language operations corresponding to “human-level mathematics” done so far is perhaps 10^{20}.

But just like for physics, all these numbers pale in comparison with those introduced by rulial sizes. We’ve talked essentially about a particular path from emes through specific axioms to theorems. But the ruliad in effect contains all possible axiom systems. And if we start thinking about enumerating these—and effectively “populating all of rulial space”—we’ll end up with exponentially more emes.

But as with the perceived laws of physics, in mathematics as done by humans it’s actually just a narrow slice of rulial space that we’re sampling. It’s like a generalization of the idea that something like arithmetic as we imagine it can be derived from a whole cloud of possible axiom systems. It’s not just one axiom system; but it’s also not all possible axiom systems.

One can imagine doing some combination of ruliology and empirical metamathematics to get an estimate of “how broad” human-equivalent axiom systems (and their construction from emes) might be. But the answer seems likely to be much smaller than the kinds of sizes we have been estimating for physics.

It’s important to emphasize that what we’ve discussed here is extremely rough—and speculative. And indeed I view its main value as being to provide an example of how to imagine thinking through things in the context of the ruliad and the framework around it. But on the basis of what we’ve discussed, we might make the very tentative conclusion that “human-experienced physics” is bigger than “human-experienced mathematics”. Both involve vast numbers of emes. But physics seems to involve a lot more. In a sense—even with all its abstraction—the suspicion is that there’s “less ultimately in mathematics” as far as we’re concerned than there is in physics. Though by any ordinary human standards, mathematics still involves absolutely vast numbers of emes.