We’ve discussed how the basic concept of space as we experience it in physics leads us to our first great physicalized law of mathematics—and how this provides for the very possibility of higher-level mathematics. But this is just the beginning of what we can learn from thinking about the correspondences between physical and metamathematical space implied by their common origin in the structure of the ruliad.
A key idea is to think of a limit of mathematics in which one is dealing with so many mathematical statements that one can treat them “in bulk”—as forming something we could consider a continuous metamathematical space. But what might this space be like?
Our experience of physical space is that at our scale and with our means of perception it seems to us for the most part quite simple and uniform. And this is deeply connected to the concept that pure motion is possible in physical space—or, in other words, that it’s possible for things to move around in physical space without fundamentally changing their character.
Looked at from the point of view of the atoms of space it’s not at all obvious that this should be possible. After all, whenever we move we’ll almost inevitably be made up of different atoms of space. But it’s fundamental to our character as observers that the features we end up perceiving are ones that have a certain persistence—so that we can imagine that we, and objects around us, can just “move unchanged”, at least with respect to those aspects of the objects that we perceive. And this is why, for example, we can discuss laws of mechanics without having to “drop down” to the level of the atoms of space.
So what’s the analog of all this in metamathematical space? At the present stage of our physical universe, we seem to be able to experience physical space as having features like being basically three-dimensional. Metamathematical space probably doesn’t have such familiar mathematical characterizations. But it seems very likely (and we’ll see some evidence of this from empirical metamathematics below) that at the very least we’ll perceive metamathematical space as having a certain uniformity or homogeneity.
In our Physics Project we imagine that we can think of physical space as beginning “at the Big Bang” with what amounts to some small collection of atoms of space, but then growing to the vast number of atoms in our current universe through the repeated application of particular rules. But with a small set of rules being applied a vast number of times, it seems almost inevitable that some kind of uniformity must result.
But then the same kind of thing can be expected in metamathematics. In axiomatic mathematics one imagines the mathematical analog of the Big Bang: everything starts from a small collection of axioms, and then expands to a huge number of mathematical statements through repeated application of laws of inference. And from this picture (which gets a bit more elaborate when one considers emes and the full ruliad) one can expect that at least after it’s “developed for a while” metamathematical space, like physical space, will have a certain uniformity.
The idea that physical space is somehow uniform is something we take very much for granted, not least because that’s our lifelong experience. But the analog of this idea for metamathematical space is something we don’t have immediate everyday intuition about—and that in fact may at first seem surprising or even bizarre. But actually what it implies is something that increasingly rings true from modern experience in pure mathematics. Because by saying that metamathematical space is in a sense uniform, we’re saying that different parts of it somehow seem similar—or in other words that there’s parallelism between what we see in different areas of mathematics, even if they’re not “nearby” in terms of entailments.
But this is exactly what, for example, the success of category theory implies. Because it shows us that even in completely different areas of mathematics it makes sense to set up the same basic structures of objects, morphisms and so on. As such, though, category theory defines only the barest outlines of mathematical structure. But what our concept of perceived uniformity in metamathematical space suggests is that there should in fact be closer correspondences between different areas of mathematics.
We can view this as another fundamental “physicalized law of mathematics”: that different areas of mathematics should ultimately have structures that are in some deep sense “perceived the same” by mathematical observers. For several centuries we’ve known there’s a certain correspondence between, for example, geometry and algebra. But it’s been a major achievement of recent mathematics to identify more and more such correspondences or “dualities”.
Often the existence of these has seemed remarkable, and surprising. But what our view of metamathematics here suggests is that this is actually a general physicalized law of mathematics—and that in the end essentially all different areas of mathematics must share a deep structure, at least in some appropriate “bulk metamathematical limit” when enough statements are considered.
But it’s one thing to say that two places in metamathematical space are “similar”; it’s another to say that “motion between them” is possible. Once again we can make an analogy with physical space. We’re used to the idea that we can move around in space, maintaining our identity and structure. But this in a sense requires that we can maintain some kind of continuity of existence on our path between two positions.
In principle it could have been that we would have to be “atomized” at one end, then “reconstituted” at the other end. But our actual experience is that we perceive ourselves to continually exist all the way along the path. In a sense this is just an assumption about how things work that physical observers like us make; but what’s nontrivial is that the underlying structure of the ruliad implies that this will always be consistent.
And so we expect it will be in metamathematics. Like a physical observer, the way a mathematical observer operates, it’ll be possible to “move” from one area of mathematics to another “at a high level”, without being “atomized” along the way. Or, in other words, that a mathematical observer will be able to make correspondences between different areas of mathematics without having to go down to the level of emes to do so.
It’s worth realizing that as soon as there’s a way of representing mathematics in computational terms the concept of universal computation (and, more tightly, the Principle of Computational Equivalence) implies that at some level there must always be a way to translate between any two mathematical theories, or any two areas of mathematics. But the question is whether it’s possible to do this in “high-level mathematical terms” or only at the level of the underlying “computational substrate”. And what we’re saying is that there’s a general physicalized law of mathematics that implies that higher-level translation should be possible.
Thinking about mathematics at a traditional axiomatic level can sometimes obscure this, however. For example, in axiomatic terms we usually think of Peano arithmetic as not being as powerful as ZFC set theory (for example, it lacks transfinite induction)—and so nothing like “dual” to it. But Peano arithmetic can perfectly well support universal computation, so inevitably a “formal emulator” for ZFC set theory can be built in it. But the issue is that to do this essentially requires going down to the “atomic” level and operating not in terms of mathematical constructs but instead directly in terms of “metamathematical” symbolic structure (and, for example, explicitly emulating things like equality predicates).
But the issue, it seems, is that if we think at the traditional axiomatic level, we’re not dealing with a “mathematical observer like us”. In the analogy we’ve used above, we’re operating at the “molecular dynamics” level, not at the human-scale “fluid dynamics” level. And so we see all sorts of details and issues that ultimately won’t be relevant in typical approaches to actually doing pure mathematics.
It’s somewhat ironic that our physicalized approach shows this by going below the axiomatic level—to the level of emes and the raw ruliad. But in a sense it’s only at this level that there’s the uniformity and coherence to conveniently construct a general picture that can encompass observers like us.
Much as with ordinary matter we can say that “everything is made of atoms”, we’re now saying that everything is “made of computation” (and its structure and behavior is ultimately described by the ruliad). But the crucial idea that emerged from our Physics Project—and that is at the core of what I’m calling the multicomputational paradigm—is that when we ask what observers perceive there is a whole additional level of inexorable structure. And this is what makes it possible to do both human-scale physics and higher-level mathematics—and for there to be what amounts to “pure motion”, whether in physical or metamathematical space.
There’s another way to think about this, that we alluded to earlier. A key feature of an observer is to have a coherent identity. In physics, that involves having a consistent thread of experience in time. In mathematics, it involves bringing together a consistent view of “what’s true” in the space of mathematical statements.
In both cases the observer will in effect involve many separate underlying elements (ultimately, emes). But in order to maintain the observer’s view of having a coherent identity, the observer must somehow conflate all these elements, effectively treating them as “the same”. In physics, this means “coarse-graining” across physical or branchial (or, in fact, rulial) space. In mathematics, this means “coarse-graining” across metamathematical space—or in effect treating different mathematical statements as “the same”.
In practice, there are several ways this happens. First of all, one tends to be more concerned about mathematical results than their proofs, so two statements that have the same form can be considered the same even if the proofs (or other processes) that generated them are different (and indeed this is something we have routinely done in constructing entailment cones here). But there’s more. One can also imagine that any statements that entail each other can be considered “the same”.
In a simple case, this means that if a=b and b=c then one can always assume a=c. But there’s a much more general version of this embodied in the univalence axiom of homotopy type theory—that in our terms can be interpreted as saying that mathematical observers consider equivalent things the same.
There’s another way that mathematical observers conflate different statements—that’s in many ways more important, but less formal. As we mentioned above, when mathematicians talk, say, about the Pythagorean theorem, they typically think they have a definite concept in mind. But at the axiomatic level—and even more so at the level of emes—there are a huge number of different “metamathematical configurations” that are all “considered the same” by the typical working mathematician, or by our “mathematical observer”. (At the level of axioms, there might be different axiom systems for real numbers; at the level of emes there might be different ways of representing concepts like addition or equality.)
In a sense we can think of mathematical observers as having a certain “extent” in metamathematical space. And much like human-scale physical observers see only the aggregate effects of huge numbers of atoms of space, so also mathematical observers see only the “aggregate effects” of huge numbers of emes of metamathematical space.
But now the key question is whether a “whole mathematical observer” can “move in metamathematical space” as a single “rigid” entity, or whether it will inevitably be distorted—or shredded—by the structure of metamathematical space. In the next section we’ll discuss the analog of gravity—and curvature—in metamathematical space. But our physicalized approach tends to suggest that in “most” of metamathematical space, a typical mathematical observer will be able to “move around freely”, implying that there will indeed be paths or “bridges” between different areas of mathematics, that involve only higher-level mathematical constructs, and don’t require dropping down to the level of emes and the raw ruliad.