If metamathematical space is like physical space, does that mean that it has analogs of gravity, and relativity? The answer seems to be “yes”—and these provide our next examples of physicalized laws of mathematics.

In the end, we’re going to be able to talk about at least gravity in a largely “static” way, referring mostly to the “instantaneous state of metamathematics”, captured as an entailment fabric. But in leveraging ideas from physics, it’s important to start off formulating things in terms of the analog of time for metamathematics—which is entailment.

As we’ve discussed above, the entailment cone is the direct analog of the light cone in physics. Starting with some mathematical statement (or, more accurately, some event that transforms it) the forward entailment cone contains all statements (or, more accurately, events) that follow from it. Any possible “instantaneous state of metamathematics” then corresponds to a “transverse slice” through this entailment cone—with the slice in effect being laid out in metamathematical space.

An individual entailment of one statement by another corresponds to a path in the entailment cone, and this path (or, more accurately for accumulative evolution, subgraph) can be thought of as a proof of one statement given another. And in these terms the shortest proof can be thought of as a geodesic in the entailment cone. (In practical mathematics, it’s very unlikely one will find—or care about—the strictly shortest proof. But even having a “fairly short proof” will be enough to give the general conclusions we’ll discuss here.)

Given a path in the entailment cone, we can imagine projecting it onto a transverse slice, i.e. onto an entailment fabric. Being able to consistently do this depends on having a certain uniformity in the entailment cone, and in the sequence of “metamathematical hypersurfaces” that are defined by whatever “metamathematical reference frame” we’re using. But assuming, for example, that underlying computational irreducibility successfully generates a kind of “statistical uniformity” that cannot be “decoded” by the observer, we can expect to have meaningful paths—and geodesics—on entailment fabrics.

But what these geodesics are like then depends on the emergent geometry of entailment fabrics. In physics, the limiting geometry of the analog of this for physical space is presumably a fairly simple 3D manifold. For branchial space, it’s more complicated, probably for example being “exponential dimensional”. And for metamathematics, the limiting geometry is also undoubtedly more complicated—and almost certainly exponential dimensional.

We’ve argued that we expect metamathematical space to have a certain perceived uniformity. But what will affect this, and therefore potentially modify the local geometry of the space? The basic answer is exactly the same as in our Physics Project. If there’s “more activity” somewhere in an entailment fabric, this will in effect lead to “more local connections”, and thus effective “positive local curvature” in the emergent geometry of the network. Needless to say, exactly what “more activity” means is somewhat subtle, especially given that the fabric in which one is looking for this is itself defining the ambient geometry, measures of “area”, etc.

In our Physics Project we make things more precise by associating “activity” with energy density, and saying that energy effectively corresponds to the flux of causal edges through spacelike hypersurfaces. So this suggests that we think about an analog of energy in metamathematics: essentially defining it to be the density of update events in the entailment fabric. Or, put another way, energy in metamathematics depends on the “density of proofs” going through a region of metamathematical space, i.e. involving particular “nearby” mathematical statements.

There are lots of caveats, subtleties and details. But the notion that “activity AKA energy” leads to increasing curvature in an emergent geometry is a general feature of the whole multicomputational paradigm that the ruliad captures. And in fact we expect a quantitative relationship between energy density (or, strictly, energy-momentum) and induced curvature of the “transversal space”—that corresponds exactly to Einstein’s equations in general relativity. It’ll be more difficult to see this in the metamathematical case because metamathematical space is geometrically more complicated—and less familiar—than physical space.

But even at a qualitative level, it seems very helpful to think in terms of physics and spacetime analogies. The basic phenomenon is that geodesics are deflected by the presence of “energy”, in effect being “attracted to it”. And this is why we can think of regions of higher energy (or energy-momentum/mass)—in physics and in metamathematics—as “generating gravity”, and deflecting geodesics towards them. (Needless to say, in metamathematics, as in physics, the vast majority of overall activity is just devoted to knitting together the structure of space, and when gravity is produced, it’s from slightly increased activity in a particular region.)

(In our Physics Project, a key result is that the same kind of dependence of “spatial” structure on energy happens not only in physical space, but also in branchial space—where there’s a direct analog of general relativity that basically yields the path integral of quantum mechanics.)

What does this mean in metamathematics? Qualitatively, the implication is that “proofs will tend to go through where there’s a higher density of proofs”. Or, in an analogy, if you want to drive from one place to another, it’ll be more efficient if you can do at least part of your journey on a freeway.

One question to ask about metamathematical space is whether one can always get from any place to any other. In other words, starting from one area of mathematics, can one somehow derive all others? A key issue here is whether the area one starts from is computation universal. Propositional logic is not, for example. So if one starts from it, one is essentially trapped, and cannot reach other areas.

But results in mathematical logic have established that most traditional areas of axiomatic mathematics are in fact computation universal (and the Principle of Computational Equivalence suggests that this will be ubiquitous). And given computation universality there will at least be some “proof path”. (In a sense this is a reflection of the fact that the ruliad is unique, so everything is connected in “the same ruliad”.)

But a big question is whether the “proof path” is “big enough” to be appropriate for a “mathematical observer like us”. Can we expect to get from one part of metamathematical space to another without the observer being “shredded”? Will we be able to start from any of a whole collection of places in metamathematical space that are considered “indistinguishably nearby” to a mathematical observer and have all of them “move together” to reach our destination? Or will different specific starting points follow quite different paths—preventing us from having a high-level (“fluid dynamics”) description of what’s going on, and instead forcing us to drop down to the “molecular dynamics” level?

In practical pure mathematics, this tends to be an issue of whether there is an “elegant proof using high-level concepts”, or whether one has to drop down to a very detailed level that’s more like low-level computer code, or the output of an automated theorem proving system. And indeed there’s a very visceral sense of “shredding” in cases where one’s confronted with a proof that consists of page after page of “machine-like details”.

But there’s another point here as well. If one looks at an individual proof path, it can be computationally irreducible to find out where the path goes, and the question of whether it ever reaches a particular destination can be undecidable. But in most of the current practice of pure mathematics, one’s interested in “higher-level conclusions”, that are “visible” to a mathematical observer who doesn’t resolve individual proof paths.

Later we’ll discuss the dichotomy between explorations of computational systems that routinely run into undecidability—and the typical experience of pure mathematics, where undecidability is rarely encountered in practice. But the basic point is that what a typical mathematical observer sees is at the “fluid dynamics level”, where the potentially circuitous path of some individual molecule is not relevant.

Of course, by asking specific questions—about metamathematics, or, say, about very specific equations—it’s still perfectly possible to force tracing of individual “low-level” proof paths. But this isn’t what’s typical in current pure mathematical practice. And in a sense we can see this as an extension of our first physicalized law of mathematics: not only is higher-level mathematics possible, but it’s ubiquitously so, with the result that, at least in terms of the questions a mathematical observer would readily formulate, phenomena like undecidability are not generically seen.

But even though undecidability may not be directly visible to a mathematical observer, its underlying presence is still crucial in coherently “knitting together” metamathematical space. Because without undecidability, we won’t have computation universality and computational irreducibility. But—just like in our Physics Project—computational irreducibility is crucial in producing the low-level apparent randomness that is needed to support any kind of “continuum limit” that allows us to think of large collections of what are ultimately discrete emes as building up some kind of coherent geometrical space.

And when undecidability is not present, one will typically not end up with anything like this kind of coherent space. An extreme example occurs in rewrite systems that eventually terminate—in the sense that they reach a “fixed-point” (or “normal form”) state where no more transformations can be applied.

In our Physics Project, this kind of termination can be interpreted as a spacelike singularity at which “time stops” (as at the center of a non-rotating black hole). But in general decidability is associated with “limits on how far paths can go”—just like the limits on causal paths associated with event horizons in physics.

There are many details to work out, but the qualitative picture can be developed further. In physics, the singularity theorems imply that in essence the eventual formation of spacetime singularities is inevitable. And there should be a direct analog in our context that implies the eventual formation of “metamathematical singularities”. In qualitative terms, we can expect that the presence of proof density (which is the analog of energy) will “pull in” more proofs until eventually there are so many proofs that one has decidability and a “proof event horizon” is formed.

In a sense this implies that the long-term future of mathematics is strangely similar to the long-term future of our physical universe. In our physical universe, we expect that while the expansion of space may continue, many parts of the universe will form black holes and essentially be “closed off”. (At least ignoring expansion in branchial space, and quantum effects in general.)

The analog of this in mathematics is that while there can be continued overall expansion in metamathematical space, more and more parts of it will “burn out” because they’ve become decidable. In other words, as more work and more proofs get done in a particular area, that area will eventually be “finished”—and there will be no more “open-ended” questions associated with it.

In physics there’s sometimes discussion of white holes, which are imagined to effectively be time-reversed black holes, spewing out all possible material that could be captured in a black hole. In metamathematics, a white hole is like a statement that is false and therefore “leads to an explosion”. The presence of such an object in metamathematical space will in effect cause observers to be shredded—making it inconsistent with the coherent construction of higher-level mathematics.

We’ve talked at some length about the “gravitational” structure of metamathematical space. But what about seemingly simpler things like special relativity? In physics, there’s a notion of basic, flat spacetime, for which it’s easy to construct families of reference frames, and in which parallel trajectories stay parallel. In metamathematics, the analog is presumably metamathematical space in which “parallel proof geodesics” remain “parallel”—so that in effect one can continue “making progress in mathematics” by just “keeping on doing what you’ve been doing”.

And somehow relativistic invariance is associated with the idea that there are many ways to do math, but in the end they’re all able to reach the same conclusions. Ultimately this is something one expects as a consequence of fundamental features of the ruliad—and the inevitability of causal invariance in it resulting from the Principle of Computational Equivalence. It’s also something that might seem quite familiar from practical mathematics and, say, from the ability to do derivations using different methods—like from either geometry or algebra—and yet still end up with the same conclusions.

So if there’s an analog of relativistic invariance, what about analogs of phenomena like time dilation? In our Physics Project time dilation has a rather direct interpretation. To “progress in time” takes a certain amount of computational work. But motion in effect also takes a certain amount of computational work—in essence to continually recreate versions of something in different places. But from the ruliad on up there is ultimately only a certain amount of computational work that can be done—and if computational work is being “used up” on motion, there is less available to devote to progress in time, and so time will effectively run more slowly, leading to the experience of time dilation.

So what is the metamathematical analog of this? Presumably it’s that when you do derivations in math you can either stay in one area and directly make progress in that area, or you can “base yourself in some other area” and make progress only by continually translating back and forth. But ultimately that translation process will take computational work, and so will slow down your progress—leading to an analog of time dilation.

In physics, the speed of light defines the maximum amount of motion in space that can occur in a certain amount of time. In metamathematics, the analog is that there’s a maximum “translation distance” in metamathematical space that can be “bridged” with a certain amount of derivation. In physics we’re used to measuring spatial distance in meters—and time in seconds. In metamathematics we don’t yet have familiar units in which to measure, say, distance between mathematical concepts—or, for that matter, “amount of derivation” being done. But with the empirical metamathematics we’ll discuss in the next section we actually have the beginnings of a way to define such things, and to use what’s been achieved in the history of human mathematics to at least imagine “empirically measuring” what we might call “maximum metamathematical speed”.

It should be emphasized that we are only at the very beginning of exploring things like the analogs of relativity in metamathematics. One important piece of formal structure that we haven’t really discussed here is causal dependence, and causal graphs. We’ve talked at length about statements entailing other statements. But we haven’t talked about questions like which part of which statement is needed for some event to occur that will entail some other statement. And—while there’s no fundamental difficulty in doing it—we haven’t concerned ourselves with constructing causal graphs to represent causal relationships and causal dependencies between events.

When it comes to physical observers, there is a very direct interpretation of causal graphs that relates to what a physical observer can experience. But for mathematical observers—where the notion of time is less central—it’s less clear just what the interpretation of causal graphs should be. But one certainly expects that they will enter in the construction of any general “observer theory” that characterizes “observers like us” across both physics and mathematics.