The ruliad in a sense contains all structurally possible mathematics—including all mathematical statements, all axiom systems and everything that follows from them. But mathematics as we humans conceive of it is never the whole ruliad; instead it is always just some tiny part that we as mathematical observers sample.
We might imagine, however, that this would mean that there is in a sense a complete arbitrariness to our mathematics—because in a sense we could just pick any part of the ruliad we want. Yes, we might want to start from a specific axiom system. But we might imagine that that axiom system could be chosen arbitrarily, with no further constraint. And that the mathematics we study can therefore be thought of as an essentially arbitrary choice, determined by its detailed history, and perhaps by cognitive or other features of humans.
But there is a crucial additional issue. When we “sample our mathematics” from the ruliad we do it as mathematical observers and ultimately as humans. And it turns out that even very general features of us as mathematical observers turn out to put strong constraints on what we can sample, and how.
When we discussed physics, we said that the central features of observers are their computational boundedness and their assumption of their own persistence through time. In mathematics, observers are again computationally bounded. But now it is not persistence through time that they assume, but rather a certain coherence of accumulated knowledge.
We can think of a mathematical observer as progressively expanding the entailment fabric that they consider to “represent mathematics”. And the question is what they can add to that entailment fabric while still “remaining coherent” as observers. In the previous section, for example, we argued that if the observer adds a statement that can be considered “logically false” then this will lead to an “explosion” in the entailment fabric.
Such a statement is certainly present in the ruliad. But if the observer were to add it, then they wouldn’t be able to maintain their coherence—because, whimsically put, their mind would necessarily explode.
In thinking about axiomatic mathematics it’s been standard to say that any axiom system that’s “reasonable to use” should at least be consistent (even though, yes, for a given axiom system it’s in general ultimately undecidable whether this is the case). And certainly consistency is one criterion that we now see is necessary for a “mathematical observer like us”. But one can expect that it’s not the only criterion.
In other words, although it’s perfectly possible to write down any axiom system, and even start generating its entailment cone, only some axiom systems may be compatible with “mathematical observers like us”.
And so, for example, something like the Continuum Hypothesis—which is known to be independent of the “established axioms” of set theory—may well have the feature that, say, it has to be assumed to be true in order to get a metamathematical structure compatible with mathematical observers like us.
In the case of physics, we know that the general characteristics of observers lead to certain key perceived features and laws of physics. In statistical mechanics, we’re dealing with “coarse-grained observers” who don’t trace and decode the paths of individual molecules, and therefore perceive the Second Law of thermodynamics, fluid dynamics, etc. And in our Physics Project we’re also dealing with coarse-grained observers who don’t track all the details of the atoms of space, but instead perceive space as something coherent and effectively continuous.
And it seems as if in metamathematics there’s something very similar going on. As we began to discuss in the very first section above, mathematical observers tend to “coarse grain” metamathematical space. In operational terms, one way they do this is by talking about something like the Pythagorean theorem without always going down to the detailed level of axioms, and for example saying just how real numbers should be defined. And something related is that they tend to concentrate more on mathematical statements and theorems than on their proofs. Later we’ll see how in the context of the ruliad there’s an even deeper level to which one can go. But the point here is that in actually doing mathematics one tends to operate at the “human scale” of talking about mathematical concepts rather than the “molecular-scale details” of axioms.
But why does this work? Why is one not continually “dragged down” to the detailed axiomatic level—or below? How come it’s possible to reason at what we described above as the “fluid dynamics” level, without always having to go down to the detailed “molecular dynamics” level?
The basic claim is that this works for mathematical observers for essentially the same reason as the perception of space works for physical observers. With the “coarse-graining” characteristics of the observer, it’s inevitable that the slice of the ruliad they sample will have the kind of coherence that allows them to operate at a higher level. In other words, mathematics can be done “at a human level” for the same basic reason that we have a “human-level experience” of space in physics.
The fact that it works this way depends both on necessary features of the ruliad—and in general of multicomputation—as well as on characteristics of us as observers.
Needless to say, there are “corner cases” where what we’ve described starts to break down. In physics, for example, the “human-level experience” of space breaks down near spacetime singularities. And in mathematics, there are cases where for example undecidability forces one to take a lower-level, more axiomatic and ultimately more metamathematical view.
But the point is that there are large regions of physical space—and metamathematical space—where these kinds of issues don’t come up, and where our assumptions about physical—and mathematical—observers can be maintained. And this is what ultimately allows us to have the “human-scale” views of physics and mathematics that we do.