23 The Physicalized Laws of Mathematics

The concept of the ruliad suggests there is a deep connection between the foundations of mathematics and physics. And now that we have discussed how some of the familiar formalism of mathematics can “fit into” the ruliad, we are ready to use the “bridge” provided by the ruliad to start exploring how to apply some of the successes and intuitions of physics to mathematics.

A foundational part of our everyday experience of physics is our perception that we live in continuous space. But our Physics Project implies that at sufficiently small scales space is actually made of discrete elementsand it is only because of the coarse-grained way in which we experience it that we perceive it as continuous.

In mathematicsunlike physicswe’ve long thought of the foundations as being based on things like symbolic expressions that have a fundamentally discrete structure. Normally, though, the elements of those expressions are, for example, given human-recognizable names (like 2 or Plus). But what we saw in the previous section is that these recognizable forms can be thought of as existing in an “anonymous” lower-level substrate made of what we can call atoms of existence or emes.

But the crucial point is that this substrate is directly based on the ruliad. And its structure is identical between the foundations of mathematics and physics. In mathematics the emes aggregate up to give us our universe of mathematical statements. In physics they aggregate up to give us our physical universe.

But now the commonality of underlying “substrate” makes us realize that we should be able to take our experience of physics, and apply it to mathematics. So what is the analog in mathematics of our perception of the continuity of space in physics? We’ve discussed the idea that we can think of mathematical statements as being laid out in a metamathematical spaceor, more specifically, in what we’ve called an entailment fabric. We initially talked about “coordinatizing” this using axioms, but in the previous section we saw how to go “below axioms” to the level of “pure emes”.

When we do mathematics, though, we’re sampling this on a much higher level. And just like as physical observers we coarse grain the emes (that we usually call “atoms of space”) that make up physical space, so too as “mathematical observers” we coarse grain the emes that make up metamathematical space.

Foundational approaches to mathematicsparticularly over the past century or sohave almost always been based on axioms and on their fundamentally discrete symbolic structure. But by going to a lower level and seeing the correspondence with physics we are led to consider what we might think of as a higher-level “experience” of mathematicsoperating not at the “molecular dynamics” level of specific axioms and entailments, but rather at what one might call the “fluid dynamics” level of larger-scale concepts.

At the outset one might not have any reason to think that this higher-level approach could consistently be applied. But this is the first big place where ideas from physics can be used. If both physics and mathematics are based on the ruliad, and if our general characteristics as observers apply in both physics and mathematics, then we can expect that similar features will emerge. And in particular, we can expect that our everyday perception of physical space as continuous will carry over to mathematics, or, more accurately, to metamathematical space.

The picture is that we as mathematical observers have a certain “size” in metamathematical space. We identify conceptslike integers or the Pythagorean theoremas “regions” in the space of possible configurations of emes (and ultimately of slices of the ruliad). At an axiomatic level we might think of ways to capture what a typical mathematician might consider “the same concept” with slightly different formalism (say, different large cardinal axioms or different models of real numbers). But when we get down to the level of emes there’ll be vastly more freedom in how we capture a given conceptso that we’re in effect using a whole region of “emic space” to do so.

But now the question is what happens if we try to make use of the concept defined by this “region”? Will the “points in the region” behave coherently, or will everything be “shredded”, with different specific representations in terms of emes leading to different conclusions?

The expectation is that in most cases it will work much like physical space, and that what we as observers perceive will be quite independent of the detailed underlying behavior at the level of emes. Which is why we can expect to do “higher-level mathematics”, without always having to descend to the level of emes, or even axioms.

And this we can consider as the first great “physicalized law of mathematics”: that coherent higher-level mathematics is possible for us for the same reason that physical space seems coherent to observers like us.

We’ve discussed several times before the analogy to the Second Law of thermodynamicsand the way it makes possible a higher-level description of things like fluids for “observers like us”. There are certainly cases where the higher-level description breaks down. Some of them may involve specific probes of molecular structure (like Brownian motion). Others may be slightly more “unwitting” (like hypersonic flow).

In our Physics Project we’re very interested in where similar breakdowns might occurbecause they’d allow us to “see below” the traditional continuum description of space. Potential targets involve various extreme or singular configurations of spacetime, where in effect the “coherent observer” gets “shredded”, because different atoms of space “within the observer” do different things.

In mathematics, this kind of “shredding” of the observer will tend to be manifest in the need to “drop below” higher-level mathematical concepts, and go down to a very detailed axiomatic, metamathematical or even eme levelwhere computational irreducibility and phenomena like undecidability are rampant.

It’s worth emphasizing that from the point of view of pure axiomatic mathematics it’s not at all obvious that higher-level mathematics should be possible. It could be that there’d be no choice but to work through every axiomatic detail to have any chance of making conclusions in mathematics.

But the point is that we now know there could be exactly the same issue in physics. Because our Physics Project implies that at the lowest level our universe is effectively made of emes that have all sorts of complicatedand computationally irreduciblebehavior. Yet we know that we don’t have to trace through all the details of this to make conclusions about what will happen in the universeat least at the level we normally perceive it.

In other words, the fact that we can successfully have a “high-level view” of what happens in physics is something that fundamentally has the same origin as the fact that we can successfully have a high-level view of what happens in mathematics. Both are just features of how observers like us sample the ruliad that underlies both physics and mathematics.