1 Mathematics and Physics Have the Same Foundations

One of the many surprising (and to me, unexpected) implications of our Physics Project is its suggestion of a very deep correspondence between the foundations of physics and mathematics. We might have imagined that physics would have certain laws, and mathematics would have certain theories, and that while they might be historically related, there wouldn’t be any fundamental formal correspondence between them.

But what our Physics Project suggests is that underneath everything we physically experience there is a single very general abstract structurethat we call the ruliadand that our physical laws arise in an inexorable way from the particular samples we take of this structure. We can think of the ruliad as the entangled limit of all possible computationsor in effect a representation of all possible formal processes. And this then leads us to the idea that perhaps the ruliad might underlie not only physics but also mathematicsand that everything in mathematics, like everything in physics, might just be the result of sampling the ruliad.

Of course, mathematics as it’s normally practiced doesn’t look the same as physics. But the idea is that they can both be seen as views of the same underlying structure. What makes them different is that physical and mathematical observers sample this structure in somewhat different ways. But since in the end both kinds of observers are associated with human experience they inevitably have certain core characteristics in common. And the result is that there should be “fundamental laws of mathematics” that in some sense mirror the perceived laws of physics that we derive from our physical observation of the ruliad.

So what might those fundamental laws of mathematics be like? And how might they inform our conception of the foundations of mathematics, and our view of what mathematics really is?

The most obvious manifestation of the mathematics that we humans have developed over the course of many centuries is the few million mathematical theorems that have been published in the literature of mathematics. But what can be said in generality about this thing we call mathematics? Is there some notion of what mathematics is like “in bulk”? And what might we be able to say, for example, about the structure of mathematics in the limit of infinite future development?

When we do physics, the traditional approach has been to start from our basic sensory experience of the physical world, and of concepts like space, time and motionand then to try to formalize our descriptions of these things, and build on these formalizations. And in its early developmentfor example by Euclidmathematics took the same basic approach. But beginning a little more than a century ago there emerged the idea that one could build mathematics purely from formal axioms, without necessarily any reference to what is accessible to sensory experience.

And in a way our Physics Project begins from a similar place. Because at the outset it just considers purely abstract structures and abstract rulestypically described in terms of hypergraph rewritingand then tries to deduce their consequences. Many of these consequences are incredibly complicated, and full of computational irreducibility. But the remarkable discovery is that when sampled by observers with certain general characteristics that make them like us, the behavior that emerges must generically have regularities that we can recognize, and in fact must follow exactly known core laws of physics.

And already this begins to suggest a new perspective to apply to the foundations of mathematics. But there’s another piece, and that’s the idea of the ruliad. We might have supposed that our universe is based on some particular chosen underlying rule, like an axiom system we might choose in mathematics. But the concept of the ruliad is in effect to represent the entangled result of “running all possible rules”. And the key point is then that it turns out that an “observer like us” sampling the ruliad must perceive behavior that corresponds to known laws of physics. In other words, without “making any choice” it’s inevitablegiven what we’re like as observersthat our “experience of the ruliad” will show fundamental laws of physics.

But now we can make a bridge to mathematics. Because in embodying all possible computational processes the ruliad also necessarily embodies the consequences of all possible axiom systems. As humans doing physics we’re effectively taking a certain sampling of the ruliad. And we realize that as humans doing mathematics we’re also doing essentially the same kind of thing.

But will we see “general laws of mathematics” in the same kind of way that we see “general laws of physics”? It depends on what we’re like as “mathematical observers”. In physics, there turn out to be general lawsand concepts like space and motionthat we humans can assimilate. And in the abstract it might not be that anything similar would be true in mathematics. But it seems as if the thing mathematicians typically call mathematics is something for which it isand where (usually in the end leveraging our experience of physics) it’s possible to successfully carve out a sampling of the ruliad that’s again one we humans can assimilate.

When we think about physics we have the idea that there’s an actual physical reality that existsand that we experience physics within this. But in the formal axiomatic view of mathematics, things are different. There’s no obvious “underlying reality” there; instead there’s just a certain choice we make of axiom system. But now, with the concept of the ruliad, the story is different. Because now we have the idea that “deep underneath” both physics and mathematics there’s the same thing: the ruliad. And that means that insofar as physics is “grounded in reality”, so also must mathematics be.

When most working mathematicians do mathematics it seems to be typical for them to reason as if the constructs they’re dealing with (whether they be numbers or sets or whatever) are “real things”. But usually there’s a concept that in principle one could “drill down” and formalize everything in terms of some axiom system. And indeed if one wants to get a global view of mathematics and its structure as it is today, it seems as if the best approach is to work from the formalization that’s been done with axiom systems.

In starting from the ruliad and the ideas of our Physics Project we’re in effect positing a certain “theory of mathematics”. And to validate this theory we need to study the “phenomena of mathematics”. And, yes, we could do this in effect by directly “reading the whole literature of mathematics”. But it’s more efficient to start from what’s in a sense the “current prevailing underlying theory of mathematics” and to begin by building on the methods of formalized mathematics and axiom systems.

Over the past century a certain amount of metamathematics has been done by looking at the general properties of these methods. But most often when the methods are systematically used today, it’s to set up some particular mathematical derivation, normally with the aid of a computer. But here what we want to do is think about what happens if the methods are used “in bulk”. Underneath there may be all sorts of specific detailed formal derivations being done. But somehow what emerges from this is something higher level, something “more human”and ultimately something that corresponds to our experience of pure mathematics.

How might this work? We can get an idea from an analogy in physics. Imagine we have a gas. Underneath, it consists of zillions of molecules bouncing around in detailed and complicated patterns. But most of our “human” experience of the gas is at a much more coarse-grained levelwhere we perceive not the detailed motions of individual molecules, but instead continuum fluid mechanics.

And so it is, I think, with mathematics. All those detailed formal derivationsfor example of the kind automated theorem proving might doare like molecular dynamics. But most of our “human experience of mathematics”where we talk about concepts like integers or morphismsis like fluid dynamics. The molecular dynamics is what builds up the fluid, but for most questions of “human interest” it’s possible to “reason at the fluid dynamics level”, without dropping down to molecular dynamics.

It’s certainly not obvious that this would be possible. It could be that one might start off describing things at a “fluid dynamics” levelsay in the case of an actual fluid talking about the motion of vorticesbut that everything would quickly get “shredded”, and that there’d soon be nothing like a vortex to be seen, only elaborate patterns of detailed microscopic molecular motions. And similarly in mathematics one might imagine that one would be able to prove theorems in terms of things like real numbers but actually find that everything gets “shredded” to the point where one has to start talking about elaborate issues of mathematical logic and different possible axiomatic foundations.

But in physics we effectively have the Second Law of thermodynamicswhich we now understand in terms of computational irreducibilitythat tells us that there’s a robust sense in which the microscopic details are systematically “washed out” so that things like fluid dynamics “work”. Just sometimeslike in studying Brownian motion, or hypersonic flowthe molecular dynamics level still “shines through”. But for most “human purposes” we can describe fluids just using ordinary fluid dynamics.

So what’s the analog of this in mathematics? Presumably it’s that there’s some kind of “general law of mathematics” that explains why one can so often do mathematics “purely in the large”. Just like in fluid mechanics there can be “corner-case” questions that probe down to the “molecular scale”and indeed that’s where we can expect to see things like undecidability, as a rough analog of situations where we end up tracing the potentially infinite paths of single molecules rather than just looking at “overall fluid effects”. But somehow in most cases there’s some much stronger phenomenon at workthat effectively aggregates low-level details to allow the kind of “bulk description” that ends up being the essence of what we normally in practice call mathematics.

But is such a phenomenon something formally inevitable, or does it somehow depend on us humans “being in the loop”? In the case of the Second Law it’s crucial that we only get to track coarse-grained features of a gasas we humans with our current technology typically do. Because if instead we watched and decoded what every individual molecule does, we wouldn’t end up identifying anything like the usual bulk “Second-Law” behavior. In other words, the emergence of the Second Law is in effect a direct consequence of the fact that it’s us humanswith our limitations on measurement and computationwho are observing the gas.

So is something similar happening with mathematics? At the underlying “molecular level” there’s a lot going on. But the way we humans think about things, we’re effectively taking just particular kinds of samples. And those samples turn out to give us “general laws of mathematics” that give us our usual experience of “human-level mathematics”.

To ultimately ground this we have to go down to the fully abstract level of the ruliad, but we’ll already see many core effects by looking at mathematics essentially just at a traditional “axiomatic level”, albeit “in bulk”.

The full storyand the full correspondence between physics and mathematicsrequires in a sense “going below” the level at which we have recognizable formal axiomatic mathematical structures; it requires going to a level at which we’re just talking about making everything out of completely abstract elements, which in physics we might interpret as “atoms of space” and in mathematics as some kind of “symbolic raw material” below variables and operators and everything else familiar in traditional axiomatic mathematics.

The deep correspondence we’re describing between physics and mathematics might make one wonder to what extent the methods we use in physics can be applied to mathematics, and vice versa. In axiomatic mathematics the emphasis tends to be on looking at particular theorems and seeing how they can be knitted together with proofs. And one could certainly imagine an analogous “axiomatic physics” in which one does particular experiments, then sees how they can “deductively” be knitted together. But our impression that there’s an “actual reality” to physics makes us seek broader laws. And the correspondence between physics and mathematics implied by the ruliad now suggests that we should be doing this in mathematics as well.

What will we find? Some of it in essence just confirms impressions that working pure mathematicians already have. But it provides a definite framework for understanding these impressions and for seeing what their limits may be. It also lets us address questions like why undecidability is so comparatively rare in practical pure mathematics, and why it is so common to discover remarkable correspondences between apparently quite different areas of mathematics. And beyond that, it suggests a host of new questions and approaches both to mathematics and metamathematicsthat help frame the foundations of the remarkable intellectual edifice that we call mathematics.