I have an intuition that physics, logic, and computation are all closely connected, and that their bounded versions are also closely connected. I think this intuition started as wishful thinking. I didn't have time to study all three of my favorite subjects, so I hoped that studying one would eventually lead to all the others. The rest of this post aims to convince you (and me) that it wasn't just wishful thinking.

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Imagine you live in the world of mathematics. The things around you are groups and manifolds and numbers. You can't touch anything. You can touch symbols floating around the objects, and only by moving these symbols around can you interact with the objects. This is Logic. Eventually you notice that the symbols are made of the same stuff as everything else, and have their own symbols. By thinking hard and manipulating these symbol symbols very carefully, you find a way to get the symbols to manipulate themselves. This is a Computer and with it you interact with the world more easily. Logic actualizes computers.

Imagine, in your platonic world, you come across a library which claims to contain proofs of every true fact of arithmetic, and only true facts. You also discover that you can build a Computer out of not just logic, but also the numbers around you. Since it's made of numbers, the library should know about it. You teach the computer to ask the library for proofs that it will finish computing and halt, and for proofs that it will run forever. You also teach the computer to disobey the library. If the library tells the computer it will halt, it should run forever. If the library tells the computer it will run forever, it should halt. If the library gives you any proof, it will have lied. If it doesn't, then there are facts it does not know, so it lied at the beginning. The library fades away, it was an illusion all along. Computers restrain logic.

Imagine you grow tired of your platonic world and you want to build a universe with planets and people and everything else. What could you build it out of? The only thing at your disposal is math. You teach your computer what numbers are needed to represent each bit of stuff, and how every bit of stuff will change at every step, then turn it on. You watch as the platonic world of numbers transmutes into the world of physics and stuff. Computers actualize physics.

Imagine looking into your universe from the outside. What would you see? If you made the universe infinite, you would see every logically possible configuration of stuff somewhere inside, constrained by the laws of physics you chose. If you made it grow in a certain way, you would expect to see the universe split into different universes like the first, but with different laws of physics. If you built in something like Quantum Mechanics, you would see the universe developing in such a way that pieces stop interacting, each split recording every "decision". Perhaps, if you build it right, and you look far enough, you would see every conceivable (consistent) mathematical structure. Physics actualizes logic.

Imagine you taught your universe how to make a particle that knew the entire future of any lightbulb it hits, and scatters to the left if the lightbulb is going to turn on, and scatters right if the lightbulb won't. Consider a turned off lightbulb in a room with a light-switch on its right, and a self-destruct switch on its left. If the lightbulb will turn on, the particle should scatter left and destroy the light before it will ever turn on. Contradiction. If the lightbulb never turns on, the particle should scatter right and turn the light on. Contradiction. The particle fades away, it was an illusion all along. Logic constrains physics.

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Imagine you live in an actual universe. There are planets and people, and everything else. You had a dream about symbols moving themselves, so you moved some particles in a clever way so that they mirror the computer in your dream. But with actual time you have to wait for your computer to finish thinking, which sometimes takes forever, and sometimes it doesn't have enough particles in the right places for it to mirror all the calculations. These computers are much more constrained by physics, so using them is much more complex. Complexity is for computers inside universes.

Imagine that you are curious. You see planets and people and computers and everything else around you, and you wonder how they work. You do some experiments and discover that there seem to be some rules that everything obeys, rules your computer can learn. Given any set up, could you figure out how it will behave? You can build computers out of the universe, so the rules should know about them. You attach a lightbulb that turns on only if the computer halts, and tell the computer to use the rules to figure out whether any photons leave the lightbulb in this configuration of particles, but you teach the computer to disobey whatever it derives. If it derives photons leaving the lightbulb, it will run forever and keep the lightbulb off. If it sees to the end of time and learns that the lightbulb never turns on, it stops and turns on the light. Both contradictions. Therefore, it can't know its behavior forever in the future, and keeps calculating forever. The laws aren't enough to decide everything quickly. Bounded Physical Laws are for universes containing computers.

Imagine you are playing a game with your friend where he gives you a number and you have to factor it. This game takes a really long time for bigger numbers, but you can't just make numbers up to trick your friend because he can check your numbers easily by multiplying. Some problems are easy-Peasy, some problems are hard. The best games involve problems that seem Not easy, but can be checked easy-Peasy (NP). Logic is a game like this. After all, a proof is easy to check, but often hard to find. Maybe, by some clever trick, all problems which are hard but can be checked easily are actually easy (P = NP). Then Math would be easy. Complexity automates Logic (or shows why it can't be automated).

Imagine you found a particle that tries to determine if a lightbulb will light up and scatter left if it will, and right if it won't, but it only has access to limited information about the lightbulb. The particle can only know with probability p that the lightbulb will turn on. To comply with the rules best it can, it will scatter left with probability p and scatter right with probability 1-p. Consider a turned off lightbulb in a room with a light-switch on its right, and a self-destruct switch on its left. If it scatters left with probability p>50% (or p<50%), then the lightbulb will never turn on with probability (1-p) > 50% (or (1-p) < 50%), a contradiction. The same thing happens if it scatters right. But, if p = 50% exactly, everything stays consistent. The particle scatters left and right randomly. Bounded Physical Laws preserve Logical Consistency (and randomize in case of inconsistency).

Imagine you find that the universe is different than you thought, it actually works on complex numbers and probabilities and "entanglement" and it turns out that you can build something that does computation in a completely different way from your computer. When you use this new machine, it seems like it can play games like factoring much faster than you could before, and you wonder if it can do all hard problems faster. If it can, you can automate math (and maybe physics and art and music and politics and...), which seems just way too powerful, so you expect it can't. If you found even newer laws, and built even stranger computers, would they ever be able to do something like that? Maybe, but it still seems unlikely. Maybe even as Bounded Physical Laws actualize complexity (by offering new designs over time), complexity constrains physics (By preventing P = NP).

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Thinking Machines live in an actual universe. They act within the constraints of a Bounded Physical Universe, they think within the constraints of Complexity.

We live in an actual universe. We see planets and people and computers and everything else, but the people and the computers look more and more the same, and because we're forcing them to look alike and not understanding why they look alike, we aren't sure how the computers will turn out. And the planets and everything else don't have any say at all, only the people do. Maybe we constrain the computers and the computers actualize us. Or maybe we actualize the computers, and the computers constrain us.

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Diagram

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Final thoughts:
These parables are pretty contrived. It would be easy to flip around the constraint/actualization directions in a lot of places, and in other places there are much deeper connections than those presented (e.g. Curry-Howard Correspondence between Logic and Computation). The point was to get across my impression that all of these fields are very closely related, and that by considering the relationships between physics and computational complexity (as bounded versions of their idealized forms) you might learn something about the bounded version of idealized Logic, which I wildly speculate to be something helpful to AI Safety, like embedded agency or something. Big thanks to MSFP for connecting me with loads of brilliant people willing to think through crazy ideas, and a special thanks to Mathijs Henquet for drawing the initial form of the diagram above.

Further Reading (not exhaustive, not always related, maybe helpful):

Logic actualizes Computers - Turing's original paper
Computers restrain Logic - Incompleteness Ex Machina by Sebastian Oberhoff
Computers actualize Physics - David Deutsch
Physics actualizes Logic - Tegmark's Universes
Logic constrains Physics - If this isn't true, I don't know what is.

Bounded Physical Laws are for universes containing Computers - Undecidability of the Spectral Gap
Physics constrains Computers and Complexity is for Computers inside universes - (The Nature of Computation by Christopher Moore is a standard reference)
Bounded Physical Laws preserve Logical Consistency - Reflective Oracles
Complexity automates Logic and Complexity and Physical laws are connected - NP Complete Problems and Physical Reality