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 ...

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