> Nearly every physical system is implementing nearly every pure algorithm, ie., every computable function.

Sure. And also about the air and neural network. This is all irrelevant, for the same reason that every possible program and every possible copyrighted work being contained in the base-10 expansion of the number PI is irrelevant. Or that a photo of every event that ever happened anywhere is contained in the space of all possible (say) 1024x1024 24-bit-per-pixel bitmaps. It's all in there, but it's irrelevant, because you have no way of determining which combinations of pixels are photos of real events. And any random sample you take is most certainly not it.

> All actual algorithms, when specified, have massive non-computational holes in them called 'i/o', device access etc.

Only if you stick to a subset of maths you use for algorithms, and forget about everything else. The only actual hole there would be in your memory, or knowledge.

Sure, I/O doesn't play nice with functional programming. It doesn't stop functional programming from being useful with real computers in the real world. We have other mathematical frameworks to describe things that timeless, stateless computation formalisms can't. You are allowed to use more than one at the same time!

> You cannot implement a web browser on 'gears' in any useful sense, in any sense in which the partices of their air arent already implementing the web browser.

Of course I can. Here is the dumb approach for the sake of proof (one can do better with more effort):

1. Find a reference for how to make a NAND gate with gears. Maybe other logic gates too, but it's not strictly necessary.

2. Find the simplest CPU architecture someone made a browser for, for which you can find or get connection-level schematics of the chip; repeat for memory and other relevant components, up to the I/O boundary. Make sure to have some storage in there as well.

3. Build electricity/rotational motion transducers, wire them to COTS display, keyboard, mouse and Ethernet ports.

4. Mechanically translate all the logic gates and connections from point 2. to their gear equivalents using table 1., and hook up to 3.

5. Set the contents of the storage to be the same as a reference computer with a web browser on it.

6. Run the machine.

Of course, this would be a huge engineering challenge - making that many gears work together, in spite of gravity, inertia, tension and wear, and building it in under a lifetime and without bankrupting the world. Might be helpful to start by building tools to make tools to make tools, etc.

But the point is, it's a dumb mechanical process, trivially doable in principle. May be difficult with physical gears, but hey, it worked in Minecraft. People literally built CPUs inside a videogame this way.

> We have rigged a system of electrical signals to immitate a formal programming langauge -- but this is an illusion.

It's the other way around: we've rigged a system of electrical signals to make physical a formal theoretical program. We can also rig a system of optical signals, or hydraulic signals, or pidgeon-delivered paper signals, to "immitate a formal programming language" and implement a formal theoretical program - and as long as those systems immitate/implement the same formal mathematical model, they're functionally equivalent and interchangeable.