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Eliezer_Yudkowsky comments on Mixed Reference: The Great Reductionist Project - Less Wrong
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It does indeed seem possible that in the long run we'll end up with one kind of stuff, either from the reduction of logic to physics, or the reduction of physics to math. It's also worth noting that my present model does have magical-reality-fluid in it, and it's conceivable that this will end up not being reduced. But the actual argument is something along the lines of, "We got it down to two crisp things, and all the proposals for three don't have the crisp nature of the two."
That seems to me more like an irreducible string of methods of interpretation. You have physics, whether you like it or not. If you want to understand the physics, you need math. And to use the math, you need logic. Physics itself does not require math or logic. We do, if we want to do anything useful with it. So it's not so much "reducible" as it is "interpretable"--physics is such that turning it into a bunch of numbers and wacky symbols actually makes it more understandable. But to draw from your example, you can't have a physical table with physically infinite apples sitting on it. Yet you can do math with infinities, but all the math in the world won't put more apples on that table.
...and since when is two apples sitting next to each other a pile??
I think you're going to have better luck figuring out how to make the third thing crisp than reducing it to the first two.
I only see one crisp thing and one thing borrowing some of the crispness of the first thing but mostly failing, in your model.
Just as mental gymnastics, what if instead we would be able to reduce physics and logic to magical reality fluid? :)
Anyway, for the "logic from physics" camp the work of Valentin Turchin seems interesting (above all "The cybernetic foundation of mathematics"). Also of notice the recent foundational program called "Univalent foundation".
I don't think you can reduce logic to anything else, since you would need to use logic to perform the reduction.
Well, since nobody have done that yet, we cannot be sure, but for example a reduction of logic to physics could look like this: "for a system built on top of this set of physics laws, this is the set of logical system available to it", which would imply that all the axiomatic system we use are only those accessible via our laws of physics. For an extreme seminal example, Turing machine with infinite time have a very different notion of "effective procedure".
How would one show the above, or even build up a system on top of physical laws without using logic?
I have (at the moment) no idea.
It's clear that such a demonstration needs to use some kind of logic, but I think that doesn't undermine the (possible) reduction: if you show that the (set of) logic available to a system depends on the physical laws, you have shown that our own logic is determined by our own laws. This would entail that (possibly) different laws would have granted us different logics. I'm fascinated for example by the fact that the concept of "second order arithmetical truth" (SOAT) is inacessible by effective finite computation, but there are space-times that allow for infinite computation (and so system inhabiting such a world could possibly grasp effectively SOATs).