Will_Sawin comments on How to Convince Me That 2 + 2 = 3 - Less Wrong
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well there's the special other thing, the reason you can't explain Peano Arithmetic to a rock, which is that axioms are static sequences of signals, but in addition you have these dynamics.
Best source on this is Lewis Carroll http://www.ditext.com/carroll/tortoise.html
These dynamics are contained within the structure of our thoughts, which is why they're preserved in a thought experiment. But we still have to actually check our thoughts, which are part of reality.
Sorry if this wasn't very coherent.