I understand what you're getting at with the 0 = 1 thing. You're saying that we can represent negation of a proposition S as S → ⊥, where ⊥ is set to 0 = 1. And the empty set is the set with no elements, which per Frege is supposedly because it contains all self-contradictory elements, except "of course nothing self-contradictory actually exists." But so the relationship between zero, the empty set, contradictions, falsity, etc. is there, it's intuitive, and what is it an intuition of?
Here's a contentious proposal which I haven't found a citation for, yet (give me time, I'm sure it's out there somewhere...):
- The real overarching debate, in the philosophy of mathematics, is not between realism and non-realism, but between ante rem and in re realism.
The one is usually known as "Platonic," the other "Aristotelian." But the point is that appeals to intuitions, fictions, modal dimensions, a priori games/symbolism, etc. are not only mostly interchangeable, in practice, with each other, but they are not even legitimately describable as "non-real," ultimately. For if we really intuit that or really narrate that or really play the game such that, or whatever, then there are intuitions, narrations, games, and the like. But they are in re things, not floating in some other world. At "worst," in the awesome light of multiversal standpoints, we can say that it is some confluence of abstract free will and pure conceptions of things that is at play here, but we still need not absolutely have our will occupying a place in another world, do we?
Still, we would like to know whether it is merely "conventional" or more like "naturally isomorphic," that 0 = 1 and negation are related as they are. Realism isn't just about "things," but about objectivity, at least to a decent extent. General murkiness aside, let's say that even if the exact details of this case are "conventional," they reflect a natural "theme" of using two different items flagging an equation sign as a primitive sentential representation of falsity. Like, 0 = 1 itself is the first false non-negative sentence possible in basic arithmetic (the first negative one would be 0 ≠ 0, I suppose).
Then that's the next layer of the dialectic: granting that either way, some sort of realism is at hand, here, are more particular cases of this phenomenon trivial or not? For it is often a sign of a theory's maturity, that it is able to pass beyond fretting over whether it is a false or a true theory, to trying to evolve into a non-trivial notion of the way things are. But so 0 = 1 is a "trivially" false sentence; or, it's a "base case" of falsity (in base-case arithmetic, or arithmetic-as-a-base-case itself!). So it might support a trivial realism, but not a very substantive one, here. (Unless, of course, triviality is turned upon itself, as a matter of inquiry: for then it is not necessarily externally trivial that it is internally trivial that 0 = 1 has a "realistic" significance.)