Without a stable analytic-synthetic distinction, Carnap cannot coherently designate mathematical calculi as purely formal while simultaneously explaining their indispensable role in confirmed empirical theories.
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Scientific theories that have been tested and verified through real-world experiments and observations, so we have good reason to believe they're true.
Empirical theories(the actual science that math supposedly explains)
Scientific theories that are based on real-world observations and experiments, not just pure logic or imagination.
Indispensable role(as used in philosophy of science)
Being absolutely necessary and impossible to replace; without it, the whole thing falls apart.
Mathematical calculi(what the statement says Carnap wants to call purely formal)
Formal systems of rules and symbols (like algebra or logic notation) used to solve problems or prove things without needing to think about what the symbols mean in the real world.
Purely formal(as describing what the inference rule is supposed to be)
Working based only on logical structure and rules, without depending on beliefs about what actually exists in the world.