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Full Idea
If we identify logic with first-order logic, and mathematics with the collection of first-order theories, then maybe we can continue to maintain the If-thenist position.
Gist of Idea
Perhaps If-thenism survives in mathematics if we stick to first-order logic
Source
Alan Musgrave (Logicism Revisited [1977], §5)
Book Ref
-: 'British Soc for the Philosophy of Science' [-], p.124
A Reaction
The problem is that If-thenism must rely on rules of inference. That seems to mean that what is needed is Soundness, rather than Completeness. That is, inference by the rules must work properly.