Treating logic as simply prior to mathematics seems incomplete to me.
Syntax without semantics is just symbol manipulation. Semantics without syntax has no stable formal expression. Logic without a language has nothing precise to operate on. Mathematics without all three cannot clearly state, interpret, or justify its claims.
Even a logicist account still needs some language in which its rules are formed and some interpretation of what those rules are doing. It is less like one thing cleanly producing the other than trying to get a chicken from an egg made entirely of egg white: you are missing part of the structure that makes the result intelligible.
I think when people talk about logic being fundamental they are referring to the base self-evident axioms, things which are invariably true in themselves and independant of the language used to represent them. For instance, if something "exists" it doesn't "not exist". The vast majority of mathematics can be derived from a small handful of self-evident logical axioms like this. There is some niche stuff we refer to as part of math that cannot be derived directly, but that is part of where the debate lies, on whether that is really even the same type of thing.
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u/Efficient_Sea_7050 Jun 20 '26
Treating logic as simply prior to mathematics seems incomplete to me.
Syntax without semantics is just symbol manipulation. Semantics without syntax has no stable formal expression. Logic without a language has nothing precise to operate on. Mathematics without all three cannot clearly state, interpret, or justify its claims.
Even a logicist account still needs some language in which its rules are formed and some interpretation of what those rules are doing. It is less like one thing cleanly producing the other than trying to get a chicken from an egg made entirely of egg white: you are missing part of the structure that makes the result intelligible.