In my 'Introduction to Logic' class, my professor told us that half of the class will be based on "mathematical" operations withing logic. After looking through the textbook, I realized that he meant things like the predicate calculus and propositional logic. I know that he probably just meant that these logic topics are "mathematical" in that they require certain symbol interpretation and manipulation, but it got me thinking about the essence of a mathematical system vs. a logical one. Set Theory, and similar foundational theories (such as Proof Theory), are based on axiomatic systems that are built upon the rules of logic. Of course, it depends on the textbook one is using to learn the mathematical system, as different logical symbols and relationships may be referenced in the system. For example, Kunen's book on Set Theory and foundations uses the first-order predicate calculus (if I remember correctly), so the
\forall symbol is defined based on relationships of other symbols. Other textbooks may strictly define the symbol as its own symbol.
However, would it be valid to define logical operations as mathematical ones? The logical "or" symbol can arguably be a mathematical symbol (union in Set Theory). But Set Theory is itself based on these logical rules, so is it then not recursive to say that logic is based on Set Theory which is based on Logic? There are also things like Number Theory or Abstract Algebra, which are not necessarily based on any logical rules (unless one formalizes the Peano axioms and such). Does it still then follow that mathematics is built on logic? And if so, are the logical operations upon which it is built considered "mathematical" operations? What even qualifies operations as being "mathematical"?
At the core of my questions is the concept of mathematical operations and whether or not such operations are valid within logical systems. I also wonder if mathematical operations simply depend on interpretation and meaning rather than something innate, meaning any mathematical operation could be a logical one depending on how we define it.
This is only an introduction for me so please excuse my ignorance.