develop tools that allow us to probe a new domain which we understand less well. After repeated probing, we develop an intuition/understanding of this new domain
but I don’t see how it can be unified with the OP’s thesis.
I’m saying the study of novel mathematical structures is analogous to such probing. At first, one can only laboriously perform step-by-step deductions from the axioms, but as one does many such deductions, intuition and understanding can be developed. This is enabled by formalization.
That certainly makes sense. For example, there are quite a few abstraction steps between the Fundamental theorem of calculus and certain commutative diagrams.
There is definitely the step of
but I don’t see how it can be unified with the OP’s thesis.
I’m saying the study of novel mathematical structures is analogous to such probing. At first, one can only laboriously perform step-by-step deductions from the axioms, but as one does many such deductions, intuition and understanding can be developed. This is enabled by formalization.
That certainly makes sense. For example, there are quite a few abstraction steps between the Fundamental theorem of calculus and certain commutative diagrams.