Yeah, I think the “varying the axioms” thing makes more sense for math in particular, not so much the other sciences. As you say, the equivalent thing in the natural sciences is more like experimentation.
Maybe we can roughly unify them? In both cases, we have some domain where we understand phenomena well. Using this understanding, we 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 as well, allowing us to develop tools to explore further domains.
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.
Yeah, I think the “varying the axioms” thing makes more sense for math in particular, not so much the other sciences. As you say, the equivalent thing in the natural sciences is more like experimentation.
Maybe we can roughly unify them? In both cases, we have some domain where we understand phenomena well. Using this understanding, we 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 as well, allowing us to develop tools to explore further domains.
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.