Well, hyperbolic geometry was counterintuitive (the relevant 2d manifold does not embed isometrically into 3d Euclidean space), but spherical was not. Euclid had all the knowledge needed to construct spherical geometry. In fact, most of it was constructed experimentally due to the need to describe the celestial sphere and to navigate the seas, it just wasn’t connected to the 4 postulates. Once this connection is made and the 5th postulate is proven independent of the first 4, but being a limit of the spherical geometry in the limit of an infinitely large sphere, at least informally, the step toward hyperbolic geometry is quite natural.
the accumulation of “data” came from working out the consequence of varying the axioms
I don’t know if that is how it worked out in this case. It does not seem to be how our understanding advances in general. In natural sciences it tends to be driven by experiment, which is enabled by a mix of knowledge and technology. The revolutionary Hodgkin and Huxley model of action potential propagation depended on advances in technology/electrophysiology (20 um silver electrodes, availability of giant squid axons), in mathematics and EE (circuit analysis) and something else, that allowed them to ask the right questions. It is not clear to me that it was a break from intuition to axioms in this case, seems unlikely.
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.
Well, hyperbolic geometry was counterintuitive (the relevant 2d manifold does not embed isometrically into 3d Euclidean space), but spherical was not. Euclid had all the knowledge needed to construct spherical geometry. In fact, most of it was constructed experimentally due to the need to describe the celestial sphere and to navigate the seas, it just wasn’t connected to the 4 postulates. Once this connection is made and the 5th postulate is proven independent of the first 4, but being a limit of the spherical geometry in the limit of an infinitely large sphere, at least informally, the step toward hyperbolic geometry is quite natural.
I don’t know if that is how it worked out in this case. It does not seem to be how our understanding advances in general. In natural sciences it tends to be driven by experiment, which is enabled by a mix of knowledge and technology. The revolutionary Hodgkin and Huxley model of action potential propagation depended on advances in technology/electrophysiology (20 um silver electrodes, availability of giant squid axons), in mathematics and EE (circuit analysis) and something else, that allowed them to ask the right questions. It is not clear to me that it was a break from intuition to axioms in this case, seems unlikely.
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.