Easier to solve the idealized general case first, so that you know what a solution even looks like
This is not generally true, in my experience. Real world problems come with constraints that often make things substantially easier. The general traveling salesman problem for high N might not generally be tractable, but a specific traveling salesman problem might have all sorts of symmetry you can exploit to make things doable.
I’m reminded of a friend in grad school who created an idealized version of some field theory problems, axiomatized it, and was demonstrating the resulting system had all sorts of really useful properties. Then his adviser showed him that the only field theory that satisfied his axioms was the trivial field theory…
This is not generally true, in my experience. Real world problems come with constraints that often make things substantially easier. The general traveling salesman problem for high N might not generally be tractable, but a specific traveling salesman problem might have all sorts of symmetry you can exploit to make things doable.
I’m reminded of a friend in grad school who created an idealized version of some field theory problems, axiomatized it, and was demonstrating the resulting system had all sorts of really useful properties. Then his adviser showed him that the only field theory that satisfied his axioms was the trivial field theory…