(December 6, 2021 at 9:04 am)polymath257 Wrote:(December 6, 2021 at 8:32 am)Jehanne Wrote: The Schrodinger equation, Einstein's equations of General Relativity and (especially) the Dirac equation have very few analytic (exact) solutions; most, to nearly all, real world sets of differential (in most cases, partial) equations do not have exact solutions either. Instead, numerical solutions are the norm. This fact should nix any notion of reducibility. For instance, ecologists do not need Quatum Mechanics to do their jobs.
Just a nit-pick. They have exact solutions, often unique ones. Those solutions just can't be given in terms of the relatively few functions we typically work with. We can also describe the properties of those solutions, often, without getting explicit solutions in terms of 'elementary functions'.
Yeah, you can use approximations to get "exact" solutions. The TSP (Traveling Salesman/Salesperson) is one example of many, converting an intractable NP problem to a more tractable polynomial one by making certain assumption/concessions. But, still, the fact remains that most problems in physics and engineering are so complex that they are solved numerically.
