(December 6, 2021 at 11:30 am)polymath257 Wrote:(December 6, 2021 at 11:21 am)Jehanne Wrote: 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.
Again, a nit-pick. We use numerical solutions because we want to test things with numbers.
Mathematically, the equations can be shown to have solutions (well, it may be quite difficult to do so--see Millennium Problems). The numerical techniques are ways to approximate those *exact* solutions.
But we do approximations even for 'standard' functions. Saying a value is e^(1.5) or sin(.32) is usually NOT what is required (even if it is exact). Instead, a numerical approximation is what is wanted.
The TSP has an exact solution. We just don't know how to find it efficiently.
And, as you know, there are a whole host of problems that have no solution (Turnings halting problem); Godel's Incompleteness theorem also puts limits on the ability of formal systems to solve certain problems.
That consciousness and intentionally are not reducible is hardly a reason to embrace one religion, or any other; it's the either/or fallacy at work here.
