Proof and evidence will always equal Science

[polymath257]

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.

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.