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Most equations that can be solved can already be solved symbolically by things like mathematica or numerically by a grab-bag of numerical techniques that every
by seanhunter 3y ago
Most equations that can be solved can already be solved symbolically by things like mathematica or numerically by a grab-bag of numerical techniques that every maths, engineering and physics student learns at some point. You don't need to have found the solution to P vs NP to do any of that.
I think most mathematicians would say that most of maths is not in solving equations because once you get the right equations, solving them is a simple mechanical process for the most part. Most of maths is figuring out what the right equations are in the first place and proving that what you have done is legitimate.
- js8 3y agoI disagree that converting from an equation to a solution algorithm is a straightforward process. It is very much where mathematics helps, for example just to ensure convergence of the method. Also, the way you model reality (choose the equation) is often done with the limitations of solvability in mind. I also disagree that maths can help you pick the right equations, just like there is no formal method that can verify whether a program specification is appropriate to the problem. Equation (or a model) is just an infinite family of SAT problems, typically parametrized on our understanding what a "number" is. If we had a really good SAT solver, then getting a solution would indeed be a really straightforward process (just pick the number representation) of solving the instance. The same is also true for checking "legitimacy" (which I interpret as consistency) of the model - SAT solver would tell you that the instance is unsatisfiable and why. But you can never formally determine that the chosen SAT instance is appropriate model for the task at hand.