Context
In mathematics, solving problems is not an end in itself, but rather a means to discover and create new knowledge, whether theorems, lemmas, concepts and even theory, above all with open and difficult problems, such as the millennium problems, among others.
For example, keeping the proportions and only for didactic and general-purpose ends, the search for solutions to a particular quadratic equation led to inventing the concept of complex numbers. Another example is the development of distribution theory to ground the numerical solutions of partial differential equations that apparently had no analytical solution (such as those of Navier Stokes). That created and accumulated knowledge was passed on from generation to generation of mathematicians in order to develop more and better theories and methods.
Recently, a group of Fields medalists published an open letter warning about this, and it will be part of the discussion in academia over the coming years.
If a machine arrives that solves that problem straight away, all that fruitful path of finding methods and ways to reach the solution of that problem is lost. Now, there is a shift in focus: mathematicians and interested parties will have to understand the solution to the problem, and from there they have to see which was the step or the inference that was missing in order to reach that solution.
Would finite element methods exist if a machine had solved all PDE problems analytically? I do not know, but I am clear that many reflections on this are coming, since the future of mathematics is at stake.
(Post written 100% by a human 😌✨️)