Decent Approximation

I have always been more interested in exact solutions to problems, and disregarded the problem of efficiency: is it practical? is it fast? Thought I understand these matters and when it comes to real world problems I'm much more sensitive to problems of efficiency, I don't really care about them when it comes to theory. I'm more interested whether there exists a solution to a bunch of problems, and whether there is a general method to solve a bunch of problems, rather than finding a fast and efficient way to solve them.

Lots of problems are known to be solvable but to solve them exactly is practically impossible, and people are more interested in decent approximations to them. I was neither interested in these approximations since there always seems to be a certain mismatch between the approximation and the nature of the corresponding solution. True, if you're a constructivist you'll say that, say, a real number is always in the process of being constructed and there's no way to specify the real number exactly, but still identifying that very real number, or the generator of that real number, is different from giving an approximation to that real number by a method that has nothing to do with the number itself. Using LLM to generate seemingly human-like responses doesn't tell us anything about intelligence and human emotions, they are really just decent - or not that decent - approximations.

By the way, later on I found that the theory of complexity actually tries to classify problems and relate seemingly totally different problems to each other, this is interesting, but it still has nothing to do with finding a practical solution to these problems. The theory of complexity tell us the nature of the problems in terms of their computational complexity, it doesn't try to practically solve them.

There is one thing that recently I've found interesting and worthy of studying when it comes to decent approximations: what is it that makes a decent approximation a decent approximation to the corresponding solution? Often these are obvious, but sometimes they are not, and why a certain approximation "converges" much more rapidly than other approximations is not at all obvious. Is it the case that when an approximation is somehow closer in its nature/essence to the corresponding solution it converges more rapidly and is a better approximation, or is it the case that whatever "nature" or "essence" has nothing to do with whether the approximation is closer to the exact solution?

I'm bewildered and don't know how and where to start.

• Modes of convergence
• Asymptotic methods