The following text is from the introduction to Carl Runge's "Graphical Methods", in which the author, well before the invention of computers, justifies the existence of numerical analysis:
"A great many if not all of the problems in mathematics may be so
formulated that they consist in finding from given data the values of certain unknown quantities subject to certain conditions.
We may distinguish different stages in the solution of a problem. The first stage we might say is the proof that the quantities sought for really exist. In many ... cases the first stage of the solution may be so easy, that we immediately pass on to the second stage of finding methods to calculate the unknown quantities sought for. Or even if the first stage of the solution is not so easy, it may be expedient to pass on to the second stage. For if we succeed in finding methods of calculation that determine the unknown quantities, the proof of their existence is
There are not a small number of men who believe the task of the
mathematician to end here. This, I think, is due to the fact that the pure mathematician as a rule is not in the habit of pushing his investigation so far as to find something out about the real things of this world. He leaves that to the astronomer, to the physicist, to the engineer. These men, on the other hand, take the greatest interest in the actual numerical values that are the outcome of the mathematical methods of calculation. Suppose the mathematician gives them a method of calculation, perfectly logical and conclusive but taking 200 years of incessant numerical work to complete. They would be justified in thinking that this is not much better than no method at all.
So there arises a third stage of the solution of a mathematical problem in which the object is to develop methods for finding the result with as little trouble as possible. I maintain that this third stage is just as much a chapter of mathematics as the first two stages and it will not do to leave it to the astronomer, to the physicist, to the engineer or whoever applies mathematical methods, for this reason that these men are bent on the results and therefore they will be apt to overlook the full generality of the methods they happen to hit on, while in the hands of the mathematician the methods would be developed from a higher standpoint and their bearing on other problems in other scientific inquiries would be more likely to receive the proper attention."