Numerical Analysis vs. Mathematics: Modern mathematics often does not deal with the practical problems which face numerical analysis. 
I hope I have shown not that mathematicians are incompetent or wrong, but why I believe that their interests, tastes, and objectives are frequently different from those of practicing numerical analysts, and why activity in numerical analysis should be evaluated by its own standards and not by those of pure mathematics.  I hope I have also shown you that much of the "art form" of mathematics consists of delicate, "noise-free" results, while many areas of applied mathematics, especially numerical analysis, are dominated by noise.  Again, in computing the process is fundamental, and rigorous mathematical proofs are often meaningless in computing situations.  Finally, in numerical analysis, as in engineering, choosing the right model is more important than choosing the model with the elegant mathematics.