How to define applied mathematics?

Question

In engineering, we use the term applied mathematics a lot. However, I've noticed that the exposure for proof is scarce. When I was an undergrad, even though the textbooks provided some details about some proofs, instructors tended to ignore them (i.e. a common practice in some engineering and science colleges, even for math courses). This practice has created an image in my head that applied mathematics means not a rigorous treatment of such problems. Moreover, it is mere apparatus, and the primary objective is to use it effectively rather than questioning its validity. This reminds me with a quote in a book authored by Paul Zarchan and Howard Musof which says "This is analogous to the quote from the recent engineering graduate who, upon arriving in industry, enthusiastically says, ''Here I am, present me with your differential equations!’’ As the naive engineering graduate soon found out, problems in the real world are frequently not clear and are subject to many interpretations." So the question is, what are the things that distinguish applied mathematics from abstract/pure mathematics in a few words? What are the actual borders? How can you tell if someone is using math properly?

Answer 1

My view:

Applied mathematics is a part of mathematics which solves practical problems coming from needs of physics, chemistry, engineering, economy, industry etc. It is almost equivalent to terms like computational mathematics or numerical mathematics.

Now, you can definitely do numerical mathematics with the same kind of rigorosity as any other part of mathematics. In practice however, it is offen not the case. They are in my view two reasons for that:

1. Many practical problems are so difficult that almost no theory exists (Example A: PDEs describing turbulent flows, like k-epsilon model, use equations, which contain ad-hoc terms in order to fit experimental data better then the standard Navier-Stokes equations, however there is no exact derivation or theory in behind of this model. Example B: models describing mechanical behaviour coupled with thermal behaviour, in difficult geometrical domains create such difficult PDEs that a) no analytical solution exist, b) no existence and uniqueness of solution of these PDEs can be proven, c) there is no available guarantee that any numerical scheme will converge. In other words - theory does not say much and you have to look only what numerically works. Also, there are numerical methods, which were created heuristically, without any theory in behind, which however work well in practice. Example: Nelder-Mead simplex method from 1965 for optimization, works well in practice, however until now no proof of convergence exists in more then one dimension.

2. Numerical modeling is a cross-field discipline done by engineers, physicists, mathematicians. Many of these people have no time to focus on theory, they need to efficiently solve practical problems, which they have. They usually use different software packages, which incorporate methods based on the known theory. Unfortunately you can use this software without knowing much about the theory.

In any case, it is always good to know what is known, what can be proven and what not. You can always find good textbooks for that.