I'd like to discuss what we think about questions that are not formulated in the language of mathematics, but ask for a mathematical formulation, or pose a question--in a thoughtful and motivated way, with some relevant domain-specific background--that requires a translation to mathematics and back.
I'm thinking of this because to me, more valuable than any given set of mathematical questions and their answers or proofs, for learning how to do mathematics, is the intuition, feel for proof writing, comfort with mathematical language, and collection of techniques that one gets upon thinking over and internalizing these problems.
In the same vein, to me, one of the central concerns in the education of an applied mathematician is developing a facility and comfort with taking a question from a scientist or engineer that is not yet mathematical, figuring out how to formulate it in the way that makes it most amenable to mathematical analysis while capturing the relevant theories and constraints governing the system within the scientific domain the question draws on, and then translating the mathematics back into something the scientist or engineer can use, understand, and build on. This skill and intuition seems more important to me for an applied mathematician than differential equations, numerical analysis, statistics, or indeed any of the mathematical tools one uses to attack a problem once it's been formulated.
Because of this, I think applied mathematicians should, optimally, develop familiarity with basic ways of thought across several fields, be exposed to the language of those fields and able to translate, and simply see a lot of examples of the process of mathematical modeling in action. I would thus love to see many questions and good answers to them on this site that pertain to the development of a mathematical model for a particular phenomenon: I think it's something that goes missing amidst all the Bessel functions and coding for a lot of applied math students, and admits the kind of nuanced, more-than-the-definition-or-solution-from-a-book answer that I've been saying we want here to questions in pure mathematics (see the discussion on when we can answer "What is X?" questions better than a textbook/encyclopaedia). It's also something that, e.g., Vivi has brought up: questions whose cores are mathematical, in that it's mathematics knowledge/experience and not some fact from the application domain that the asker is missing.
As long as the asker can provide what he believes are the relevant concepts/issues from the application domain that should be factored into the model, I think these questions should be asked and encouraged. But I know it's hard to determine boundaries for what belongs here, so I'd like to discuss this.
Relevant background: Several times, questions about modeling something have been asked on MO. Often they are eventually closed, but with several valuable answers posted first, and more lamentation than usual that this is actually a great question, just not suitable for MO. (There's also been discussion about the conspicuous dearth of applied mathematicians on MO, and whether a percieved bias against applied math might be driving potential participants away). I'd like, at minimum, a place for those modeling questions that were considered excellent but slightly off-topic for MO.
This question on "mathematics in nature" was discussed on meta.MO and eventually closed, but it was indicated that this is the kind of thing that, reformulated to be a better question, MO people would like to see on a site like math.SE. In particular, the question as stated is poor because it might as well be "give some examples of when math shows up in the real world," but I'd be curious what you all think of whether, say, more targeted questions that might elicit the responses given to this question would work on math.SE.