In doing so, I note that a line is drawn between math majors and non-math majors. This led me to wonder about applied math majors. I checked the website of my Alma Mater in New Zealand and saw that these days, they offer the possibility to major in pure mathematics, including taking a first course on proofs, or majoring in applied mathematics, which does not appear to require taking such a course. Since the original question seemed primarily about North American universities, I wonder if this situation also exists in North America.
It is tempting to ask the spin-off question whether applied mathematics majors should have a separate course on proofs, and whether they should be classified under maths majors or non-math majors, but I suspect that such a question would be perceived as argumentative. The word "should" is probably a warning sign.
I am also sensitive to the point of view that that separating mathematics into pure and applied is somewhat artificial, although at times useful.
I would much appreciate your thoughts on whether this question can be turned into an acceptable non-argumentative question.
Short answer: I think it can be turned into an acceptable question , but why bother?
Long answer: I would prefer that a goal be set first. Why ask the question? If you are going to act on the answer, most likely you will pursue finding/creating such a course, in which case you really want a different question. Otherwise you are asking a question that (in my opinion) isn't even for academic purposes in the traditional sense: you are asking a question just to get an answer.
I think it more reasonable to ask what has been done rather than what should be done. If you are indeed thinking of such a course, you might list a couple of goals and then ask what other purpose such a specialized course might serve. That has a better taste in my mouth than what you propose in your post.
Gerhard "Ask Me About Course Purposes" Paseman, 2013.01.25
Part of my reason for considering such a question was indeed mere curiosity and the suspicion of a false dichotomy between math and non-math.
As it turns out, I currently teach mathematics to engineering students, rather than applied mathematics majors as such. The focus is much more on applications than on proofs, as is often the case when teaching non-math majors. That said, I think a case can be made for teaching proofs to engineers. As a matter of fact, they do run into the basics of logic in their digital electronics courses. Furthermore, proofs can arise in system verification and validation.
I am also pleasantly suprised that much so-called pure mathematics, not just traditional applied math, turns out to be applicable to engineering. Yet pure mathematics is often not very accessible to engineers without training in reading and understanding theorems and proofs. This could also be a reason to teach engineers a course in proofs, although perhaps not a primary reason.
I think I will refrain, at least for the time being, from asking a soft question about teaching applied mathematicians or engineers to write proofs. I certainly retain an interest in applicable mathematics and making it more accessible to those who could benefit from using it. Perhaps I can take a leaf out of Gregory Chirikjian's book, so-to-speak, as in his "Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods" (see the Preface on, for instance, Amazon.com).