Once again I come to MO for help with something I'm writing for the public.

Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a community, the way we decide on our goals, whatever -- would you recommend that  non-mathematicians adopt, at least in certain contexts?  

In other words:  if you can imagine a situation in which someone came to you for advice, and you said, "Look, I think you should be a little more like a mathematician about this and...."  what would be the end of the sentence?

In my experience, mathematicians will frequently argue (in general, not just in mathematics) by passing to an extreme case at the beginning.    Non-mathematicians (again in my experience) sometimes object to such a mode of argument as invalid or irrelevant because such extreme hypotheticals are clearly unrealistic.  

I think that the mathematical idea of first setting all the parameters to their maximal, or minimal, values, and understanding that case, before trying to tune them to a more realistic choice of values (and seeing how the solution/context changes with the parameters) can sometimes be valuable (even though it involves as a first step considering a situation that may be very unrealistic).
