Research in Mathematics (Formalities vs. Results) This question is intended to solve an internal dilemma I have been having lately. I assume that most of those who have gone through the process of mathematical research have, at some point, considered this as well. Therefore, I would be greatly appreciative of any input one has on the subject.
Recently I have seen a dichotomy in mathematics research. This dichotomy lies in the difference between formally and completely justified mathematics, and mathematics who's assumptions are correct but not as well justified.
I may be wrong on these assumptions, and if so please clarify. In mathematics, we are forced to make a choice, in some sense. Do we accept the axioms given to us as truly the basis for mathematics constructed thereafter? Or do we assume nothing and thoroughly prove everything? I have come to this assessment for many reasons. Particularly, I have two professors this semester. One professor is very informal, and in a sense provides less reasoning then satisfactory sometimes. On the other side, there is an extremely formal professor who covers every point in depth (i.e. "First we must prove the existence, then we must prove property, then we must prove the uniqueness, now apply this theorem and use the proof as justification." -Prof. "But sir, it's the division algorithm... Is it 'necessary' to prove existence and then justify the use of the algorithm once we have proven it's truth?" - Student.)
For one that knows they want to participate in math research themselves, how would one resolve this conflict? Is it a philosophy in and of itself? Do we make a choice of this philosophy and then preach it as is? Or do we simply accept that the conflict resides and simply brush it to the side in order to achieve ground breaking results (whether based on false realities in the first place or not).
Thank you in advance. If this question has been asked before I could not find it, and will accept direction in any way possible.
 A: I think there are two different issues here.  As @almagest said, in practice most mathematicians will avoid unduly attention to detail as it would most likely be considered pedantry by their peers who are supposed to read the papers.  Of course, that doesn't mean that all papers have to be alike and there are certainly good and bad authors.  Unfortunately, expository skills are usually not rewarded in research, so most people's aim is not to write papers that are easy to read and understand.  (And this is not the same as attention to detail.  I just wanted to emphasize that there certainly are different styles out there.)  There are a couple of famous mathematicians who were notorious for being extremely "hand-wavy" although their results are considered important or even momentous.  (For example, Rota in "Indiscrete Thoughts" says about Lefschetz that he had never given a completely correct proof, but had never made a wrong guess either... :)
A good exercise would be to read a couple of current research papers by different authors in your area of choice to get a feeling for how "formal" they are.
On the other hand, you mentioned Russell's paradox and that's a completely different cup of tea as this is about foundations.  This is an area where attention to detail and formality are sometimes crucial because tiny deviations in the axioms or definitions might result in big differences in the resulting theories.  An early and influental example is Hilbert's "Grundlagen der Geometrie" which in a way adds nothing new to Euclid's geometry but rather examines its logical foundations and tries to clarify them.
