After more than ten years in the world of academic mathematics, I have some experience with cranks and crackpots. For a while I was getting regular emails with beautiful graphics about squaring the circle, and I’ve seen talks where the validity or interest of the result hinged on, essentially, the incorrect use of a mathematical term (one such was the closest I’ve seen to a riot in a mathematical setting). I’ve read a number of abstracts submitted to logic conferences, for publication in the program without an accompanying talk. Logic gets the cream of the crackpot crop, in my opinion.
Out of this I’ve formed Opinions about traits one should avoid, lest one look like a crank.
The easiest is the true crackpot calling card: claiming something is true that was definitively proved false long ago, such as the ability to square the circle. Typically this comes with some argument about near-sightedness, essentially, where if you just expand your viewpoint you can use techniques of the greater universe to solve all problems. Sometimes these have explicit mystical/religious tones, or allusions to historical figures who were denied but were correct. They are my favorite to read but they are the most easily dismissed.
There are three more, however, that are potential traps for legitimate scientists. Cranks will often name-drop (leading one in a respected position to be careful about replying to their questions lest it be made to look as though you are collaborators). In a standard math paper there is no need to mention anyone who is not attached to the current or cited results, except perhaps in an acknowledgement for “helpful conversation.”
Cranks will frequently feel the need to show off possession of a huge body of knowledge, even if it’s not entirely relevant. This is especially true of results that are more abstract and/or difficult, such as the Axiom of Choice. I did, however, see a legitimate paper refer to the Axiom of Choice (with an exclamation point) before admitting that it wasn’t actually necessary for the object at hand. Including it did nothing but muddy the waters.
Finally, the trait that I see with nearly every crank is urgency in claiming credit and differentiating one’s own work from that previously existing, almost to the point of defensiveness. This is easy to find in certain legitimate mathematicians’ work as well, though, if they are trying to prove themselves or otherwise have a chip on their shoulders. I read a paper where, after giving an approach that was of a similar bent, the author was quick to point out that it was similar at a high level only, and not in actual methodology.
The mathematicians I admire the most are quick to give credit and unworried about claiming it. They put in their papers only what is necessary to understand the results at hand and their context and importance, perhaps with interesting sidebars but always labeled as such. Of course, this is easier to do when you are confident of your abilities and the quality of your work, but I think emulating this appearance will help anyone come across as confident and competent. That is, decidedly not a crank.