One becomes a mathematician, even if only temporarily, if one becomes interested in an investigation into purely mathematical matters (e.g. in finding a rigorous proof of a statement) rather than in using mathematics as a computational tool in one's (original) area of interest, such as physics or biology. Even when one happens to invent a new mathematical method that works, one does not, in general, instantly become a mathematician - unless, of course, they lose the focus and turn their attention to making the method just discovered more efficient or better substantiated from the purely logical standpoint.