Did the ancient Greeks, who used geometric axioms, count as mathematicians under Wilson's "definition" involving formal symbolic manipulation? I guess I would say that mathematics is the study of thinking. And how to be a better thinker. But this is not a mathematical definition :) And I regard CS, physics, and lots of other fields such as biology, chemistry, psychology, economics as all involved and all as fair game for mathematicians. And if not, that is because of a failure of either those fields, or of mathematicians. You know, I once worked at a place called NEC research institute which was supposedly about combining physics, computer science, chemistry, and biology. It self destructed, which is another story. But anyhow, as far as I could tell I was the only one there who tried to do all of them. Mostly, everybody quickly segregated into "their own field" and avoided the others. I also was one of the only ones there (depending on the date) who called himself a mathematician. DId this mean that I was about the sole exemplar of what they wanted? Actually, far as I could tell my approach was not appreciated. More precisely, some appreciated it others not, but the weighted sum was more toward "not." I also think that even within the set of people who call themselves mathematicians, my universalist approach is usually not taken. Usually instead the mathematicians themselves specialize into subfields. This may or may not be a good thing (probably both) but the potential of mathematics as the unifying glue connecting every field, has not been exploited as much as it should. Who are some good universalist mathematicians? List your faves. Von Neumann, Poincare, Feynman spring to mind. Neil Sloane, who sometimes posts on this list, could be argued to be a specialist, but I think he's succeeded in having an impact felt across many fields. I personally would regard Francis Crick as a mathematician, although nobody called him that, including himself. Maybe what I want is what they used to call "natural philosopher" but I think few if any people call themselves that anymore. Newton certainly tried. He also was interested in biology and chemistry, but as far as I know never published in those areas because he never succeeded in his own view in deciphering them. But he tried hard.