Could be argued. Quite a few notable mathematicians in history were hobbyists, more or less. As to a more current example, Grigori Perelman refused to be a professional mathematician at some point.
Fact is, some think that the moment you're getting paid for it you kind of stop being a mathematician.
True, though I always respect people who are good at the subject, it would be amazing to see the world through the eyes of a mathematician for a day and ask myself, is the subject a truth, or is it just another man made system designed to explain things based on the only way we can explain them, or whether its a truth which transcends different galaxies.
Does 1+1 really equal 2 for every lifeform outside of this galaxy, assuming it exists, which it most probably does.
If not then, probably everything we think we know about numbers is one giant, wasted misconception built on the assumptions of others and merely designed to serve us.
That would bother me as I would imagine the motive for every math guy is discovery.
I would rather put my time into something with a simpler, and more certain outcome.