How many mathematicians are there?

In an article written a few years ago ("Mathematicians in France and in the world"), Jean-Pierre Bourguignon estimates that there are around 80 000 mathematicians worldwide, with the AMS having about 15 000 members.

For France he says 4000 work in academia ("a reliable estimate") and about 2000 in the private sector. Since there are about 60 million inhabitants there, that's 1 mathematician per 10 000 inhabitants.

I am surprised that nobody mentioned the Math Reviews authors database.

Currently it has about 650000 authors. I suppose that about 50% of them are dead, which gives an estimate of 300-400 K living mathematicians. Of course, as it was noticed in the question, it is hard to establish a criterion, whom do we call mathematicians. I think about 1/3 or 1/2 of the people in this database are those who published only one paper.

Such sources as Math Genealogy project are much less reliable, because they do not include most Soviet, Chinese and other mathematicians. But it also includes people who defended a PhD in mathematics, published one paper (or even did not publish anything) and switched to some other activity.

Current count of Mathematics Genealogy Project is 137672 (I am assuming that the PhD students that graduated are ranked as "research mathematicians"). But the problem is.. Mathematics Genealogy is mostly for universities of developed countries. There could be some really good university in Russia, China or Korea out there that doesn't give us the correct statistics. Another problem is.. Mathematics Genealogy Project counts even the dead mathematicians (like Hilbert, Hasse, Kepler and so on).. and I am assuming you want a report of living mathematicians.. but hey, I'm quite surprised by the number even 200k is pretty low for the living!