Although we are not so numerous as other respected professionals, like for example lawyers, I wonder if we could come up with a reasonable estimate of our population.
Needless to say, the question more or less amounts to the definition of"mathematician".
Since I should like to count only research mathematicians (and not, say, high-school teachers) some criterion of publishing should be applied. But it should not be too strict in order not to exclude Grothendieck, for example, who has not published any mathematics for a long time.
An excuse for asking a question so soft as to verge on the flabby is that it might be considered an exercise in Fermi-type order of magnitude estimation.
Best Answer
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!