How  many mathematicians are there? 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.
 A: 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.
A: Only somewhat related, I think it was Frank Adams who commented on the varying percentage of a country's population in math(s) e.g Hungary and Roumania.
A: 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.
A: 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! 
A: Typing "how many mathematicians" into Wolfram|Alpha, yields the information that there are 3160 mathematicians in the United States. The source listed is the US Bureau of Labor Statistics, and this site in particular (broken link http://www.bls.gov/oco/ocos043.htm, Wayback Machine). That site contains such gems about our profession as: "Mathematicians usually work in comfortable offices."
Cool.
Edit: newer link that isn't broken, yet (Wayback Machine)
A: By definition mathematician is a person who holds phd in mathematics. Per Wikipedia only 2.94% of population in US out of 310 million hold phd in anything.  And math phd's are very rare. By estimation I would assume that only 0.001% of total population would hold phd in mathematics. So that does end up as around 3100 mathematicians in US. Many of the math teachers in high school or community colleges may not be holding phd in math. So technically they are not mathematicians. 
