Mathematician
From Academic Kids

A mathematician is a person whose area of study and research is mathematics.
Contents 
Roles
Mathematicians not only study, but also research, and this must be given prominent mention here, because a misconception that everything in mathematics is already known is widespread among persons not learned in that field. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics).
Problem solving
Contrary to popular belief, mathematicians are not necessarily any better at adding or subtracting numbers, or figuring the tip on a restaurant bill, than members of any other profession  in fact some of the best mathematicians are notoriously bad at these tasks! On the other hand, there are also mental calculators  prodigies at performing such calculations  some of whom have moreover been great mathematicians.
Logic and patterns
Mathematicians are typically interested in finding and describing patterns that may have originally arisen from problems of calculation, but have now been abstracted to become problems of their own. From much published research work of mathematicians, it may look as if the primary approach of a mathematician is to start with some given assumptions, often called axioms, and then proceed to prove other ideas (theorems) that follow from the assumptions according to exact rules of logic. That, however, is the finished product that gets published; it is not work in progress.
Motivation
Mathematicians have different or mixed motivations. Generally, mathematicians are not motivated by monetary greed; the fruits of labor are not usually patentable and in many cases are of purely intellectual interest. Instead, many mathematicians are motivated by either the sheer pleasure of mathematical activity or by expectation of greater respect by fellow mathematicians, or both.
Differences
Mathematicians differ from philosophers in that the primary questions of mathematics are assumed (for the most part) to transcend the context of the human mind; the idea that "2+2=4 is a true statement" is assumed to exist without requiring a human mind to state the problem. Not all mathematicians would strictly agree with the above; the philosophy of mathematics contains several viewpoints on this question.
Mathematicians differ from physical scientists such as physicists or engineers in that they do not typically perform experiments to confirm or deny their conclusions; and whereas every scientific theory is always assumed to be an approximation of truth, mathematical statements are an attempt at capturing truth. If a certain statement is believed to be true by mathematicians (typically as special cases are confirmed to some degree) but has neither been proven nor disproven to logically follow from some set of assumptions, it is called a conjecture, as opposed to the ultimate goal  a theorem that is ultimately true. Unlike physical theories, which may be expected to change whenever new information about our physical world is discovered, mathematical theories are "static"  once a statement achieves the lauded position of a theorem, it remains true forever. There still exists experimental mathematics, where the truth of conjectures is probed by testing them on a number of examples, generally using computers.
History
Mathematics requires one to spend a long time just sitting and thinking to find something of a new approach. Hence, mathematicians need enough free time to pursue their interest. In fact, in history, mathematicians often come from a wealthy family. "Legend says that Archimedes was part of the royal family of Syracuse. The Marquis de l'Hopital (16611704) was rich enough to hire Johann Bernoulli to instruct him in the new calculus that was then sweeping Europe." (Dunham, 1994) Nevertheless, there has been the occasional unwealthy, unpaid amateur mathematician who has made important contributions to the field.
Demographics
As is the case in many scientific disciplines, the field of mathematics has been disproportionately dominated by men. Among the minority of prominent female mathematicians are Emmy Noether (1882  1935), Sophie Germain (1776  1831), Sofia Kovalevskaya (1850  1891), Rozsa Peter (1905  1977), Julia Robinson (1919  1985), Mary Ellen Rudin, Eva Tardos, ɭilie du Chⴥlet and Marianna Csornyei. Also, there is a perception that East Asians, like the Chinese, Korean and Japanese are more inclined to technical and mathematical studies. Recent TIMSS (Trends in Mathematics and Science Study)results show that Asian students perform better in Math and Science than their American counterparts.
Quotes
Template:Wikiquote ...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
 St. Augustine, De Genesi ad Litteram (actually "mathematicians" in this context refers mainly to astrologers and such)
A mathematician is a machine for turning coffee into theorems.
Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so ?tzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. (Mathematicians are [like] a sort of Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.)
Jokes
Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows:
 An engineer, a physicist, and a mathematician are all staying at a hotel one night when a fire breaks out. The engineer wakes up and smells the smoke; he quickly grabs a garbage pail to use as a bucket, fills it with water from the bathroom, and puts out the fire in his room. He then refills the pail and douses everything flammable in the room with water. He then returns to sleep.
 The physicist wakes up, smells the smoke, jumps out of bed. He picks up a pad and pencil and makes some calculations, glancing frequently at the flames. He then measures exactly 15.6 liters of water into the garbage pail, and throws it on the flames, which are extinguished. Smiling, he returns to sleep.
 Finally the mathematician wakes up. He too grabs a pad and begins furiously writing; glancing at the flames; and then writing more. After a while he gets a satisfied look on his face; entering the bathroom, he produces a match, lights it, and then extinguishes it with a bit of running water. "Aha! A solution exists," he murmurs  and returns to his slumbers.
Another joke goes thus:
 Three men are flying in the hot air balloon and suddenly they realize that they are lost. Luckily they see a man plowing the earth and ask, "Where are we?". The man on the ground thinks for a minute and then answers, "You are in the hot air balloon". One of the men in the air then says to his friends, "He was a mathematician  he thought before answering, his answer was totally right and totally useless"
And another:
 An astrologer, a chemist, and a mathematician are on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by. The astrologer excitedly exclaims, "Ah, this shows Scottish sheep are black!" The chemist didactically corrects him: "No, no, it just shows some Scottish sheep are black." The mathematician then says, "Actually, we can only be sure there is at least one Scottish sheep of which at least one side is black"
And finally:
 An experiment is being made. A physicist (or an engineer) and a mathematician are asked to boil hot water, but the kettle is in the living room. The physicist goes to the living room, takes the kettle, returns to the kitchen and puts it on the stove and boils the water. The mathematician does the same. In the second stage, the kettle is in the kitchen and the two are again asked to boil hot water. The physicist simply puts the kettle on the stove and boils the water. However, the mathematician takes the kettle, puts it in the living room and declares: "We have already solved this problem!"
Links and references
References
 A Mathematician's Apology, by G. H. Hardy. Memoir, with foreword by C. P. Snow.
 Reprint edition, Cambridge University Press, 1992; ISBN 0521427061
 First edition, 1940
 Dunham, William. The Mathematical Universe. John Wiley 1994.
See also
 mental calculator
 List of mathematicians
 List of amateur mathematicians
 Astronomers, Physicists, Philosophers, Scientists
 Mathematical Association of America
External links
 The MacTutor History of Mathematics archive (http://wwwhistory.mcs.stand.ac.uk/history/index0.html), a very complete list of detailed biographies.
 The Mathematics Genealogy Project (http://genealogy.math.ndsu.nodak.edu/), which allows to follow the succession of thesis advisors for most mathematicians, living or dead.