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A mathematician is a person whose area of study and research is mathematics.
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, physics or quantum mechanics).-
Contrary to popular belief, mathematicians are not typically 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.
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.
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.
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'Hospital (1661-1704) 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.
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, Émilie du Châtelet and Marianna Csornyei.
...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.
A mathematician is a machine for turning coffee into theorems.
Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows:
Another one involves an astrologer, a chemist, and a mathematician on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by.
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