A mathematician is a person whose area of study and research is mathematics. In the United States, most mathematicians are employed by private firms in various capacities. Others may be employed as professors at universities, by research institutions, or by various military or civilian government agencies. 
While some people believe mathematics is fully understood (as it is often presented this way in elementary textbooks), in fact, there is ongoing research into many areas of mathematics. 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).
Unlike the other sciences, research in mathematics generally does not consist of performing experiments. Rather, mathematics is about problem-solving, where truths are deduced from other known truths. Computer experiments and other numerical evidence might be a part of this process, but in the end, mathematics research is about constructing proofs of theorems.
In particular, calculation is not a big part of mathematics research, and mathematicians need not have any extraordinary ability in adding or multiplying numbers. See mental calculators to read about prodigies at performing such calculations.
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. Problems have come from physics, economics, games, generalizations of earlier mathematics, and some problems are simply created for the challenge of solving them. Although much mathematics is not immediately useful, history has shown the eventually applications are found. For example, number theory originally seemed to be without purpose, but after the invention of computers it gained countless applications to algorithms and cryptography.
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 proven 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 is considered a theorem, it remains true forever.
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), Rózsa Péter (1905 - 1977), Julia Robinson (1919 - 1985), Mary Ellen Rudin, Eva Tardos, Émilie du Châtelet, Mary Cartwright and Marianna Csörnyei.
...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.
- -Saint Augustine, De Genesi ad Litteram (actually "mathematicians" in this context refers mainly to astrologers and such)
Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen 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.)
Some humans are mathematicians; others aren't.
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 a hot air balloon and suddenly they realize that they are lost. Luckily they see a man plowing a field and ask, "Where are we?". The man on the ground thinks for a minute and then answers, "You are in a 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"
- 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"
- 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
- 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.
- mental calculator
- List of mathematicians
- List of amateur mathematicians
- Astronomers, Physicists, Philosophers, Scientists
- American Mathematical Society
- Mathematical Association of America
- The MacTutor History of Mathematics archive, a very complete list of detailed biographies.
- The Mathematics Genealogy Project, which allows to follow the succession of thesis advisors for most mathematicians, living or dead.bg:Математик
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