A mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts. Some notable mathematicians include Sir Isaac Newton, Johann Carl Friedrich Gauss, Archimedes of Syracuse, Leonhard Paul Euler, Georg Friedrich Bernhard Riemann, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, David Hilbert, Joseph-Louis Lagrange, Georg Cantor, and Pierre de Fermat.
Some scientists who research other fields are also considered mathematicians if their research provides insights into mathematics—one notable example is Edward Witten. Conversely, some mathematicians may provide insights into other fields of research—these people are known as applied mathematicians.
Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of a person's understanding in mathematics; should s/he pass, s/he is permitted to work on a doctoral dissertation.
There are notable cases where mathematicians have failed to reflect their ability in their university education, but have nevertheless become remarkable mathematicians. Fermat, for example, is known for having been "Prince of Amateurs", because he never did research in university and took Mathematics as a hobby.
Mathematicians do research in fields such as logic, set theory, category theory, abstract algebra, number theory, analysis, geometry, topology, dynamical systems, combinatorics, game theory, information theory, numerical analysis, optimization, computation, probability and statistics. These fields comprise both pure mathematics and applied mathematics, as well as establish links between the two. Some fields, such as the theory of dynamical systems, or game theory, are classified as applied mathematics due to the relationships they possess with physics, economics and the other sciences. Whether probability theory and statistics are of theoretical nature, applied nature, or both, is quite controversial among mathematicians. Other branches of mathematics, however, such as logic, number theory, category theory or set theory are accepted to be a part of pure mathematics, although they do indeed find applications in other sciences (predominantly computer science and physics). Likewise, analysis, geometry and topology, although considered pure mathematics, do find applications in theoretical physics - string theory, for instance.
Although it is true that mathematics finds diverse applications in many areas of research, a mathematician does not determine the value of an idea by the diversity of its applications. Mathematics is interesting in its own right, and a majority of mathematicians investigate the diversity of structures studied in mathematics itself. Furthermore, a mathematician is not someone who merely manipulates formulas, numbers or equations - the diversity of mathematics permits for researchers in other areas too. In fact, the theory of equations and numbers (formulas to a lesser extent in theoretical mathematics, but to some extent in applied mathematics), can lead to deep questions. For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask of the geometric properties of the graph. Thus one can understand equations by a pure understanding of abstract topology or geometry - this idea is of importance in algebraic geometry. Similarly, a mathematician does not restrict his study of numbers to the integers; rather he considers more abstract structures such as rings, and in particular number rings in the context of algebraic number theory. This exemplifies the abstract nature of mathematics and how it is not restricted to questions one may ask in daily life.
In a different direction, mathematicians ask questions about space and transformations, but which are not restricted to geometric figures such as squares and circles. For instance, an active area of research within the field of differential topology concerns itself with the ways in which one can "smoothen" higher dimensional figures. In fact, whether one can smoothen certain higher dimensional spheres remains open - it is known as the smooth Poincaré conjecture. Another aspect of mathematics, set-theoretic topology and point-set topology, concerns objects of a different nature to those in our universe, or in a higher dimensional analogue of our universe. These objects behave in a rather strange manner under deformations, and the properties they possess are completely different to those objects in our universe. For instance, the "distance" between one point on such an object, and another point, may depend on the order in which you consider the pair of points. This is quite different to ordinary life, in which it is accepted that the straight line distance from person A to person B is the same (and not different to) that between person B and person A.
Another aspect of mathematics, often referred to as "foundational mathematics", consists of the fields of logic and set theory. Here, various ideas regarding the ways in which one can prove certain claims are explored. This theory is far more complex than it seems, in that the truth of a claim depends on the context in which the claim is made, unlike basic ideas in daily life where truth is absolute. In fact, although some claims may be true, it is impossible to prove or disprove them in rather natural contexts.
Category theory, another field within "foundational mathematics", is rooted on the abstract axiomatization of the definition of a "class of mathematical structures", referred to as a "category". A category intuitively consists of a collection of objects, and defined relationships between them. While these objects may be anything (such as "tables" or "chairs"), mathematicians are usually interested in particular, more abstract, classes of such objects. In any case, it is the relationships between these objects, and not the actual objects which are predominantly studied.
The Nobel Prize is never awarded for work in the field of theoretical mathematics. Instead, the most prestigious award in mathematics is the Fields Medal, sometimes referred to as the "Nobel Prize of Mathematics". The Fields Medal is considered more of a prestige than a mere reward in that it is only awarded every four years, and the amount of money awarded is small in comparison to that of the Nobel Prize. Furthermore, the recipient of the Fields Medal must be (roughly) under 40 years of age at the time the medal is awarded. Other prominent prizes in mathematics include the Abel Prize, the Nemmers Prize, the Wolf Prize, the Schock Prize, and the Nevanlinna Prize.
Mathematics differs from natural sciences in that physical theories in the sciences are tested by experiments, while mathematical statements are supported by proofs which may be verified objectively by mathematicians. If a certain statement is believed to be true by mathematicians (typically because special cases have been confirmed to some degree) but has neither been proven nor dis-proven, it is called a conjecture, as opposed to the ultimate goal: a theorem that is proven true. Physical theories may be expected to change whenever new information about our physical world is discovered. Mathematics changes in a different way: new ideas do not falsify old ones but rather are used to generalize what was known before to capture a broader range of phenomena. For instance, calculus (in one variable) generalizes to multivariable calculus, which generalizes to analysis on manifolds. The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint without making what was proved before in any way incorrect. While a theorem, once proved, is true forever, our understanding of what the theorem really means gains in profundity as the mathematics around the theorem grows. A mathematician feels that a theorem is better understood when it can be extended to apply in a broader setting than previously known. For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.
The number of Doctoral degrees in mathematics awarded each year in the United States has ranged from 750 to 1230 over the past 35 years. In the early seventies, degree awards were at their peak, followed by a decline throughout the seventies, a rise through the eighties, and another peak through the nineties. Unemployment for new doctoral recipients peaked at 10.7% in 1994 but was as low as 3.3% by 2000. The percentage of female doctoral recipients increased from 15% in 1980 to 30% in 2000.
As of 2000, there are approximately 21,000 full-time faculty positions in mathematics at colleges and universities in the United States. Of these positions about 36% are at institutions whose highest degree granted in mathematics is a bachelor's degree, 23% at institutions that offer a master's degree and 41% at institutions offering a doctoral degree.
The median age for doctoral recipients in 1999-2000 was 30, and the mean age was 31.7.
While the majority of mathematicians are male, there have been some demographic changes since World War II. Some prominent female mathematicians are Hypatia of Alexandria (ca. 400 AD), Labana of Cordoba (ca. 1000), Ada Lovelace (1815–1852), Maria Gaetana Agnesi (1718–1799), Emmy Noether (1882–1935), Sophie Germain (1776–1831), Sofia Kovalevskaya (1850–1891), Alicia Boole Stott (1860-1940), Rózsa Péter (1905–1977), Julia Robinson (1919–1985), Olga Taussky-Todd (1906–1995), Émilie du Châtelet (1706–1749), and Mary Cartwright (1900–1998).
The Association for Women in Mathematics is a professional society whose purpose is "to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity and the equal treatment of women and girls in the mathematical sciences." The American Mathematical Society and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics.
The following are quotations about mathematicians, or by mathematicians.