A mathematician is a person whose primary area of study or research, or both, is the field of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematicians are concerned with particular problems related to logic Logic is the study of reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, space Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime. In mathematics one examines ', transformations, numbers A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels , and more general ideas which encompass these concepts. Some notable mathematicians include Sir Isaac Newton Sir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is considered by many scholars and members of the general public to be one of the most influential men in human history. His 1687 publication of the Philosophiæ Naturalis Principia Mathematica (usually called the Principia), Muhammad ibn Mūsā al-Khwārizmī Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī (c. 780, Khwārizm – c. 850) was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad, Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, Archimedes of Syracuse Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is, Leonhard Paul Euler Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər in English (German pronunciation: [ˈɔʏlɐ]); the common English pronunciation /ˈjuːlər/ EW-lər is incorrect, Georg Friedrich Bernhard Riemann Georg Friedrich Bernhard Riemann (German pronunciation: [ˈʁiːman]; September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity, Euclid of Alexandria Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is the most successful textbook and one of the most influential works in the history of mathematics,, Jules Henri Poincaré Jules Henri Poincaré (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe]) was a French mathematician, theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime, Srinivasa Ramanujan Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician and self taught genius who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions, Alexander Grothendieck, David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of, Joseph-Louis Lagrange Joseph-Louis Lagrange , born Giuseppe Lodovico (Luigi) Lagrangia, was an Italian-born mathematician and astronomer, who lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. On the recommendation of Euler and D'Alembert, in 1766, Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the, Évariste Galois Évariste Galois (October 25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory, a major branch of and Pierre de Fermat Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the.

Some scientists who research other fields are also considered mathematicians if their research provides insights into mathematics—one notable example is Edward Witten Edward Witten is an American theoretical physicist with a focus on mathematical physics. He is a professor at the Institute for Advanced Study. He is a leading researcher in superstring theory, supersymmetric quantum field theories and other areas of mathematical physics. Although Witten is regarded by many as one of the greatest living physicists,. Conversely, some mathematicians may provide insights into other fields of research—these people are known as applied mathematicians Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory ; and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the.

Education

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 student's understanding of mathematics; the students who pass are 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 of his extraordinary achievements with little formal mathematics training.^[1]

Motivation

Mathematicians do research in fields such as logic Logic is the study of reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, category theory In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows, abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae, number theory Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study, analysis Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle, though analysis as a formal concept is a relatively recent development, geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by, topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation, dynamical systems The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake, combinatorics Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory),, game theory Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology , engineering, political science, international relations, computer science, and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, or games, in which an individual's success, information theory Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception it, numerical analysis Numerical analysis is the study of algorithms that use numerical approximation for the problems of continuous mathematics (as distinguished from discrete mathematics), optimization In mathematics and computer science, optimization, or mathematical programming, refers to choosing the best element from some set of available alternatives, computation Computation is a general term for any type of process, algorithm or measurement; this often includes but is not limited to digital data. This includes phenomena ranging from human thinking to calculations with a more narrow meaning. Computation is a process following a well-defined model that is understood and can be expressed in an algorithm,, probability Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of and statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments. These fields comprise both pure mathematics Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards and applied mathematics Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory ; and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the, 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 Computer science or computing science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe, and transform information. Computer science and physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves). Likewise, analysis, geometry and topology, although considered pure mathematics, do find applications in theoretical physics - string theory String theory is a developing theory in particle physics which attempts to reconcile quantum mechanics and general relativity. String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines , possessing only the dimension of length, but not height or width. The 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 allows for research concerning how concepts in one area of mathematics can be used in other areas too. For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask about the geometric properties of the graph. Thus one can understand equations by a pure understanding of abstract topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation or geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by - this idea is of importance in algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a. Similarly, a mathematician does not restrict his study of numbers to the integers The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6; rather he considers more abstract structures such as rings In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions—namely, the set must be an abelian group under addition and a monoid under, and in particular number rings in the context of algebraic number theory In mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field. 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 In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds concerns itself with the ways in which one can "smooth" higher dimensional figures. In fact, whether one can smooth certain higher dimensional spheres remains open - it is known as the smooth Poincaré conjecture In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds. Originally conjectured by Henri Poincaré, the claim concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-. Another aspect of mathematics, set-theoretic topology and point-set topology, concerns objects of a different nature from objects 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 from those of objects in our universe. For instance, the "distance" between two points on such an object, may depend on the order in which you consider the pair of points. This is quite different from ordinary life, in which it is accepted that the straight line distance from person A to person B is the same as 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.

There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics (e.g. John Nash). The most prestigious award in mathematics is the Fields Medal, sometimes referred to as the "Nobel Prize of Mathematics". The Fields Medal 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. While the Fields Medal is awarded to younger mathematicians for a major achievement, these other prizes are awarded for outstanding work over a longer period or for work that has had lasting influence.

Differences with scientists

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. 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 proved nor disproved, it is called a conjecture, as opposed to the ultimate goal: a theorem that has been proved. 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.

Doctoral degree statistics for mathematicians in the United States

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.^[2] 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. The median age for doctoral recipients in 1999-2000 was 30, and the mean age was 31.7.

Mathematicians are employed as professors or researchers in colleges and universities but many are employed in non-academic settings in such areas as support of engineering and scientific research, quantitative analysis in economics, and statistics. See references below for more information on employment of mathematicians.

Women in mathematics

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.

Quotations about mathematicians

The following are quotations about mathematicians, or by mathematicians.

A mathematician is a device for turning coffee into theorems.

—Attributed to both Alfréd Rényi^[3] and Paul Erdős

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.)

—Johann Wolfgang von Goethe

Each generation has its few great mathematicians...and [the others'] research harms no one.

—Alfred W. Adler (1930- ), "Mathematics and Creativity"^[4]

In short, I never yet encountered the mere mathematician who could be trusted out of equal roots, or one who did not clandestinely hold it as a point of his faith that x squared + px was absolutely and unconditionally equal to q. Say to one of these gentlemen, by way of experiment, if you please, that you believe occasions may occur where x squared + px is not altogether equal to q, and, having made him understand what you mean, get out of his reach as speedily as convenient, for, beyond doubt, he will endeavor to knock you down.

—Edgar Allan Poe, The purloined letter

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

—G. H. Hardy, A Mathematician's Apology

Some of you may have met mathematicians and wondered how they got that way.

—Tom Lehrer

It is impossible to be a mathematician without being a poet in soul.

—Sofia Kovalevskaya

See also

• List of amateur mathematicians
• List of female mathematicians
• List of mathematicians
• Mathematical joke

Notes

1. ^ http://mathsforeurope.digibel.be/pierredefermat.html
2. ^ Archived January 14, 2006 at the Wayback Machine.
3. ^ Biography of Alfréd Rényi
4. ^ Alfred Adler, "Mathematics and Creativity," The New Yorker, 1972, reprinted in Timothy Ferris, ed., The World Treasury of Physics, Astronomy, and Mathematics, Back Bay Books, reprint, June 30, 1993, p, 435.

References

• A Mathematician's Apology, by G. H. Hardy. Memoir, with foreword by C. P. Snow.
  • Reprint edition, Cambridge University Press, 1992; ISBN 0-521-42706-1
  • First edition, 1940
• Paul Halmos. I Want to Be a Mathematician. Springer-Verlag 1985.
• Dunham, William. The Mathematical Universe. John Wiley 1994.

External links

• Occupational Outlook – Mathematicians. Information on the occupation of mathematician from the US Department of Labor.
• Sloan Career Cornerstone Center: Careers in Mathematics. Although US-centric, a useful resource for anyone interested in a career as a mathematician. Learn what mathematicians do on a daily basis, where they work, how much they earn, and more.
• The MacTutor History of Mathematics archive. A comprehensive list of detailed biographies.
• The Mathematics Genealogy Project. Allows to follow the succession of thesis advisors for most mathematicians, living or dead.
• Weisstein, Eric W., "Unsolved Problems" from MathWorld.
• Middle School Mathematician Project Short biographies of select mathematicians assembled by middle school students.

Categories: Mathematical science occupations | Mathematicians

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