Mathematician

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Mathematicians

A mathematician is a person whose primary area of study and research is mathematics. In other words, a mathematician is a person who contributes to the field of mathematics. People who apply mathematics to other fields, but do not contribute directly to mathematics, are generally not considered mathematicians.

Overview

Unlike the other sciences, fundamental research in much of mathematics 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 can result in new problems and are sometimes used to solve them, though most of the time they are just used as indicators that the work is on the right track - numerical evidence is not proof to a mathematician. In the end mathematics research is about constructing proofs of theorems, and most journals would reject a paper consisting solely of numerical data. Some outstanding open problems in mathematics, such as the Birch and Swinnerton-Dyer conjecture, developed after analyzing numerical work on a computer.

Not only is calculation not a big part of some areas of mathematics research, but people who have had an important influence on mathematics do not necessarily have any extraordinary ability in adding or multiplying numbers. For instance, Albert Einstein, whose ideas had a significant impact in geometry, had great difficulties with mathematics when he was a youth. See mental calculators to read about prodigies performing impressive mental calculations.

Fields of work

Mathematicians are employed by private firms in various capacities or as professors at universities or other educational institutions, by research organizations, or by military or civilian government agencies. ^* The largest employer of mathematicians in the United States, for instance, is the National Security Agency. Finally, because mathematics is useful in a wide range of fields, many who consider themselves mathematicians are involved in other subjects, such as physics and computer science.

Mathematics can be divided into many different areas, but broadly speaking mathematicians speak of pure mathematics and applied mathematics.

Pure mathematics traditionally includes algebra, geometry, and (some areas of) analysis, while applied mathematics involved the use of differential equations or other aspects of analysis to solve practical problems. Throughout the physical and social sciences and the business world, much use is made of probability and statistics. However, with the advent of the computer, even parts of algebra (number theory and combinatorics) and geometry (elliptic curves) are used in applied situations.

Problems in mathematics

Some people incorrectly believe mathematics is fully understood, but 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). One of the most exciting recent developements was the proof of Fermat's Last Theorem, following 350 years of the brightest mathematical minds attempting to settle the problem.

There are many famous open problems in mathematics, many dating back tens if not hundreds of years. Some examples include the Riemann hypothesis (from 1859), the Poincaré conjecture (1904) and Goldbach's Conjecture (1742).

Motivation

Mathematicians are typically interested not in calculating, but in finding and describing patterns, or creating proofs that justify a theorem mathematically. Problems have come from physics, economics, games, computer science and generalizations of earlier mathematics. Some problems are simply created for the challenge of solving them. Although much mathematics is not immediately useful, history has shown that eventually applications are found. For example, number theory originally seemed to be without purpose to the real world, but after the development of computers it gained important applications to algorithms and cryptography.

There are no Nobel Prizes awarded to mathematicians. The award that is generally viewed as having the highest prestige in mathematics is the Fields Medal. This medal, sometimes described as the "Nobel Prize of Mathematics" is awarded once every four years to up to four young (under 40 years old) awardees at a time. Other prominent prizes include the Abel Prize, the Wolf Prize, the Schock Prize, and the Nevanlinna Prize.

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. Nonetheless, many of the great philosophers were mathematicians, such as Rene Descartes.

Whereas physical theories in the sciences are usually 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. Physical theories may be expected to change whenever new information about our physical world is discovered. Mathematics changes in a different way: new ideas don't 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.

Demographics

As is the case in many scientific disciplines, the majority of mathematicians are male. There was a little change after World War II. Among the 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, Hypatia of Alexandria and Marianna Csörnyei.

Doctoral degree statistics for mathematicians in the United States

The number of doctoral degrees 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 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.

Quotes

The following are quotes about mathematicians, or by mathematicians.

A mathematician is a machine for turning coffee into theorems.

—Alfréd Rényi ^*

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 ^* 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

Some humans are mathematicians; others aren't.

—Jane Goodall (1971) In the Shadow of Man

"I'm not a magician, I'm a mathemagician!"

—Simpsons

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.

External links

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.

Mathematical science occupations | Mathematicians

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