2008/9 Schools Wikipedia Selection. Related subjects: Mathematics
Problems in mathematics
The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals. One of the most exciting recent developments was the proof of Fermat's last theorem by Andrew Wiles, 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) and Goldbach's conjecture (1742). The Millennium Prize Problems highlight longstanding, important problems in mathematics and offers a US$1,000,000 reward for solving any one of them. One of these problems, the Poincaré conjecture (1904), was proven by Russian mathematician Grigori Perelman in a paper released in 2003; peer review was completed in 2006, and the proof was accepted as valid.
Mathematicians are typically interested in finding and describing patterns, or finding (mathematical) proofs of theorems. Most problems and theorems come from within mathematics itself, or are inspired by theoretical physics. To a lesser extent, problems have come from economics, games and computer science. 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 as many as four young (under 40 years old) awardees at a time. Other prominent prizes 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 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.
While the majority of mathematicians are male, there have been some demographic changes since World War II. Some prominent female mathematicians are Ada Lovelace (1815 - 1852), Maria Gaetana Agnesi (1718-1799), Emmy Noether (1882 - 1935), Sophie Germain (1776 - 1831), Sofia Kovalevskaya (1850 - 1891), Rózsa Péter (1905 - 1977), Julia Robinson (1919 - 1985), Olga Taussky-Todd (1906 - 1995), Émilie du Châtelet (1706 – 1749), Mary Cartwright (1900 - 1998), and Hypatia of Alexandria (ca. 400 AD). The AMS and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics.
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. Archive copy at the Internet Archive 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.
The following are quotations about mathematicians, or by mathematicians.
- A mathematician is a machine for turning coffee into theorems.
- —Attributed to both Alfréd Rényi 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.)
- Some humans are mathematicians; others aren't.
- — Jane Goodall (1971) In the Shadow of Man
- Each generation has its few great mathematicians...and [the others'] research harms no one.
- —Alfred Adler, "Mathematics and Creativity"
- Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
- —Bertrand Russell, The Study of Mathematics
- 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
- Another roof, another proof.
- — Paul Erdős
- 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