The Math Book

by DK

  See also: Algebra • The fundamental theorem of algebra

  GOTTLOB FREGE

  1848–1925

  The son of the principal of a girls’ school in Wismar, northern Germany, Frege studied mathematics, physics, chemistry, and philosophy at the universities of Jena and Göttingen. He then spent his whole working life teaching mathematics in Jena. He lectured in all areas of mathematics, specializing in calculus, but wrote mostly on the philosophy of the subject, bringing the two disciplines together to almost single-handedly invent modern mathematical logic. He once commented that “Every good mathematician is at least half a philosopher, and every good philosopher at least half a mathematician.” Frege mixed little with students or colleagues and remained largely unrecognized in his lifetime, although he was a major influence on the work of Bertrand Russell, Ludwig Wittgenstein, and other mathematical logicians.

  See also: The logic of mathematics • Fuzzy logic

  SOFYA KOVALEVSKAYA

  1850–91

  Moscow-born Kovalevskaya was the first woman in Europe to gain a doctorate in mathematics, the first woman to join the editorial board of a scientific journal, and the first woman to be appointed a professor of math. She achieved all this despite being barred from a university education in her native Russia because of her gender. Aged 17, Sofya eloped with paleontologist Vladimir Kovalevsky to Germany, where she studied at the University of Heidelberg and then Berlin, where she received tuition from German mathematician Karl Weierstrass. Her doctorate was awarded for three papers, the most significant being on partial differential equations. Sofya ended her career as a professor of math at the University of Stockholm, where she died of influenza aged just 41.

  See also: Calculus • Newton’s laws of motion

  GIUSEPPE PEANO

  1858–1932

  Brought up on a farm in the northern Italian region of Piedmont, Peano studied at the University of Turin, where he gained his doctorate in math in 1880. Almost immediately, he began to teach infinitesimal calculus at the same institution, where he was appointed a full professor in 1889. Peano’s first textbook, on calculus, was published in 1884, and in 1891 he began work on the five-volume Formulario Mathematico (Formulation of Mathematics), which contained the fundamental theorems of math in a symbolic language largely developed by him. Many of the symbols and abbreviations are still in use today. He devised axioms for natural numbers (Peano axioms), developed natural logic and set theory notation, and contributed to the modern method of mathematical induction, used as a proof technique.

  See also: Calculus • Non-Euclidean geometries • The logic of mathematics

  NIELS VON KOCH

  1870–1924

  Born in Stockholm, Sweden, Koch studied at the universities of Stockholm and Uppsala, later becoming professor of mathematics at Stockholm. He is best known for the fractal—Von Koch’s “snowflake” curve—he described in a 1906 paper. This fractal is constructed from an equilateral triangle in which the central third of each side is replaced by the base of another equilateral triangle, with this process continuing indefinitely. If all the triangles face outward, the resulting curve takes on the appearance of a snowflake.

  See also: Fractals

  ALBERT EINSTEIN

  1879–1955

  Einstein was an outstandingly gifted physicist and mathematician. Born in Germany, he moved with his family to Italy when young and studied in Switzerland. In 1905, he was awarded his doctorate by the University of Zurich and published groundbreaking papers on Brownian motion, the photoelectric effect, special and general relativity, and the equivalence of matter and energy. In 1921, he was awarded the Nobel Prize for his contribution to physics, and he continued to develop the understanding of quantum mechanics in the years that followed. Because of his Jewish background, he did not return to Germany after Hitler came to power in 1933, but settled in the United States, becoming a citizen there in 1940.

  See also: Newton’s laws of motion • Non-Euclidean geometries • Topology • Minkowski space

  L. E. J. BROUWER

  1881–1966

  Luitzen Egbertus Jan Brouwer (known to his friends as “Bertus”) was born in Overschie, Netherlands. He graduated in mathematics in 1904 from the University of Amsterdam, where he also taught from 1909 to 1951. Brouwer criticized the logical foundations of mathematics as espoused by David Hilbert and Bertrand Russell and helped to found mathematical intuitionism, based on the view of math governed by self-evident laws. He also transformed the study of topology by associating it with algebraic structures, in his fixed-point theorem.

  See also: Topology • 23 problems for the 20th century • The logic of mathematics

  EUPHEMIA LOFTON HAYNES

  1890–1980

  Born in Washington, DC, Lofton Haynes was the first African-American woman to gain a doctorate in mathematics. After graduating from Smith College, Massachusetts, with a math degree in 1914, she then embarked on a teaching career, and in 1930 established the math department at Miner Teachers College, which later merged with the University of the District of Columbia. Her doctorate was awarded by The Catholic University of America in 1943 for a dissertation on set theory. In 1959, Lofton Haynes received a Papal medal for her contributions to education and community activism, and in 1966 she was the first woman to chair the District of Columbia State Board of Education.

  See also: The logic of mathematics

  MARY CARTWRIGHT

  1900–98

  The daughter of an English country vicar, Cartwright was one of the first mathematicians to investigate what would later be known as chaos theory. She graduated from the University of Oxford in 1923 with a degree in mathematics. Seven years later, her doctoral thesis was examined by mathematician John E. Littlewood, with whom she would have a long academic collaboration, especially on the study of functions and differential equations. In 1947, Cartwright became the first female mathematician to be elected a Fellow of the Royal Society in London. She had a long association with Girton College, Cambridge, from 1930 to 1968, during which time she taught, researched, and served as Mistress of the college.

  See also: The butterfly effect

  JOHN VON NEUMANN

  1903–57

  The son of affluent Jewish parents in Budapest, Hungary, von Neumann was a child prodigy, able to divide eight-digit numbers in his head at the age of six. He began to publish major mathematical papers in his late teens and started teaching math at the University of Berlin aged 24. In 1933, he moved to the United States to take up a post at the Institute of Advanced Learning, Princeton, New Jersey, and became a US citizen in 1937. During a lifetime of mathematical study, von Neumann contributed to virtually every area of the discipline. He was a pioneer of game theory, based on the “two-person zero-sum game,” whereby one side wins what the other loses. The theory provided insights into complex systems in daily life such as economics, computing, and the military. He also created a design model for modern computer architecture, and worked in quantum and nuclear physics, contributing to the atomic bomb during World War II.

  See also: The logic of mathematics • The Turing machine

  GRACE HOPPER

  1906–92

  Born Grace Murray in New York City, Hopper was a pioneering computer programmer. After gaining a doctorate from Yale University in 1934, she taught for several years before the outbreak of World War II. When her application to enlist in the US Navy was rejected, she joined the Naval Reserve and began her transition to computer science. After the war, while employed as a senior mathematician at a computer company, she developed the Common Business-Oriented Language (COBOL), which became the most widely used programming language. Hopper retired from the Navy Reserve in 1966, but was called back on active duty the following year, not retiring until 1986, by which time she held the rank of rear admiral. She coined the word “bug” for a computer glitch after a moth flew into circuits on which she was working.

  See also: The mechanical computer • The Turing
machine

  MARJORIE LEE BROWNE

  1914–79

  Only the third African-American woman to earn a PhD in math, Browne was born in Tennessee at a time when it was hard for women of color to pursue an academic career. With the support of her father, a railroad clerk, she graduated from Howard University, Washington DC, in 1935, and, after teaching briefly in New Orleans, continued her studies at the University of Michigan, gaining her doctorate in 1949. Two years later, she was appointed chair of the mathematics department at North Carolina Central University. Marjorie gained a reputation for being an excellent teacher, and for her research, especially in topology.

  See also: Topology

  JOAN CLARKE

  1917–96

  London-born Clarke achieved a double first in math at the University of Cambridge on the eve of World War II but was denied a full degree because of her gender. Her mathematical prowess had been recognized, however, and when the Bletchley Park project was established to decipher the German Enigma Code, Clarke was recruited. At Bletchley, she became one of the leading cryptanalysts, working closely with Alan Turing, to whom she was engaged for a short time. Although they did the same work as the male code-breakers, Clarke and the other Bletchley women were paid less. The Bletchley Park operation was hugely successful, cutting short the length of the war and saving countless lives. After the war, Clarke worked at the British government’s surveillance center, GCHQ. Because so much of Clarke’s work was secret, the full extent of her accomplishments is still unknown.

  See also: The Turing machine • Cryptography

  KATHERINE JOHNSON

  1918–

  A child math prodigy, Katherine Johnson (born Coleman) was a pioneer of computing and the American space program. Her calculations on flight trajectories were critical in enabling Alan Shepard to become the first American in space (1961), John Glenn to be the first American to orbit Earth (1962), Apollo 11 to land on the Moon (1969), and the Space Shuttle program to launch (1981). Johnson graduated in 1937 from West Virginia State College and became one of the first African-Americans to enroll in a graduate program at West Virignia University. She worked for the National Advisory Committee for Aeronautics (NACA) from 1953 as part of a group of African-American women mathematicians known as the West Area Computers, who later inspired the film Hidden Figures (2016). Johnson then worked for the National Aeronautics and Space Administration (NASA) from 1958 as part of its Space Task Group. In 2015, President Obama awarded Johnson the Presidential Medal of Freedom.

  See also: Calculus • Newton’s laws of motion • Non-Euclidean geometries

  JULIA BOWMAN ROBINSON

  1919–85

  Born Julia Bowman in St. Louis, Missouri, Robinson gained her mathematics doctorate at the University of California, Berkeley, in 1948. She developed a fundamental theorem of elementary game theory (see John von Neumann) in 1951, but is best known for her work on solving the tenth of David Hilbert’s list of 23 mathematical problems, drawn up in 1900—whether there is an algorithm that could find a solution to any Diophantine equation (one that uses whole numbers and finite unknowns). Robinson proved, along with other mathematicians, such as Yuri Matiyasevich, that such an algorithm could not exist. Robinson was appointed a professor at Berkeley in 1975, and in 1976 she was the first woman to be elected to the American National Academy of Sciences.

  See also: Diophantine equations • 23 problems for the 20th century

  MARY JACKSON

  1921–2005

  An aerospace engineer, Mary Jackson (born Winston) worked on the US space program and campaigned for better opportunities in engineering for women and people of color. After graduating in math and physical sciences from Hampton University, Virginia, Jackson taught for a while, then in 1951 started work in the West Area Computing Unit of the National Advisory Committee for Aeronautics (NACA). The unit, known as the West Area Computers, comprised female African-American mathematicians, including Katherine Johnson. From 1958—when Jackson became NASA’s first female black engineer—to 1963, she worked on Project Mercury, the program that put the first Americans into space.

  See also: Calculus • Newton’s laws of motion • Non-Euclidean geometries

  ALEXANDER GROTHENDIECK

  1928–2014

  Considered by many to be the greatest pure mathematician of the second half of the 20th century, Grothendieck was unorthodox in every respect. Born in Germany to anarchist parents, at the age of 10 he fled the Nazi regime as a refugee to France, where he spent most of his life. His huge output—much of it never published—included revolutionary advances in algebraic geometry, the devising of the theory of schemes, and contributions to algebraic topology, number theory, and category theory. Grothendieck’s radical political activities included delivering math lectures just outside Hanoi while the city was being bombed during the Vietnam War.

  See also: Non-Euclidean geometries • Topology

  JOHN NASH

  1928–2015

  American mathematician John Nash is best known for establishing the mathematical principles of game theory (see John von Neumann). After graduating from Carnegie Mellon University in 1948 and being awarded a doctorate from Princeton University in 1950, he joined the Massachusetts Institute of Technology (MIT), where he researched partial differential equations and began the work on game theory that won him the Nobel Prize for Economics in 1994. For much of his life, Nash fought paranoid schizophrenia, as dramatized in the film A Beautiful Mind (2001).

  See also: Calculus • The logic of mathematics

  PAUL COHEN

  1934–2007

  New Jersey-born Cohen was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1966 for solving the first of David Hilbert’s list of 23 unresolved mathematical problems—that there is no set whose number of elements is between that of the integers and that of the real numbers. Cohen graduated and later received his doctorate, in 1958, from the University of Chicago before moving to the Massachusetts Institute of Technology (MIT), Princeton University, and finally Stanford University, where he became professor emeritus in 2004.

  See also: 23 problems for the 20th century

  CHRISTINE DARDEN

  1942–

  Along with Katherine Johnson and Mary Jackson, Darden is one of the African-American women whose work as mathematicians made key contributions to the work of NASA’s space programs. After graduating from Hampton University, Darden taught at Virginia State University before moving in 1967 to NASA’s Langley Research Center. There, she built her reputation as an aeronautical engineer, specializing in supersonic flight. In 1989, she was appointed leader of the Sonic Boom Team, working on designs to reduce noise pollution and other negative effects of supersonic flight.

  See also: Calculus • Newton’s laws of motion • Non-Euclidean geometries

  KAREN KESKULLA UHLENBECK

  1942–

  In 2019, Uhlenbeck became the first woman to be awarded the Abel Prize for Mathematics. Born in Cleveland, Ohio, in 1942, she gained a PhD in mathematics from Brandeis University, Waltham, Massachusetts in 1968, and went on to achieve notable breakthroughs in mathematical physics, geometrical analysis, and topology. A champion of gender equality in science and mathematics, in 1990 she became the first woman since Emmy Noether to give a plenary speech at the International Congress of Mathematics. In 1994, she founded the Women and Mathematics Program at the Institute of Advanced Study in Princeton, New Jersey.

  See also: Topology

  EVELYN NELSON

  1943–87

  The Krieger–Nelson Prize, awarded by the Canadian Mathematical Society for outstanding research by a female mathematician, is named in honor of Evelyn Nelson and fellow Canadian Cecilia Krieger. Nelson began a career of teaching and research at McMaster University after obtaining her doctorate there in 1970. She published more than 40 research papers in a 20-year career that was cut short by cancer. Her main contributions were to universal algebra (the study of algeb
raic theories and their models) and algebraic logic, applying these to the field of computer science.

  See also: The fundamental theorem of algebra • The logic of mathematics

  YURI MATIYASEVICH

  1947–

  While studying for his doctorate at the Steklov Institute of Mathematics in Leningrad (now St. Petersburg), Matiyasevich became fascinated by the challenge of solving David Hilbert’s tenth problem. Just as he was about to give up, he read the paper “Unsolvable Diophantine problems” (1969) by American mathematician Julia Robinson, and a solution fell into place. In 1970, Matiyasevich provided the final proof that the tenth problem is unsolvable because there is no general method of determining whether Diophantine equations have a solution. In 1995, he was appointed professor at St. Petersburg University, first as chair of software engineering and later as chair of algebra and number theory.