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Thoralf Albert Skolem (pronounced "SKOHlum") was born on May 23, 1887, in Sandsvaer, Norway. He was the son of Even Skolem and Helene Vaal. His father was an elementary school teacher, but most of the family were farmers. Skolem passed the concluding examinations of the Norwegian Gymnasium in 1905 and immediately went to the University of Oslo to study mathematics and science. He was graduated with the highest distinction in 1913 after completing a dissertation on the algebra of logic. By this time, he had already spent four years as an assistant to physicist Olaf Kristian Bernhard Birkeland, with whom he published his first scientific papers. Skolem spent another two years as Birkeland's assistant before going abroad to Göttingen, Germany, in 1915 to study for a year. From 1916 to 1918, he was a research fellow at the University of Oslo, and then he was appointed to a newly created position as assistant professor. Mathematical logic was in a state of chaos at the beginning of the twentieth century. Although the main concepts were already introduced and some interesting results were found, not enough was known to indicate which concepts were fundamental. Mathematics was thought of as an axiomatic system in which certain self-evident statements were taken as axioms and results were deduced by using logic. It was hoped that all the true mathematical statements could be derived by manipulating symbols according to explicit rules, but no one had the slightest clue how to prove that this was actually so. In his early works, Skolem was one of the first to indicate that it may not be possible to derive all the true mathematical statements. He published an important paper in 1920; from it emerged what is now known as the Löwenheim-Skolem theorem. Skolem proved that if a countable set of sentences is satisfiable, then it is satisfiable within a countable domain. INSET: Thoralf Albert Skolem.

Valerie Elaine Taylor was born on May 24, 1963. The daughter of Willie and Ollie Thompson Taylor, she grew up in Chicago, where her father owned an engineering company. Taylor credited her father with inspiring her to pursue a scientific career. She helped him with his work on weekends and decided that she wanted to be an engineer. Taylor enrolled at Purdue University in 1981 and, although she was lonely as one of the few African Americans and women in her classes, she excelled in her studies, being elected to the engineering honorary Tau Beta Pi and serving as president of Purdue's student chapter from 1983 to 1984. She also was a campus and community leader, participating in programs to encourage minority students in high school and college to study science. Taylor led computer workshops to familiarize students with technology. During summers, Taylor worked for International Business Machines (IBM) and Exxon, gaining practical engineering experience. She completed a bachelor's degree in electrical engineering in 1985 and began graduate work at Purdue, working on processor problems and presenting papers at conferences. In 1986, she earned a National Science Foundation Minority fellowship. Taylor joined the Institute of Electrical and Electronics Engineers and the National Society of Professional Engineers. She won a paper award from the National Society of Black Engineers in 1987. After receiving her master's degree in electrical engineering from Purdue in 1986, Taylor moved to Oakland, California, to begin doctoral work at the University of California, Berkeley. By 1991, she finished her Ph.D. in electrical engineering and computer sciences, focusing on processors. In November, 1991, Taylor filed for a patent for a computer process that she designed for her dissertation. Patent #5,206,822, "Method and Apparatus for Optimized Processing of Sparse Matrices," was approved in 1993. INSET: Valerie E. Taylor.

Siméon-Denis Poisson (pronounced "pwah-SOHN") was born in 1781. He was the son of the head of local government at Pithiviers. After completing his elementary education, Poisson was sent to an uncle, a surgeon, at Fountainebleau for training in the medical arts. The first lessons were in bleeding and blistering, then common treatments for a variety of ailments. Poisson showed little interest in the curative arts, but he did demonstrate a genuine talent for mathematics. Poisson was admitted to the École Polytechnique in Paris. He quickly attracted the notice of the instructors at the school because of his skills in higher mathematics. Recognizing Poisson's reasoning abilities in mathematics, the faculty members wisely allowed him to follow studies that were intellectually satisfying to him. In 1800, less than two years after he entered the school, he wrote two mathematical memoirs critically examining the methodologies and soundness of logic in two papers by the mathematician E. Bouzout. One memoir, on the number of integrals of an equation of finite differences, drew the attention of senior faculty members, who were so impressed that they recommended the paper be published. This was looked on as a major honor for the young Poisson, who was still a student. Other honors soon came his way. Well-known mathematicians drew him into their circle and invited him to attend special lectures in higher mathematics. His peers often gathered in Poisson's room after an unusually difficult lecture on some complex theorem to have him repeat it and explain it to them. The officials at the school recognized the value of his volunteer services and appointed him to an official post to carry on, with a stipend, the duties that he had performed as an unpaid volunteer. INSET: Siméon-Denis Poisson.

Pierre-Simon Laplace (pronounced "lah-PLAHS") was born in Beaumont-en-Auge in Normandy on March 23, 1979. His father, Pierre, was a business agent for the parish and was also in the cider business. His mother, Marie-Anne Sochen, came from a prosperous farm family. Laplace attended a local Benedictine school from the age of seven to sixteen and then enrolled at the University of Caen. Following his father's wishes, Laplace began studying for a career with the church, but his interests eventually took him into mathematics. At age nineteen, he left for Paris with a letter of recommendation to Jean Le Rond d'Alembert, one of the foremost mathematicians of the day. According to tradition, d'Alembert presented Laplace with a mathematical problem and told him to come back in a week. Instead, Laplace returned with the solution the next day. D'Alembert presented him with a second problem, and the results were the same. D'Alembert not only became Laplace's mentor but also secured an appointment for him as a professor of mathematics at the École Militaire in Paris. After only five years in Paris, Laplace won election to the Académie des Sciences. The academy's secretary wrote that never had it received so many important papers on such varied and difficult topics in such short time from so young a candidate. Laplace had submitted thirteen papers on such diverse topics as adapting integral calculus to the solution of difference equations, the mathematics of chance and games, and problems of mathematical astronomy. INSET: Pierre-Simon Laplace.

Joseph-Louis Lagrange (pronounced "law-GRAHNZH") was born and raised in Italy, although his family came from the French aristocracy. His father served as a public official but was not wealthy. He hoped that his son would become a lawyer, but Lagrange became devoted to mathematics early in his studies. By the age of seventeen, he was studying analysis. In 1754, he began to correspond with Leonhard Euler and the geometer Giulio da Fagnano. These men helped make others aware of Lagrange's talents. In 1755, a royal decree named Lagrange a professor at the Royal Artillery School in Turin. He taught classes on mechanics and analysis. During this time, he researched the calculus of variations, which deals with maximizing and minimizing the areas of planar figures. In addition, Lagrange advanced the technique of integration by parts. He developed the principle of virtual velocities and applied it to planetary movement in several prize-winning papers. He also helped found the Royal Academy of Sciences of Turin and was later elected to the Berlin Academy of Sciences. In 1766, Lagrange was offered more money and better scientific facilities to work at the Berlin Academy of Sciences. He became director of the mathematics section, a position that Euler had vacated. Lagrange wrote sixty-three papers during his twenty-one years in Berlin. He also married his cousin, Vittoria Conti. Both Lagrange and his wife suffered from poor health throughout their lives. She passed away in 1783 after a long illness. The couple had no children. INSET: Joseph-Louis Lagrange.

On April 25, 1903, Andrey Nikolayevich Kolmogorov (pronounced "kul-muh-GAW-ruf") was born in Tambov, Russia. His mother, an aristocratic woman named Mariya Yakovlevna Kolmogorova, died giving birth to him. Little is known of his father. He was raised by his mother's sister, Vera Yakovlevna Kolmogorova, on the family estates until the Russian Revolution of 1917. After the revolution, Kolmogorov worked for a while as a railway conductor before entering Moscow University in 1920; he would remain there in various capacities for the rest of his life. This was a time of hardship at the university. Scholarships consisted of rations of bread and fat, and the lecture halls went unheated through the Russian winter. Kolmogorov supplemented his income by teaching in an experimental school. He persisted and advanced rapidly in his studies, taking an interest in Russian history and poetry as well as mathematics. After completing his doctorate in 1925, Kolmogorov was permitted to remain at the university for four more years as a research student. In 1929, Pavel Sergeevich Aleksandrov obtained a permanent position for him at the Institute for Mathematics and Mechanics at Moscow University, and, in 1931, Kolmogorov was promoted to full professor. Aleksandrov and Kolmogorov became close friends and remained so until Aleksandrov's death in 1982. Much of Kolmogorov's early work concerned mathematical logic--he developed a theory called intuitionistic logic--and probability. Although methods had long existed for calculating some specific probabilities, no general axiomatic theory of probability existed before Kolmogorov's 1929 paper on a general theory of measure and the calculus of probabilities. He continued this line of research with papers on random processes and summarized his work in 1933 with a monograph Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability, 1950). INSET: Andrey Nikolayevich Kolmogorov.

Josiah Willard Gibbs was the only son among five children born to Mary Anna Van Cleve and Josiah Willard Gibbs, Sr. The senior Gibbs was a noted philologist who served as a professor of sacred literature at Yale Divinity School from 1826 until his death in 1861. The younger Gibbs was a withdrawn and intellectually intense youth of delicate health. He attended Yale College, where he received prizes in Latin, Greek, and mathematics. After graduating in 1858, he enrolled in Yale's newly formed graduate school of engineering, earning a Ph.D. in 1863. He spent the next three years tutoring in Latin and natural philosophy at Yale. When he returned to New Haven in June, 1869, having assimilated the intellectual techniques of his teachers, Gibbs was more interested in theory and broad applications. American science of the time was preoccupied with practical devices. In 1871, two years before publishing his first paper, Gibbs was appointed professor of mathematical physics at Yale, a position that he occupied until his death. During the first nine years of his tenure at Yale, Gibbs served without salary, supporting himself by his inheritance from his father. It was during these unsalaried years that he wrote the papers on thermodynamics that are now recognized as his greatest scientific contribution. INSET: J. Willard Gibbs.

Évariste Galois (pronounced "gahl-WAH") was born on October 25, 1811, at Bourg-la-Reine, near Paris, France. He was the middle of three children born to Nicholas-Gabriel and Adélaide-Marie Demante Galois. Nicholas operated a school that his family had owned for some time and later became mayor of Bourg-la-Reine for fifteen years. Adélaide was a well-educated woman who taught her children at home for several years. Évariste himself was taught at home until he was twelve years old. In 1823, he was sent to the Lycée Louis-le-Grand in Paris. This school had grades divided into classes, where a new student would start in the sixth class and a finishing student would be in the first class. Évariste made great progress in his education there. Nevertheless, the headmaster did not want to promote him to the first class at age fifteen, since he was thought to be too young. Nicholas Galois enrolled his son in the first class anyway, but a teacher's report that Évariste was "too immature" for this level sent him back to the second class at midyear. By the time that he was sixteen, Galois was becoming absorbed in mathematics. School reports indicated problems with his behavior and with the fact that he was not completing his lessons. He wanted to think only about mathematics--and mathematics that was advanced even beyond the level of most of his teachers. In the spring of 1829, the famous mathematician Augustin-Louis Cauchy presented a paper by Galois to the great French scientific academy, the Académie des Sciences. Until this time, it was almost unheard of that the great Cauchy would present someone else's paper to the academy, much less that of a mere seventeen-year-old. Yet, the mathematical thoughts of Galois were advancing beyond those of any of his contemporaries. INSET: Évariste Galois.

Paul Erdös (pronounced "EHR-dihsh") was born in Budapest, Hungary, on March 26, 1913. His parents, who were Jewish, were teachers of mathematics. His talent for mathematics was evident early in life. By the age of four, he had already discovered negative numbers for himself and could multiply two three-digit numbers in his head. Erdös's two sisters died only days before he was born, and his parents were very protective of him. Consequently, he was educated at home by his mother until 1922. He then spent four years in school at Tavaszemezö Gymnasium and one year at Saint Stephen's School, where his father was a high school teacher. In 1930, Erdös entered the University in Budapest. During his second year, he found a simple proof of a theorem first proved by Pafnuty Lvovich Chebyshev, which resulted in a published paper. He received his doctorate from the university in 1934. While in the United States, Erdös wrote many papers by himself and with other American mathematicians, as well as collaborating by mail with those outside the country. Most of the work that he did during these years was in number theory and combinatorics. After leaving Princeton permanently in 1950, Erdös began his neverending travels. He has been described by many as a mathematical pilgrim. In 1954, he visited Amsterdam to attend the International Congress of Mathematicians, but he could not get a reentry permit to the United States because of the feelings that had been stirred up against communism during the McCarthy era. Not wanting to be stopped by a piece of paper, he found a temporary home in Israel. INSET: Paul Erdös.

Alessandro Giuseppe Antonio Anastasio Volta's father, Filippo, was a minor nobleman of Lombardy, a district in north Italy. He married Maddena dei Conti Inzaghi, who came from another established family of the region. Alessandro seemed dull-witted as a child, not speaking until the age of four. He developed rapidly into a model student, however, and continued classical studies until he was sixteen. He had the command of several languages by then, but he was most strongly attracted to the study of the natural sciences, especially physics, chemistry, and electricity. Volta began publishing scientific papers at the age of twenty-four. While fishing one day on a lake near his home, he noticed bubbles rising to the surface, especially in the shallows and marshes. He began collecting samples of the vapors rising above the bubbles and learned that they comprised a flammable gas. These marsh vapors are now known as methane gas, which is produced when organic material decays. Volta published his findings in what became a widely read scientific paper. Volta was almost fifty when he married Signorina Teresa Peregrini, the youngest daughter of a local nobleman. The couple had three sons: Zanino, Flaminio, and Luigi. Zanino was elected mayor of Como, while Flaminio and Luigi became publishers of a periodical devoted to applied science and industry. Volta's scientific productivity declined significantly, however, once he became a father. While examining the conclusions of Luigi Galvani, who had postulated the existence of "animal electricity," Volta found instead that the natural current seen in this experiment resulted from contact between two dissimilar metals. Volta also observed that current flowed when metal touched some fluids. He learned that the effect could be strengthened when dissimilar metals, such as zinc and silver, were interleaved with moist paper pads. INSET: Count Alessandro Volta.

Julian Seymour Schwinger was born in New York City in 1918. His father, a clothing designer, was a successful businessman, and Schwinger grew up in an upper-middle-class environment. A true child prodigy, Schwinger was graduated from Townsend Harris High School, one of the premier academic preparatory schools in the country, at the age of sixteen. Following his graduation, he enrolled at the City College of New York, but he transferred to Columbia University after a year upon the suggestion I. I. Rabi, a noted physicist at Columbia who recognized Schwinger's unusual aptitude in theoretical physics. As an undergraduate at City College, Schwinger had already written one research paper on quantum electrodynamics. He published several additional papers as an undergraduate and, in fact, completed his doctoral dissertation before receiving his bachelor of science degree in physics from Columbia in 1936. He continued at Columbia for a year, working with Rabi, before spending eighteen months at the University of Wisconsin. In 1939, following his return to New York, Schwinger was awarded a Ph.D. in physics from Columbia University. He spent the next two years at the University of California, Berkeley, working with J. Robert Oppenheimer and other theoreticians. During this time, his research focused on the electromagnetic properties of deuterium and many particle systems. In 1941, Schwinger accepted an offer of an instructor position at Purdue University. INSET: Julian Seymour Schwinger.

Frederick Sanger was born in Rendcombe, Gloucestershire, England, on August 13, 1918. The son of a physician, he spent his early education at Bryanston School. Sanger attended Cambridge University for his entire university education, earning first a bachelor of arts degree in 1939 and then a doctorate in 1943 through St. John's College at Cambridge. He continued his biochemistry research at Cambridge as a research fellow until 1951, at which time he joined the staff of the Medical Research Council. During the 1940's and 1950's, much chemical research focused on the hereditary molecule deoxyribonucleic acid (DNA) and proteins, the products of DNA. Sanger's primary research interest at this time was in determining the exact amino acid sequence of a protein. In the early 1940's, he experienced a significant breakthrough toward that end. In 1945, he devised a method to use the chemical 2,4-dinitrofluorobenzene, later known as Sanger's reagent, to label one end of the protein chain. Sanger could then use acid to break up the protein into smaller fragments. Other scientists had designed a method to separate the individual amino acids from a mixture called paper chromatography. Sanger planned to break down the sequence of amino acids partially; attach his reagent to one end, break these labeled fragments down to individual amino acids, and separate them using paper chromatography. In this way; he could identify which amino acid was labeled. By painstakingly repeating this procedure many times, he could generate overlapping fragments of the protein and then, as if working a jigsaw puzzle, deduce the order of the amino acids. The protein hormone insulin had been isolated some twenty-five years earlier and was known to consist of two connected chains of fifty amino acids. Sanger chose this protein, which he obtained from the pancreatic tissue of cattle, for his sequence experiment. INSET: Frederick Sanger.