Biographies

Thales (634 BC - 546 BC)

Pythagoras (560 BC - 480 BC)

Zeno (490 BC - 425 BC)

Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School , one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC.

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Hippocrates (470 BC - 410 BC)

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Eudoxus (400 BC - 347 BC)

Greek philosopher, astronomer, and mathematician who accepted Plato's notion of the rotation of the planets around the Earth on crystalline spheres, but noticed discrepancies with observations. He tried to adjust Plato's model by postulating that each crystalline sphere had its poles set to the next sphere. His model contained no mechanical explanation; it was simply a mathematical description.

Archimedes (287 BC - 212 BC)

Aristotle (384 BC - 322 BC)

Plato (427 BC - 347 BC)

Euclid (ca. 325 - ca. 270 BC)

Euclid's greatest accomplishment was the Elements , his 13-chapter book outlining everything he knew about geometry. He based all of his geometrical theorems on just five postulates, making the work very rigorous and complete, but for two millenia mathematicians wondered if the fifth postulate (the so-called "Parallel Postulate") could in fact be derived from the other four. This was finally answered (in the negative) by Lobachevsky, Bolyai, and Gauss, leading to the branch of mathematics we now call Non-Euclidean Geometry .

Al-Karaji, Abu Bekr ibn Muhammad ibn al-Husayn (953 - ca. 1029)

Al-Karaji's work centered around algebra and polynomials, giving rules for arithmetic operations to manipulate polynomials. Woepcke describes his work as introducing the "theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle. Additionally, Al-Karaji used induction to prove his results.

Fibonacci, Leonardo de Pisa (ca. 1170 - ca. 1240)

The Fibonacci numbers are taught to schoolchildren around the world as a fun and interesting mathematical diversion. The numbers have an intimite relationship with the golden ratio the Greeks so admired, and with Euclid's GCD algorithm, making them more significant than Fibonacci probably anticipated when he posed his original problem involving rabbits. Besides the numbers named after him, Fibonacci wrote important books that revived classical Greek mathematical knowledge, on such subjects as the place-value system and geometry.

Pascal, Blaise (1623 - 1662)

In his short life, Pascal made contributions to many fields of science. He was possibly the second person ever to create a mechanical calculator, and ran many barometric experiments. However, much of work focused on mathematics, including geometry (particularly conic sections), probability theory, and the "Arithmetical Triangle" (known today as "Pascal's triangle").

Leibniz, Gottfried (1646-1717)

Euler, Leonhard (1707-1783)

Leonhard Euler made contributions to just about every area of mathematics. Some of his more famous results included many theorems about Complex Numbers (introducing the symbol i for the square root of -1), his theorem on Eulerian circuits that led to the development of Graph Theory, and his discovery of the closed form for the Fibonacci numbers.

Gauss, Karl Friedrich (1777-1855)

Karl Friedrich Gauss was one of the most prolific and influential mathematicians in all of history (second only to Euler if anyone), in spite of the fact that he was a perfectionist and did not publish many of his discoveries. He made lasting contributions not just to number theory, geometry, and calculus, but also physics and astronomy. Among other things he is known for discovering how to sum consecutive integers at a very young age, and for his Prime Number Theorem.

Galois, Evariste (1811-1832)

Cantor, Georg (1845-1918)

Georg Cantor was a German mathematician who introduced the mathematical world to the amazing concept of measuring the cardinality of infinite sets, but unfortunately most of his contemporaries did not like his ideas because they were so radical and disturbing. He discovered that the real numbers are not countable, but the rationals are. When not working on mathematics, Cantor tried to prove that Francis Bacon actually wrote Shakespeare's plays.

Polya, George (1887 - 1985)

Kolmogorov (1903 - 1987)

In the context of the course, Kolmogorov is associated with randomness and incompressibility. However, he made many contributions in other areas, ranging from set theory and Markov processes to probability and physics. His outstanding career brought many honors from many countries, including Russian, American, and British organizations.

Church, Alonzo (1903-1995)

Godel, Kurt (1906-1978)

Turing, Alan Mathison (1912-1954)

Alan Turing was one of the fathers of computer science. His theoretical contributions included the formalization of an abstract machine ("Turing Machine"), the proof that the halting problem was uncomputable, and a philosophical treatise on Artificial Intelligence (the "Turing Test"). Just as significantly, though, he was a key player in the Allies' breaking of the German Enigma cipher in WWII - with a combination of pure mathematics and an ingenious machine Turing helped build, the Allies were able to crack every new German cipher within hours.

Shannon, Claude (1916 - 2001)

In 1936, Claude Shannon graduated from University of Michigan with Bachelor degrees in Electrical Engineer and Mathematics. This dual interest was this key to his later work. He then went to M.I.T. and spent time working with the Bush differential analyzer, a mechanical device that solved differential equations. He recognized that the machine was a two-valued system, ideally suited for Boolen algebra. His ideas were refined at Bell Labs and subsequent years at M.I.T. leading to a vacuum tube version called the Bush Rapid Selector. Out of his research came our modern notions of computers, information theory, and communication networks.

Robinson, Julia (1919-1985)

Julia Robinson was the first female mathematician elected to the National Academy of Sciences and the first woman president of the American Mathematical Society. Initially, she enrolled in college to receive public school teaching credentials, but became involved with number theory. In particular, she demonstrated that the notion of an integer can be derived from the notion of rationals. Most of her work was focussed on solving Hilbert's tenth problem, finding an effective method to solve diophantine equations.

Karatsuba, Anatolii Alexeevich (1937-)

Anatolii Karatsuba is a Russian mathematician who is best known for discovering the first multiplication algorithm that runs in less than O(n²) time - specifically his algorithm is O(n ^{log₂ 3}) or about O(n ^1.58 ). He has also written papers on such diverse areas as finite automata theory and the Riemann Zeta function. He enjoys mountaneering and is currently the head of the Laboratory of Analytic Number Theory at the Steklov Mathematical Institute in Moscow.

Cook, Stephen (1939 - )

Cook was born in Buffalo, NY and completed his undergraduate education at the University of Michigan. He earned his Ph.D. in Mathematics from Harvard, then moved on to teach at U.C. Berkeley until 1970. Since then he has taught at the University of Toronto. His principal research area is computational complexity, with excursions into programming language semantics, parallel computation, and especially the interaction between logic and complexity theory. Cook is best known for his formulation of the notion of NP-completeness and his proof that satisfiability is NP-complete. He has published about one hundred papers in theoretical computer science, including fundamental results in parallel complexity.

Chaitin, Gregory (1947 - )

"Gregory Chaitin is at the IBM Watson Research Center in New York. In the mid 1960s, when he was a teenager, he created algorithmic information theory (AIT), which combines, among other elements, Shannon's information theory and Turing's theory of computability. In the three decades since then he has been the principal architect of the theory. Among his contributions are the definition of a random sequence via algorithmic incompressibility, and his information-theoretic approach to Gödel's incompleteness theorem. His work on Hilbert's 10th problem has shown that in a sense there is randomness in arithmetic, in other words, that God not only plays dice in quantum mechanics and nonlinear dynamics, but even in elementary number theory. His latest achievement has been to transform AIT into a theory about the size of real computer programs, programs that you can actually run. He is the author of six books: Algorithmic Information Theory published by Cambridge University Press; Information, Randomness & Incompleteness and Information-Theoretic Incompleteness, both published by World Scientific; and The Limits of Mathematics, The Unknowable and Exploring Randomness, all published by Springer-Verlag. In 1995 he was given the degree of doctor of science honoris causa by the University of Maine, and he was elected to the IBM Academy of Technology. In 1998 he was named visiting professor at the University of Buenos Aires."
-quoted from G J Chaitin Homepage