Kurt Friedrich Gödel (1906 – 1978) was an Austrian, and then American, logician, mathematician, and philosopher. In his family, young Kurt was known as Herr Warum ("Mr. Why") because of his insatiable curiosity. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic. Gödel made an immense impact upon scientific and philosophical thinking in the 20th century. He, as well as Aristotle and Gottlob Frege, are considered to be one of the most significant logicians in history. Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects. Although Kurt had originally prefered languages, he later became more interested in history and mathematics. One year after finishing his doctorate at the University of Vienna, when he was 25 years old, Gödel published his two incompleteness theorems in 1931. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. Gödel's life course may have been set after he attended a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems. In 1928, Hilbert and Wilhelm Ackermann published "Grundzüge der theoretischen Logik" (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: "Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?" This became the topic that Gödel chose for his doctoral work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem).
Gödel became a full professor at the Institute for Advanced Study at Princeton in 1953 and an emeritus professor in 1976. In that time period, "Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed time-like curves, to Albert Einstein's field equations in general relativity.He is said to have given this elaboration to Einstein as a present for his 70th birthday. His "rotating universes" would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation)." In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned and would eat only food that his wife had prepared for him. Late in 1977, she was hospitalized for six months. In her absence, he refused to eat and eventually starved himself to death. He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974. A biography of Gödel was published by John Dawson in 2005. Source: Kurt Gödel. (n.d.). Retrieved January 7, 2015, from http://en.wikipedia.org/wiki/Kurt_Gödel
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Maurits Cornelis Escher (1898-1972) was a Dutch graphic artist. He made 448 lithographs, woodcuts and wood engravings and over 2000 drawings and sketches. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, and tessellations. In addition to being a graphic artist, M.C. Escher illustrated books, designed tapestries, postage stamps and murals. Escher was born in Leeuwarden, Holland. He was a sickly child, and his grades were generally poor except for drawing which he excelled at. Escher's father (an architect) enrolled him in the School for Architecture and Decorative Arts in Haarlem as he wanted his son to become an architect. After only one week, Escher switched to decorative arts after having showed his drawings and linoleum cuts to his graphic teacher ,Samuel Jessurun de Mesquita, who encouraged him to continue with graphic arts. In his early years, Escher sketched landscapes and nature. His first artistic work was completed in 1922 and featured eight human heads divided in different planes. That year, Escher traveled through Italy (Florence, San Gimignano, Volterra, Siena, Ravello) and Spain (Madrid, Toledo, Granada). He was impressed by the Italian countryside and by the Alhambra, a fourteenth-century Moorish castle in Granada. The intricate decorative designs at Alhambra, which were based on geometrical symmetries featuring interlocking repetitive patterns sculpted into the stone walls and ceilings, were a powerful influence on Escher's works. In his graphic art, Escher portrayed mathematical relationships among shapes, figures and space. The mathematical influence in his work emerged around 1936, when he journeyed to the Mediterranean. Escher described his journey through the Mediterranean as "the richest source of inspiration I have ever tapped." As a result of the journey, he became interested in order and symmetry. Although Escher did not have mathematical training (his understanding of mathematics was largely visual and intuitive) his artwork is especially liked by mathematicians and scientists. They enjoy his use of polyhedra and geometric distortions. Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept. In 1959, Coxeter published his finding that these works were extraordinarily accurate: "Escher got it absolutely right to the millimeter." Escher also studied topology. He learned additional concepts in mathematics from the British mathematician Roger Penrose. From this knowledge he created Waterfall and Up and Down (shown in the book "Gödel, Escher, Bach: An Eternal Golden Braid"), featuring irregular perspectives similar to the concept of the Möbius strip. His artistic expression was created from images in his mind, rather than directly from observations and travels to other countries. More than a few of the worlds which he drew were built around impossible objects such as the Necker cube and the Penrose triangle (which Escher incorporated into his famous lithograph "Waterfall"). Escher's first print of an impossible reality was Still Life and Street, 1937. Escher played with architecture, perspective and impossible spaces. In 1955, he was awarded the Knighthood of the Order of Orange Nassau. In 1958, he published a book entitled Regular Division of the Plane, with reproductions of a series of woodcuts based on tessellations of the plane, in which he described the systematic buildup of mathematical designs in his artworks. He emphasized, "Mathematicians have opened the gate leading to an extensive domain."
In 1969, Escher's business advisor, Jan W. Vermeulen, established the M.C. Escher Stichting (M.C. Escher Foundation), and transferred into this entity almost all of Escher's work. These works were lent by the Foundation to the Hague Museum. Vermeulen also wrote a biography of Escher. Source: M. C. Escher. (n.d.). Retrieved January 6, 2015, from http://en.wikipedia.org/wiki/M._C._Escher Who is Bach?? Johann Sebastian Bach (1685 - 1750) was a German musician and composer. His family was a "musical family" as his father and uncles were professional musicians. Throughout Europe, Bach was most renowned for his skill with the organ and his ability to improvise. Only in the first half of the 19th century was he recognized as a great composer. Nowadays Bach is considered one of the greatest composers that ever were. Musikalisches Opfer (Musical Offering) On the 7th of May (1747), Bach visited King Fredrick II of Prussia and was asked by him to improvise a 3-voice fugue on a complex and long theme. After having done so, Fredrick challenged him to improvise 6-voice fugue on that theme. Bach said that he'll work on it and send it to the king. Two months later, Bach published a set of pieces titled "Musikalisches Opfer" based on the theme given by Fredrick and dedicated to him. Bach hasn't specified the order in which the pieces are to be played nor the instruments that are meant to play them. One of the canons, namely "Canon per Tonos" seems to end but then starts again with another key. It starts with the key C, passes through the keys D,E,F,G, and B then returns back to C without the listener noticing any abruptness. The canon seems to go on for ever in this pattern and hence it was named "An Endlessly Rising Canon". The Musical Offering: |