Dangerous knowledge

In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide. The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity. Ludwig Boltzmann's struggle to prove the existence of atoms and probability eventually drove him to suicide. Kurt Gödel, the introverted confidant of Einstein, proved that there would always be problems which were outside human logic. His life ended in a sanatorium where he starved himself to death. Finally, Alan Turing, the great Bletchley Park code breaker, father of computer science and homosexual, died trying to prove that some things are fundamentally unprovable. The film also talks to the latest in the line of thinkers who have continued to pursue the question of whether there are things that mathematics and the human mind cannot know. They include Greg Chaitin, mathematician at the IBM TJ Watson Research Center, New York, and Roger Penrose. Dangerous Knowledge tackles some of the profound questions about the true nature of reality that mathematical thinkers are still trying to answer today.

Georg Cantor (1845-1918) was a German mathematician, born in Saint Petersburg. He taught at the University of Halle and was professor there after 1872. His early work with the Fourier Series led to his development of a theory of irrational numbers. Cantor also formulated the theory of sets, upon which modern mathematical analysis is based. This theory extended the concept of number by introducing infinite or, as he called them, transfinite numbers. Cantor's work was largely responsible for the subsequent critical investigation of the foundations of mathematics and mathematical logic.

Ludwig Boltzmann (1844-1906)was an Austrian physicist, who helped lay the foundation for the field of physics known as statistical mechanics. Boltzmann was born in Vienna and educated at the universities of Vienna and Oxford. He was a professor of physics at various German and Austrian universities for more than 40 years. During the 1870s Boltzmann published a series of papers that showed that the second law of thermodynamics could be explained by statistically analyzing the motions of atoms. In these papers Boltzmann utilized the central principle of statistical mechanics: that large-scale, visible phenomena, such as the second law of thermodynamics, can be explained by statistically examining the microscopic properties of a system, such as the motions of atoms. Boltzmann also formulated the law of thermal radiation, named for him and the Austrian physicist Josef Stefan. The Stefan-Boltzmann law states that the total radiation from a blackbody, which is an ideal surface that absorbs all radiant energy that strikes it, is proportional to the fourth power of the absolute temperature of the blackbody. Boltzmann also made important contributions to the kinetic theory of gases. Boltzmann's work was strongly attacked by scientists of his time. However, much of Boltzmann's work was substantiated by experimental data soon after he committed suicide in 1906.

Kurt Gödel (1906-78), American logician, known primarily for his research in philosophy and mathematics. He was born in Brünn, Austria-Hungary (now Brno, Czech Republic). He was educated at Vienna University and taught at that institution from 1933 to 1938. He immigrated to the United States in 1940 and became an American citizen in 1948. Gödel was a member of the Institute for Advanced Studies, Princeton, New Jersey, until 1953, when he became professor of mathematics at Princeton University. Gödel became prominent for a paper, published in 1931, setting forth what has become known as Gödel's proof. This proof states that the propositions on which the mathematical system is in part based are unprovable because it is possible, in any logical system using symbols, to construct an axiom that is neither provable nor disprovable within the same system. To prove the self-consistency of the system, methods of proof from outside the system are required. Gödel also wrote The Consistency of the Continuum Hypothesis (1940) and Rotating Universes in General Relativity Theory (1950).

Alan Turing (1912-1954), British mathematician considered one of the most important founders of computer science and artificial intelligence. He was the first to describe in detail a machine that could carry out mathematical operations and solve equations. His work brought together symbolic logic, numerical analysis, electrical engineering, and a mechanical vision of human thought processes.

Alan Mathison Turing was born in London. He had a conventional school education. Early on he developed a passion for science. He became a student in mathematics, but also gathered ideas from philosophy and physics. Early letters show how a concern for the problem of the mind was given emotional weight by the death of a dearly loved school friend.

In 1935 Turing entered King’s College at Cambridge University. Inspired by problems left outstanding by the work of Kurt Gödel in the foundations of mathematics, Turing began combining symbolic logic with an original analysis of mental activity. His paper “On Computable Numbers” (1937) introduced the abstract Turing machine to define the concept of a fixed computational method or algorithm. He also introduced the “universal” Turing machine, a single machine capable of performing any instructions given to it. He thereby solved the major mathematical problem called “decidability.” His work opened up the new theory of computability (what a computing device can do), and laid out the principles of the modern computer. He earned his Ph.D. at Princeton University in the United States in 1938, then returned to England.

Turing’s mathematical career was overtaken by World War II (1939-1945). He became a cryptographer for the British Foreign Office and excelled at applying scientific ideas to code-breaking (cryptography). Most famously, he constructed a machine to help break the German naval Enigma cipher. His successful deciphering of the code provided a tool to track German ships in the Atlantic, an advantage critical to the victory of the Allies in the war. Turing was awarded the Order of the British Empire (OBE) for his work.

In 1946 Turing used his experience with electronic technology to translate his abstract universal Turing machine into a detailed design for a digital computer. American researchers had already made similar proposals, but Turing’s design was far ahead in grasping the scope of new technology. The National Physical Laboratory, where he was working, failed to take practical steps to build his proposed machine. In 1948 Turing moved to the University of Manchester to pioneer the use of the computer developed there.

Turing held that a computer was capable in principle of doing anything that the brain can do. His 1950 paper “Computing Machinery and Intelligence” set forth a theory that remains a cornerstone of the science of artificial intelligence. The test that Turing proposed involved a computer communicating with human judges via a teleprinter link. If the computer’s responses to questions were indistinguishable from those of a human being, Turing claimed, the computer should be regarded as exhibiting intelligence. The Turing test for machine intelligence had a lasting influence in the philosophy of mind and still provokes discussion.

In 1951 Turing was named a Fellow of the Royal Society. The next year he began to publish his work on the mathematical aspects of pattern and form development in living organisms.

Events in Turing’s personal life effectively ended his career, however. In 1952 he stood trial for having had a homosexual relationship, then a crime in Britain. He was classed as a security risk and accepted injections of estrogen to avoid imprisonment. He was open about his homosexuality and unashamed of it. However, homosexuality was then considered a mental illness and he sought Jungian therapy. In 1954 Turing died of cyanide poisoning. Although the circumstances of his death were not clear, the verdict was suicide.

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