John Von Neuman, Anant Singh, CSC 170 Project 3

John Von Neumann

Manhattan Project

Background

John von Neumann was a Hungarian-American mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.

Von Neumann was a child prodigy. As a 6 year old, he could multiply and divide two 8-digit numbers in his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly in front of her, the 6 year old von Neumann asked her: "What are you calculating?"

Formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins. Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of 8, von Neumann was familiar with differential and integral calculus, but he was particularly interested in history, reading his way through Wilhelm Oncken's 46-volume Allgemeine Geschichte in Einzeldarstellungen. A copy was contained in a private library Max purchased. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor.

Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Lutheran Church, the majority of its pupils were Jewish. The school system produced a generation noted for intellectual achievement, that included Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913). Collectively, they were sometimes known as Martians. Wigner was a year ahead of von Neumann at the Lutheran School. When asked why the Hungary of his generation had produced so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that von Neumann was the only genius.

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő.[22] On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.[24] Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest.[22] By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition.[25] At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.

Achievements

He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer. He published over 150 papers in his life: about 60 in pure mathematics, 20 in physics, and 60 in applied mathematics, the remainder being on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain.

His analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

During World War II he worked on the Manhattan Project, developing the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, and later as one of its commissioners. He was a consultant to a number of organizations, including the United States Air Force, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project, and the Lawrence Livermore National Laboratory. Along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.

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Ordinal Numbers

Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.The standard definition, suggested by John von Neumann, is: each ordinal is the well-ordered set of all smaller ordinals. In symbols, λ = [0,λ).[3][4] Formally: A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S. The natural numbers are thus ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving bijective function between them.

Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T, S is an element of T if and only if S is a proper subset of T. Moreover, either S is an element of T, or T is an element of S, or they are equal. So every set of ordinals is totally ordered. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered.

Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the axiom of union.

The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its strict ordering by membership. This is the Burali-Forti paradox. The class of all ordinals is variously called "Ord", "ON", or "∞".

An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its subsets has a maximum.

Symbol	Elements	Description
0	{}	empty set
1	{0}	set of one element
2	{0,1}	set of two elements
3	{0,1,2}	set of three elements
ω	{0,1,2,...}	set of all finite ordinals

Advancements in Computer Science

In 1945, mathematician John von Neumann undertook a study of computation that demonstrated that a computer could have a simple, fixed structure, yet be able to execute any kind of computation given properly programmed control without the need for hardware modification. Von Neumann contributed a new understanding of how practical fast computers should be organized and built; these ideas, often referred to as the stored-program technique, became fundamental for future generations of high-speed digital computers and were universally adopted. The primary advance was the provision of a special type of machine instruction called conditional control transfer--which permitted the program sequence to be interrupted and reinitiated at any point, similar to the system suggested by Babbage for his analytical engine--and by storing all instruction programs together with data in the same memory unit, so that, when desired, instructions could be arithmetically modified in the same way as data. Thus, data was the same as program.

As a result of these techniques and several others, computing and programming became faster, more flexible, and more efficient, with the instructions in subroutines performing far more computational work. Frequently used subroutines did not have to be reprogrammed for each new problem but could be kept intact in "libraries" and read into memory when needed. Thus, much of a given program could be assembled from the subroutine library. The all-purpose computer memory became the assembly place in which parts of a long computation were stored, worked on piecewise, and assembled to form the final results. The computer control served as an errand runner for the overall process. As soon as the advantages of these techniques became clear, the techniques became standard practice. The first generation of modern programmed electronic computers to take advantage of these improvements appeared in 1947.

This group included computers using random access memory (RAM), which is a memory designed to give almost constant access to any particular piece of information. These machines had punched-card or punched-tape input and output devices and RAMs of 1,000-word. Physically, they were much more compact than ENIAC: some were about the size of a grand piano and required 2,500 small electron tubes, far fewer than required by the earlier machines. The first- generation stored-program computers required considerable maintenance, attained perhaps 70% to 80% reliable operation, and were used for 8 to 12 years. Typically, they were programmed directly in machine language, although by the mid-1950s progress had been made in several aspects of advanced programming. This group of machines included EDVAC and UNIVAC, the first commercially available computers.

List of Selected Works

1923. On the introduction of transfinite numbers, 346–54.
1925. An axiomatization of set theory, 393–413.
1932. Mathematical Foundations of Quantum Mechanics, Beyer, R. T., trans., Princeton Univ. Press. 1996 edition: ISBN 0-691-02893-1.
1937. von Neumann, John (1981). Halperin, Israel, ed. Continuous geometries with a transition probability. Memoirs of the American Mathematical Society. 34. ISBN 978-0-8218-2252-4. MR 634656.
1944. Theory of Games and Economic Behavior, with Morgenstern, O., Princeton Univ. Press, online at archive.org. 2007 edition: ISBN 978-0-691-13061-3.
1945. First Draft of a Report on the EDVAC TheFirstDraft.pdf
1948. "The general and logical theory of automata," in Cerebral Mechanisms in Behavior: The Hixon Symposium, Jeffress, L.A. ed., John Wiley and Sons, New York, N. Y, 1951, pp. 1–31, MR 0045446.
1960. von Neumann, John (1998). Continuous geometry. Princeton Landmarks in Mathematics. Princeton University Press. ISBN 978-0-691-05893-1. MR 0120174.
1963. Collected Works of John von Neumann, Taub, A. H., ed., Pergamon Press. ISBN 0-08-009566-6
1966. Theory of Self-Reproducing Automata, Burks, A. W., ed., University of Illinois Press. ISBN 0-598-37798-0

Citations

"John Von Neumann." History of Computing Science: John Von Neumann. N.p., n.d. Web. 23 Feb. 2017.

"John Von Neumann." Wikipedia. Wikimedia Foundation, 22 Feb. 2017. Web. 23 Feb. 2017.

"Ordinal Number." Ordinal Number -- from Wolfram MathWorld. N.p., n.d. Web. 23 Feb. 2017.

"Ordinal Number."Wikipedia. Wikimedia Foundation, 22 Feb. 2017. Web. 23 Feb. 2017.