Mathematician: The Life and Work of John von Neumann - Research Paper Example

Mathematician: The Life and Work of John von Neumann Introduction John von Neumann (1903-1957) was a mathematician whose work impacted various disciplines. His theory of games significantly helped the field of economics. Further, he is renowned for his work as one of the original professors at the Electronic Computer Project at the Institute for Advanced Study, Princeton. During his comparatively short life span he made invaluable contributions to quantum physics, logic, applied mathematics, and computer science. The mathematician along with F.J. Murray developed the theory behind the quantum concept “ring of operators” now known as von Neumann algebras. A pioneer in the field of computer science, “he addressed design questions such as how to construct cellular automata and the use of the bit in computer memory” (Von Neumann 82). Von Neumann was a member of the Atomic Energy Commission in 1957, when he passed away due to cancer. Thesis Statement: The purpose of this paper is to investigate John von Neumann’s contributions to the world of mathematics Discussion John von Neumann was a Hungarian-born American mathematician. The wide range of topics on which he worked included “mathematics, physics, computer engineering, and mathematical economics” (Ayoub 169). During World War II, he served as military consultant, and later took part in the development of the hydrogen bomb. After World War II he continued working as a consultant to the government, and was known to have “hawkish” policies unpopular with colleagues. Nuemann participated in the greatest scientific breakthrough of the first half of the twentieth century: “the scientific understanding of the atom” (Macrae 12) which he mathematized. In the subsequent electronic revolution, he had a major role in developing computers. Research Von Neumann was engaged in research activities, among which he was greatly interested in hydrodynamical turbulence and “analysis of the underlying non-linear partial differential equations” (Ayoub 170). For insights into this difficult field, numerical analysis appeared to be the only method. Consequently, the requirement to carry out elaborate calculations compelled him to work on new techniques for performing these calculations, which particularly included investigation of the growing field of electronic computers. Neumann’s work contributed extensively to the formulation of techniques and methodologies in use today. He led the construction of an electronic computer in Princeton. With Oscar Morgenstern, the mathematician undertook research in economics, and “wrote a seminal work on mathematical economics” (Ayoub 170). Other important contributions of Neumann include work on the extension of operators in Hilbert space from bounded to unbounded ones, almost periodic functions on groups, algebras of bounded operators in Hilbert space, defining “state” in quantum theory, as an element in Hilbert space, and the introduction of analytic parameters in a locally Euclidean group in the compact case (Bochner 439) Mathematics as a Multiple Phenomenon The writings of John von Neumann reveal that he considered mathematics as a multiple phenomenon, also falling into a great number of widely differing subdivisions. The great variety of fields provides considerable freedom to the mathematician with respect to his work. Further, Neumann states that some of the best inspirations in pure mathematics have come from the natural sciences. Significantly, geometry was the main part of ancient mathematics, and continues to be one of the main parts of modern mathematics. The symbolism of algebra was invented for mathematical use, but had strong empirical associations. On the other hand, modern, abstract algebra has developed in directions that have fewer empirical connections, which is also true for topology, real function theory and real point-set theory. Differential geometry and group theory were conceived as abstract, nonapplied disciplines; however, they became useful in physics, although they continue to be used in their nonapplied form. Von Neumann’s Perspective on Whether Mathematics is Empirical To address this question, it is essential to elaborate on the term “empirical”, which can be interpreted as 1) Whether the formulation of a mathematical concept is meant to solve a real world problem such as the area of an irregularly-shaped piece of land. Or, 2) is the invention one which finds its inspiration indirectly in the “real” world, which is similar to the creation of artists and composers, though the tools differ. John Von Neumann was oriented towards the first view, but was concerned about the logical basis of empiricism. He also expressed anxiety about mathematical pursuits without a firm foundation based on empirical sources. This is due to the fact that if a mathematical discipline veers extensively from reality, “it runs the risk of degenerating into a sterile inquiry leading to contrived and sterile forms of structures” (Ayoub 172). Neumann’s work also reflects the history of mathematical concepts that various mathematical concepts have been expressed in reality long after they were created. The mathematician believed that the essential characteristic about mathematics was its peculiar relationship to the natural sciences “or to any science which interprets experience on a higher than purely descriptive level” (Ayoub 172). In this context, geometry which is one of the main branches of mathematics, originated as a natural empirical science. A major criterion of success of modern empirical sciences is whether they have become accessible to the mathematical method or the closely mathematical methods of physics. Scientific progress in the natural sciences is increasingly associated with successive pseudomorphoses pressing towards mathematics. Conclusion This paper has highlighted John von Neumann’s contributions to the world of mathematics. He left a rich legacy for the growth of mathematics and related disciplines like physics, computer engineering, and mathematical economics, which will continue to benefit generations to come. Neumann has conducted research and written extensively on his mathematical concepts including the relationship of mathematics to the natural sciences, why mathematics is a multiple phenomenon, and whether it is empirical in nature. He has received various honours and awards from all over the world. Works Cited Ayoub, Raymond George. Musings of the masters: An anthology of mathematical reflections. The United States of America: Mathematical Association of America Publications. (2004). Bochner, S. John von Neumann 1903-1957. A biographical memoir. The National Academy of Sciences. (1958): pp.438-459. Retrieved on 25th November, 2010 from: www.nap.edu/html/biomems/jvonneumann.pdf Macrae, Norman. John von Neumann: The scientific genius who pioneered the modern computer, game theory, nuclear deterrence, and much more. Edition 2. The United States of America: American Mathematical Society Publications. (2000). Von Neumann, John. The computer and the brain. Retrieved on 25th November, 2010 from: http://cas.buffalo.edu/classes/dms/rtscholz/THEBRAIN.PDF Read More