John Von Neumann is a Hungarian mathematician that grew up in a banking family in Budapest, Hungary. Neumann made significant contributions in various fields of mathematical logic and physics. His interest is diverse and wide-ranging and he set to do valuable contributions in other fields like economics, strategic thinking and computer technology (National Academy of Sciences, 2005). His contribution to the field of mathematics is known as his most fundamental contribution in the world of knowledge. His earliest significant contributions in mathematics can be summed up under two headings, specifically the axiomatic set theory and Hilbert’s proof theory (Macrae, 1999).
Von Neumann’s axiomatic system of set theory satisfied two major conditions which are; (a) it allows the development of the theory of the whole series of cardinal numbers and (b) its axioms are finite in number and expressible in the lower calculus of functions. Furthermore, in deriving the theorems on sets from his axioms, he developed the first satisfactory formulation and derivation of definition by transfinite induction (Wigner, Mehra & Wightman, 2001). Another contribution he made in mathematics is the Hilbert’s Proof Theory under the Hilbert program of David Hilbert. His discovery is consisted of the theories of the Hilbert space and of operators in that space. The developments he made in the axiomatic set theory is the forerunner of the work Kurt Gödel, who further developed the set theory. The majority of contributions of Neumann in different fields paved way for today’s principles and theory.
Kurt Gödel is the known mathematician and logician that is a native of Vienna, Austria. He is best known for his contribution in the development of the set theory for his proof of “Gödel’s Incompleteness Theorems” (O’Connor & Robertson, 2003). Gödel revealed certain unforeseen limitations in the axiomatic procedure in the set theory and disproved certain beliefs like all the important areas in mathematics can be completely axiomatized. His discovery proved that in a formal system, there are questions that exists that are neither provable nor disprovable on the basis of the axioms that define the system (Eves, 1997).
Through his theorems, Gödel showed that there are problems that cannot be solved by any set of rules and procedures instead, the set of axioms must always be extended. The historical context of the theorem of Kurt Gödel was its discovery under the Hilbert program of David Gilbert which attempt to condense all mathematical truths in a formal system with finite description. The formalist program hoped to be able to replace truths with a finite notion of provability. But in 1931, Gödel show impossibility to such pursuit as there are questions that cannot be proved or disproved as explained in his theorem (Svokil, 1993). This is the major impact of Kurt Gödel’s work in the field of mathematics, that everything cannot be proved or disproved. It opened new doors for the further developments in various field of mathematics.
Alexander Grothendieck is a German-born, French mathematician who also made valuable contribution in mathematics. He is one of the pioneers of the topos theory and the new algebraic geometry and number theory. In 1966, he was awarded the Fields Medal for his fundamental contributions to mathematics (“Alexander Grothedieck”, n.d.). Grothedieck is considered as one of the most influential mathematicians of the twentieth century. The Fields Medal is intended for his contributions in Homological Algebra and Algebraic Geometry. However, he also made significant contribution in Functional Analysis (Velasco & Rodriguez-Palacios, 2007).
Among the manifold of Grothendieck’s contributions to algebraic geometry is his emphasis on the search for a universal cohomology theory for algebraic varieties and a conjucntured description of it in terms of motives. Different authors describe this particular property of cohomology theory for algebraic varieties as motivic cohomology (Cartier, Grothendieck & Katz, 2006). Grothendieck work on theories that are relevant not only in the specific field of mathematical logic and category theory but also to algebraic geometry, number theory, bioinformatics, institutional ontology classification and computer software/programming. He significantly expanded the knowledge on the algebraic geometry through his major contributions.
Alexander Grothendieck. (n.d.). PlanetMath.org. Retrieved May 28, 2009, from http://planetmath.org/encyclopedia/AlexanderGrothendieck.html.
Cartier, P., Grothendieck, A. & Katz, N.M. (2006). The Grothendieck Festschrift Volume 3: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck. New York, New York: Birkhauser Boston.
Eves, H. (1997). Foundations and Fundamental Concepts of Mathematics. Mineola, New York: Dover Publications, Inc.
Macrae, N. (1999). John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. American Mathematical Society Bookstore.
National Academy of Sciences. (2005). John Von Neumann. Retrieved May 28, 2009, from http://www.nationalacademies.org/history/members/neumann.html.
O’Connor, J.J. & Robertson, E.F. (2003, October). Kurt Gödel. Retrieved May 28, 2009, from http://www.gap-system.org/~history/Biographies/Godel.html.
Svokil, K. (1993). Randomness & Undecidability in Physics. Singapore: World Scientific Publishing Co. Pte. Ltd.
Velasco, M.V. & Rodriguez-Palacios, A. (2007). Advanced Courses of Mathematical Analysis 2: Proceedings of the 2nd International School Granada, Spain 20 – 24 September 2004. Singapore: World Scientific Publishing Co. Pte. Ltd.
Wigner, E.P., Mehra, J. & Wightman, A.S. (2001). Historical and Biographical Reflections and Syntheses. Berlin, Germany: Springer-Verlag Berlin Heidlberg.