Mathematics, Godel, numbering, theorem, incompleteness, arithmetic
History of Science, Technology, and Medicine | Mathematics | Number Theory | Physical Sciences and Mathematics
By the beginning of the twentieth century there was a great change occurring in the way mathematicians thought about mathematics and mathematical truth. Before this time, proving something in mathematics was thought to be proving something about the world. For example, the Geometry used by the ancient Greeks and formalized in Euclid's Elements, was though to be an actual model of how things really are rather than just a logical conclusion from a set of axioms. This is because the axioms of Geometry were though to be intuitively obvious, and so any conclusions from them would have to be true.
This particular view was shattered when, in the 19th century, several mathematics, including, Gauss, Lobatchevsky, Bolyai, and Reimann, simultaneously found other logically valid possible geometries. Each of the new geometries were shown to be as possible as the original geometry. That is, each was shown to be consistent if one assumes that the original Euclidean Geometry is in fact consistent. This discovery would mark the end of the old way of doing things. No longer could one assume that Euclidean Geometry is consistent because it was the geometry of the universe since there were other possible candidates for this distinction.
Department 1 Awarding Honors Status
Parker, C. C. (1997). Godel's Incompleteness Theorem (Undergraduate honors thesis, University of Redlands). Retrieved from https://inspire.redlands.edu/cas_honors/703