Kurt_Godel

Perhaps one of the greatest contributors to mathematics in the 20th century, Kurt Gödel made famous by the incompleteness theorem, which stated that within any axiomatic mathematical system there are things that cannot be proved or disproved on the basis of the axioms within that system, therefore was not able to be complete or consistent. His proof made him on even grounds with one of the greatest logicians, Aristotle. Kurt Gödel was born on April, 28, 1906, in Austria-Hungary (Czech Republic now). Where, he was living in a newly formed country of Czechoslovakia because of World War I in 1918. Later, he was accepted into the University of Vienna, where he earned his doctorate in mathematics in 1929 and joined its staff the following year. During the early 1900s, Vienna was considered intellectually dominant in the world and earned the title of Vienna Circle, a gathering of scientists, mathematicians, and philosophers who believed in logical positivism, or a naturalistic, empiricist, and antimetaphysical view. Gödel was introduced to the group by his mentor, Hans Hahn. He held views of Platonism, theism, and mind-body dualism, which the Vienna Circle felt a bit contempt towards. Moreover, Gödel was subjected to paranoia, a mental instability, which he gained as a child and grew as he aged. “Über die Vollständigkeit des Logikkalküls” (On the completeness of the Calculus of Logic), or the completeness theorem was published abridged in 1930, Gödel proved one of the most important local results of his time. It established that classical first-order logic, or predicate calculus is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems. “It stated that no consistent system of axioms whose theorems can be listed by an “effective procedure” is capable of proving all facts about natural numbers,” as described by “The Exploratorium.” Moreover, a year later he published the incompleteness theorem, (also the second part of the incompleteness theorems) which was considered more famous as the completeness theorem’s counterpart. It established the result that it is impossible to use the axiomatic method to construct a mathematical theory, which entails all of the truths in that branch of mathematics. “The second incompleteness theorem shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself.(The Exploratorium)” In England, a similar proof was drawn up two decades before by Alfred North Whitehead and Bertrand Russell as the Principia Mathematica in three volumes. For example, it was impossible to derive with an axiomatic mathematical theory that contains everything about the natural numbers. This was important in that before 1931, many mathematicians tried to accomplish that problem by constructing axiom systems that could prove all mathematical truths. The following mathematicians, Whitehead, Russell, Gottlob Frege, and David Hibert spent a large portion of their mathematical careers precisely trying to solve this problem; however Gödel’s theorem destroyed their entire work on axiomatic research. After the publication of the incompleteness theorem, Gödel became a famous intellectual figure and traveled around the world, especially to the United States, where he lectured at Princeton University in New Jersey. This was an important point in his life for he met Albert Einstein, which began a close friendship between two famous individuals of the time. However, Gödel’s mental illness began to deteriorate as he aged. He suffered from depression and a nervous breakdown because of the homicide of Moritz Schlick, one of the leaders of the Vienna Circle and an important friend to Gödel. Nazi Germany annexed Austria on March 12, 1938. Gödel found himself to be in a bind because he was associated with Jewish individuals in the Vienna Circle and was subjected to youth attacks on the streets of Vienna. Moreover, there was danger of being conscripted into the German army. On September 20, 1938, Gödel married Adele Nimbursky. When World War two broke out a year later, he and his wife fled to the rest of Europe and taking the trans-Siberian railroad, fled across Asia, and sailed across the Pacific Ocean, and then taking another train across the United States to Princeton, New Jersey. There, with Einstein’s aid he formed the Institute for Advanced Studies, where he spent the rest of his life working and teaching. He became an official United States resident/citizen in 1948, where Einstein attended his hearing because of Gödel’s instability and unpredictable behavior. In 1940 a few months after his arrival at Princeton, Gödel published another mathematical work called “Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory,” which proved that the continuum hypothesis and the axiom of choice are consistent with the standard axioms of set theory. Gödel’s conjecture was that it could not be proven false under those theories. Gödel’s other work also made contributions to physics in 1949, where his work showed that Einstein’s theory of general relativity allows for the possibility of time travel. In Gödel’s later years, Gödel wrote about philosophical issues, which was his interest throughout his life. It was said that there’s a little-known fact that he proved the incompleteness theorem in the first place for the sake of establishing a philosophical view on mathematical Platonism. This is the view that mathematical sentences, for example “4 + 4 = 8” is a description of a collection of objects (numbers),that are nonphysical, nonmetal, and existed outside of space and time in a mathematical realm, also dubbed as “Platonic Heaven.” Thus, Gödel thought that if he proved the incompleteness theorem, then he would’ve been able to show that there were improvable mathematical truths. However, this went beyond the regular Platonism because it showed that mathematical truth was the true objective of Platonism. In 1964, Gödel published, “What is Cantor’s Continuum Problem,” a philosophical paper that proposed an answer to Platonism. Platonism is often argued that its ideas are false because it makes mathematical knowledge impossible and beyond the human sensory perception. Moreover, Platonism went beyond the human perception of the external world into these “mathematical realms.” However, Gödel responded to this by claiming that humans possess a faculty of mathematical intuition in addition to the original five senses. This allowed people to grasp the numbers or “to see them in the mind’s eye.” He claimed that this mathematical intuition made it possible to acquire knowledge of mathematical objects that existed outside of space and time. However, Gödel’s philosophical views have not been accepted by the public audience. Ever since Gödel was a child, he was subjected to constant periods of illness, which included a rheumatic fever and eventually paranoia and depression. This ultimately led to his downfall, as he became crazed and thought that his paranoia was due to sabotage by poisoning. After the hospitalization of his wife, he was starved to death because he refused to eat any food that wasn’t tested by his wife. Overall, Gödel was a genius as a youth. He went on from an intern for the Vienna Circle to becoming a dominant member of the group. He brought the world a storm, when he proved the impossible with the incompleteness theorems in 1931. This brought a few groups to their demise because of his idea and brought Gödel world recognition. This led to his travel to the United States, where he met Einstein, which became an important time in his life because it created a long lasting friendship between two great minds. Einstein influenced Gödel’s view and brought him further into philosophy. When Nazi Germany annexed Austria, Gödel travelled to the United States to avoid being conscripted into the army. Here Einstein helped Gödel into citizenry and the establishment of the Institute for Advanced Studies. Overall, Einstein had a large influence on Gödel. Gödel’s purpose was always Platonism because he always wanted to prove of its validity. For example, the incompleteness theorems were to prove that something impossible is real and therefore the mathematical realm was also real. Later on, he proved this point when he wrote about philosophical papers and ideas that argued that mathematical intuition was a sixth sense that allows perception of mathematical reals. He earned many prestigious awards for his incompleteness theorems. In turn, it brought him to his travels to the United States, where it developed his later career in the United States and also created a refuge. Gödel was a prominent figure in that his attempts at proofs were to prove the unattainable and the nonphysical. His mathematical views has contributed to society by showing that an open mind is important. Gödel was a genius that contributed to the future.