Godel's world

8/21/2007

➴ Gödel world a stray

The collapse of the entire Hilbert's formalization of the mathematical world (though I admire Hilbert, I have more admiration for Godel), Godel's incompleteness throughout the scientific theory of community, from philosophy to mathematics, cognitive science , we can say is "flooding" applied to the biological sciences. Last see a discussion on the possibility of computer breakthrough incompleteness limit moderators answer got me thinking a bit, told him not to the completeness limit is a form of system, if you can create forms in the W3C forum outside the system, maybe. Back to Gödel theory's look, in fact, had a similar discussion, only fur I used to see those people quoted before, how they have to be a middleman to add something different . In accordance with the progress of my leisure time reading, I carefully read the biography of Godel and understand records down, the future reference.

American educator Flexner New Jersey department store heir, 30 million of the early 1930s an attempt to establish a "useless knowledge useful" school. He actually created a pure thinker of the halls of the ivory tower is called the Institute for Advanced Study in Princeton, red brick wall IAS Website: http://www.ias.edu. With his own thoughts, that is, mathematicians or pure theorist good feed, Oh, some people, some books, blackboards, pens and paper became crazy thinkers all. 1932 IAS recruited two scientists well-known in the world, Veblen, and Einstein. Latter is needless to say, I can only say that Nazi Germany, the achievements of science and technology in the United States (U.S. educator, once said: "Hitler shake the tree, where I picking apples in the following." Cheap those Yankees ). The former famous for no less than Albert Einstein, he is the mentor of the Church, in accordance with the data I have just viewed, mathematician genealogy 5545 disciples. (Church, achievements, I will not say. Church or Turing's instructors. Really in every generation there is talent in the ah.) At the same time, von Neumann, Hungary also went to New Jersey, and began manufacturing a computer, which makes the Flexner "pure theory" advocated by the University's commitment to evaporate. Einstein was to be an annual salary with the Flexner 3000, the results of Flexner give his annual salary is $ 16,000. At that time, Einstein's fame great that people at that time called the Institute for Advanced Study in Princeton as "Einstein College".

Godel to Princeton lectures are not very well known (1934), at least with Einstein far worse. In Princeton, Einstein often walk with Godel discussion, they also can be said that the friendship between A, Straus recalled, according to Einstein's assistant, two very different, and has established a profound friendship, on the other side are very much appreciated. Godel's thinking in the ordinary view is weird, it is estimated and Einstein himself on the number. I also recorded a joke. Einstein with Straus in 1953, said: "Do you know, Godel was completely crazy." Straus, answered him: "He can make matters worse what?" Einstein said: "He actually voted for Eisenhower . "

23 years old, Godel proved the incompleteness of the theory was later shocked the whole world, that year is 1930. This achievement is his pillar, and Heisenberg's uncertainty theory, Einstein's theory of relativity together to shake the foundation of pure science as a whole. The completeness of the emergence of the field of mathematics is a nightmare: The math is no longer comprehend the universe and all of the language is no longer the language of God, made for part of the uncertain trend of thought of the 20th century, the scientific community. Godel's world, there is no chance to the emergence of anything; which system is built in does not change based on; truth is fictional. Even writers Godel is the devil of the mathematical community.

➴ Gödel objectivity of the world's two math / science

Genius in the ordinary view both mad. Gödel is stubborn and arrogant in the eyes of others, the relationship with his best scientists Einstein. Princeton once at a dinner, an astrophysicist Bahcall self-introduction to Gödel, told Gödel himself a physicist, Godel faint replied one was later regarded as one of his famous words: "I do not believe in natural science." Indeed, the the Gödel hate random things in the world of logic, he has to put an end to the regular things. Philosopher Nagel recalled that Gödel did not believe in evolution, because he is an extreme dualism. Gödel even with Nagel that Stalin does not believe in evolution, but Stalin was a very clever man, let each other speechless. Chomsky (learn compiled grammar of Noam Chomsky, linguist), I also tried Gödel powerful. Chomsky asked Godel What are the Gödel letter I am afraid that since Leibniz in the 17th century to the time no one dare to say: "I tried to follow the law of the a priori proof of nature.." Love Einstein also said that Gödel do not believe that experience, but he and his friendship with Gödel is indeed the most profound, both of them basically, apart from the other party outside, no longer to talk with other scientists.

The cattle people always like to stay together and cattle. . . Einstein in his later years to the Princeton office, but in order to be able to talk and Gödel. 1955 Death of Einstein, Godel basically do not talk to what people speak. . . He kind of the feeling of academic wandering peak (1945 naturalized U.S. citizenship). . . Einstein with Godel, persistent and stubborn, his later years, ideas began to be conservative, even been accepted quantum physics, because in essence, quantum physics is random. But in fact quantum physics is just a logical extension of his theory of relativity. . . But more precisely, Einstein was the most maverick exclusion of subjectivity: subjectivity to explain his theory of relativity is not physical, but independent of the human, subjective time and space. His theory of relativity is not proposed as a measure of things (this is the Men of ordinary time and space and the theory of relativity, but in fact with Einstein's own understanding is different..) Him with Gödel attached to the same mathematical Platonism against Protagoras' (this school think talent is the measure of all things). Platonism emphasized that the theory is independent of human existence, human studies do not study, they are always there. The irony is that there are people holding them the banner of an attack on objectivism, apparently consequently Meinong Dong. . . Einstein in his later years to fall back on the philosophy of like faith to escape their academic sense of exclusion, which is why he with "Frankenstein" the Godel very common reason. . .

Godel's theory is not his conception of mathematics, but his philosophy. This is a staggering conclusion. But think about it, it was his theory of change in many areas of logic and mathematics. Clearly, the direction of my research, a logical field, the use of model theory in the proof of Godel used. In addition, recursion theory began his proof. Unfortunately, many areas advocated by Hilbert, Gödel theory seemed so insignificant, so slowly abandoned. The paradox is that the reasoning in the reef and Godel's logic, even his own there is the paradox. . (Actually, a lot of logical paradox because of the formation of self-reference, not the letter over the paradox set) However, it is the ingenious application of these paradoxes in his proof of these lead us to look at the problem from another angle: human mathematical mind can bypass the limitations of man-made systems, to capture the essence of truth / knowledge / certainty!

Write so today. . . Later you can see the young logician's unusual life. . . .

➴ Gödel world Vienna circle

Russell, the great genius of the philosopher, mathematician and writer, found the paradox, namely: the sentence is wrong, the same effect on the cornerstone of the theory of mathematics: set theory. Very simple, assume that we build an all does not belong to their own collections of collections, then this collection is not itself an element? Obviously, it is their own element only if and only if it does not belong to itself. . . Previously mentioned, the paradox is that the reasoning in the reef, Russell discovered a reef on the mathematical basis to give the hit. Even more tragic is the founder of of mathematical logic, logic master Fligor by "harm" old masterpiece "arithmetic principle (German Grundgesetze der Arithmetik, the basic rules of arithmetic) has just come out on Russell to his a letter to the knockout, Fligor this arithmetic system is also based on set theory, Russell wrote to tell him the "perfect" system is indeed contradictory, inconsistent. The Fligor the adequate representation of all arithmetic principle, but it is the inconsistency is precisely the form of the worst outcome. Why the inconsistency there is so devastating? Because of an inconsistent system can infer any conclusions, so it might just as well what have not been available. Why do you say? The school are aware of several mathematical logic and certainly before the inference rules (the Modus ponens English, Chinese translation, I do not know how translated, called ponens forget.), P -> q. Such a proposition is true only in the P and q is not set up a false case, In other words, as long as P is false, then all the conclusions q true or false does not matter, because this proposition is always true. . . That is why the contradictions of the system can infer any conclusions.

Contradictions in the system based on Russell's paradox and Fligor, Russell and his teacher Whitehead (the Whitehead should be translated as bald is better, huh, huh) in their co-famous "Principia Mathematica Principia of Mathematica in addition construct a system , all add to the domain of all entities in a sequence, a small sequence of the individual only as a sequence greater than its individual members, so the collection will not be able to do their own members, although it can do the collections of collections members (BTW, really hard to pronounce). Russell's paradox can not be in the system. However, such a system is actually no theoretical support, such as Why do you allow some collection but does not allow some collection. Russell, their sole purpose in order to prevent the Russell paradox, nothing. Of course, the result is that the system is consistent. The paradox of this system and Cantor (cantor learn computational complexity still remember his perfect diagonalization to prove), is also banned. Cantor had proved that a subset of the set (power set) certainly has set itself a greater cardinality than this. The question is, Suppose we construct a collection of complete works (universal set), namely: the collection of all collections. Obviously, which cardinality is bigger than the complete works, unless the world is crazy. However, according to the Cantor proof, has a larger base than the Complete Works Complete Works of a subset of. . . Again, this paradox can also be excluded from the outside of set theory. It can be seen, when the mathematician who discovered the paradox helpless, they have only to add some inexplicable rules limit the paradox. However, such a system always makes me feel a sense of crisis, and the internal crisis. . . Similarly, such a sense of crisis has also become a research focus of the philosophers: actually there Luo Suxun MG (his hereditary viscount title) can not be resolved, it really is worth a try ah ~ this group of people there later become famous philosophical masters Wittgenstein (because he was nothing but an aeronautical engineering undergraduates, Russell went to Cambridge to study philosophy). However, Wittgenstein told Russell to later chilled on Russell that this arrogant philosopher: Lucifer proud. What is Lucifer? Lucifer is the devil. Visible the Wittgenstein personality is not so good, of course, estimated philosophers generally are so jumpy.

Having said that, with Gödel, what does it matter? And Gödel, these people are a master of the relationship with the Vienna Circle. Austria was gathered a lot of master-level characters, they discuss problems in the cafe the edge of the University of Kreise another form is set up according to different research interests, the German meaning of the circle (Circle). Circle which one of the most famous Vienna Circle (Der-Wiener-Kreise), the philosophical circle created by the philosopher Moritz, Schlick (Adams gram), also known as Sri Lanka gram ring, by invitation only people can participate in a regular meeting discussion. Gödel in the years 1926-1928 are invited to participate as an undergraduate. This circle of mathematicians Hans, Hans Hahn, Sri Lanka gram pulled from the German character of Vienna, his wife, Olga Neurath is also a member of the Vienna Circle. Hans in Vienna circle and not a small contribution, he not only introduced to the members Fligor theory, back to them to explain Russell's masterpiece: Principia Mathematica 3 volume
Hans also has Godel number theory to the mathematical logic in this direction when the thesis advisor. Hans introduce undergraduate Gödel and his other students, Karl in Menger to the Vienna circle, while two outstanding students in Sri Lanka gram Herbert Feigl, Friedrich Waismann, into this organization. Those people are a master of brilliant achievements, of course, is inseparable from the impact on them to join the Vienna Circle.

It is worth noting that the Vienna Circle was prevalent in the empirical the logicism view (positivism). Declaration published by the Vienna Circle in 1929 proposed that "man is the measure of all things, anything to match the Their point of view, mathematics with a similar logic, without any real content, mathematics, but is the syntax (Syntactic). Light in front, and Mathematics of the Platonists, that is, Godel belongs to the school, the view is the opposite. However, these declarations did not the stimulate Plato send Godel him because it is silent and will not stir up any trouble. Next, back to Wittgenstein. This arrogant German went to Cambridge University to get the favor of Russell's, rely on your strength to become a master of Cambridge's most revered philosophy. Unfortunately, Wittgenstein often lamented his Cambridge colleagues and students can not always understand his thinking, enough arrogance. Speaking of Wittgenstein, first talk to a genius Weining Ge: Otto Weininger. The psychological bit bt Vienna philosophical genius biggest problem is to look down on women, and advocated a genius, the Weining Ge's theory can be attributed to the dark and cruel choice: man [woman did something, sit and wait to die] is either a genius , or choose to die (genius or death). Wittgenstein young when read the Wei Ningge writings, has become neurotic, for example, Wittgenstein wanted to learn his idol Weining Ge as to commit suicide, the result has been to attempt suicide nine years, of course, not as good as his idol, powerful, The Weining Ge shot successful suicide. Wittgenstein famous, even in part, in World War I trenches completed the Tractatus, Tractatus Logico - Philosophicus also not underestimate the impact of the Vienna Circle. Wittgenstein returned to Vienna, the Vienna circle of those veterans, including Sri Lanka gram, talk to Wittgenstein talks like a pilgrimage, their thinking is consistent with Wittgenstein at any time, completely and thoroughly cult of personality. The Vienna Circle Wittgenstein believers and even imitate the gestures of Wittgenstein. From here we can see the harm of the cult of personality, and even mind the best thinking of the philosopher to the crazy personality cult, when IQ is reduced to the extent of the idiot. Wittgenstein to become emperor of the Vienna Circle philosophers do not fit with his thought has been kicked out. In the deity of worship Wittgenstein Vienna circle, students Godel still quiet, sit quietly side, or absorb or eliminate these thinking, drawn from his own epoch-making conclusions.

A rational ➴ the world, incomplete world

Intermittent, until recently, after reading this little book. Then Godel did not want to write down, but after reading the mood too complicated, as if there are numerous ideas, but no one can truly and accurately express the

A following the world's greatest philosopher since Aristotle, died on January 14, 1978. Came to America from Europe in the help of friends from the fireworks filled with humanity crazy removed, began to become a U.S. citizen with regret to him that he did not achieve his target was buried in the tomb of Princeton, and his life has not changed much. A genius brain has known the mystery of the universe, but attached to a weak body than most ordinary people are. Rationality, knowledge and truth, thinking they did not wake up the contemporary so-called scientists, but was intentionally or unintentionally marginalized, to deepen his loneliness and sense of isolation, directly or indirectly led to the originally should prolific genius, philosopher, mathematician, and even the genius physicist no chance, there is no incentive to seek the secret of life, truth, and the universe. Those years in Princeton, in addition to harassment against Nazi Germany, in my opinion, did not get any other benefits.

Godel's life, according to the description of the famous Chinese logician Hao, WANG, in his opinion is a complete life. Godel himself to break the mystery of mathematics at the same time expect to find a mathematical real (reality) authentic cornerstone of mathematics. But he did not find, or that he is not able to find many reasons. He expect to find in the philosophical world the nature and principles of this material world. He often envious of Plato, Descartes, Husserl's philosophy master in life there are a few moments suddenly opens to see the material world world was much different in a mortal eye, or be said to be They have found a new look at the world. However, Godel did not. Until his death, he did not wait until his eureka moment of opportunity. Godel to realize their ideals in the world of mathematics and philosophy ended in failure, and in the world of physics, he eventually or not to wait for the result he wanted.

Princeton physicist Godel their discussion, the genius mathematician and philosopher actually have so many physical theory. In fact, Godel has been, almost from his university began, the physical is very preferences. Estimates plus those years in Princeton, with his close friends, Einstein often talk about physics theory, rigorous reasoning, philosophers did not give up and interrupt the love of physics. 70 birthday of Albert Einstein, Godel, Einstein wrote an article to celebrate his 70th birthday. In the article, Godel new theoretical model of a structure to satisfy the field equations of Einstein's general theory of relativity. Call this model the theory, because in such a model, time is cyclical. The time is no end to it like a space, you get on a rocket starting from now on you over the past, the future back to the present. The logician's thesis is impossible to go wrong, otherwise he will never out. Einstein also a bit of a shock, apparently so many years of conversation, Godel did not talk about such a time model to explain Einstein's general theory of relativity. Einstein showed that the model formally, but in the physical reality can be excluded. In other words, Einstein apparently did not consider there is evidence to support this model. Godel, has always been a real theorists, but for this physical model, began collecting test data. He even often asked whether the Princeton physicist found evidence of galaxy cycle. However, the same leave until Godel Paul afflicted the world, even until now, did not allow him to realize their evidence of the physics goals. Godel, with his incomplete life, so leave this world. Perhaps, in the model of his time, he just travel only at some future time, he will again experience the world.