Apr 282016
 

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Today is the birthday (1906) of Kurt Gödel, Austrian mathematician, logician and philosopher. Gödel is one of my great intellectual heroes, even though he is hardly a household name. His incompleteness theorems, for which he is famous, are the bedrock of my general thinking about the nature of human thought and belief. I’ll try to discuss his work in plain language even though in so doing I will inevitably oversimplify it.

Gödel  was born in Brünn when it was part of Austria-Hungary (now Brno, Czech Republic) into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh). At the time of his birth the city had a German-speaking majority] which included his parents. His father was Catholic and his mother was Protestant and the children were raised Protestant.

Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I. According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, “Gödel always considered himself Austrian and an exile in Czechoslovakia.” He chose to become an Austrian citizen at age 23. When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became a U.S. citizen.

In his family, young Gödel was known as Herr Warum (“Mr. Why”) because of his insatiable curiosity. He attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages, and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. In his teens, Gödel studied Gabelsberger shorthand, Goethe’s Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics and initially intended to study theoretical physics. He also attended lectures on mathematics and philosophy, and participated in the Vienna Circle of philosophers with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell’s Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was “a science prior to all others, which contains the ideas and principles underlying all sciences.”

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Gödel chose to study completeness in logic for his doctoral work. Put simply, completeness is a quality of a logical system – such as arithmetic or geometry – whereby every statement within the system can be proven to be true without resort to statements outside the system. Mathematicians had been trying for millennia, without success, for example, to prove that Euclid’s 5 axioms, or postulates – statements that classical geometry rests on – could be shown to be true using reasoning within geometry. If you accept these axioms, there are countless theorems you can derive from them, such as the Pythagorean theorem, but mathematicians were not happy that the axioms themselves could not be proven to be true. They seem to be true – self evidently – but no one could prove them to be true.

In 1929, at the age of 23, Gödel completed his doctoral dissertation under Hans Hahn’s supervision. In it, he established the completeness of the first-order predicate calculus (Gödel’s completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science. So far so good. BUT . . . in 1931, while still in Vienna, Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der “Principia Mathematica” und verwandter Systeme (called in English “On Formally Undecidable Propositions of “Principia Mathematica” and Related Systems”). In that article, he proved that arithmetic is incomplete. This was an astounding and revolutionary idea. Again in simple terms, Gödel proved that although arithmetic is a powerful system capable of great things, it rests on premises that cannot be proven. This led to his basic notion of the incompleteness of mathematics summed up in two statements:

If a system is consistent, it cannot be complete.

The consistency of the axioms cannot be proven within the system.

This notion is usually expressed simply as that “in any logical system there will always be at least one statement which is true but cannot be proven to be true.”

I just love it. All the smarty pants in the world who hammer religion because it is based on faith whereas science is based on “proven fact” don’t know what they are talking about. Yes, religion is based on faith, SO IS SCIENCE !!! Logical proof has limits and there’s no way round this.

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In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend, and  delivered an address to the annual meeting of the American Mathematical Society. In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled “On undecidable propositions of formal mathematical systems.” Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.

Gödel married Adele Nimbursky (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he was. Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

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After the Anschluss in 1938, Austria became a part of Nazi Germany. Germany abolished his university post, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down. His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the trans-Siberian railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel accepted a position at the Institute for Advanced Study (IAS).

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Albert Einstein was also living at Princeton during this time, and Gödel and Einstein developed a strong friendship. They took long walks together to and from the Institute for Advanced Study, deep in conversation. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his “own work no longer meant much, but he came to the Institute merely to have the privilege of walking home with Gödel.”

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution that could allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend’s unpredictable behavior might jeopardize his application. Fortunately, the judge turned out to be Phillip Forman, who knew Einstein and had administered the oath at Einstein’s own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.

Gödel became a permanent member of the Institute for Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976. During his many years at the Institute, Gödel’s interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed time-like curves, to Albert Einstein’s field equations in general relativity. He is said to have given this elaboration to Einstein as a present for his 70th birthday. His “rotating universes” would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation).

In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz’ version of Anselm of Canterbury’s ontological proof of God’s existence. This is now known as Gödel’s ontological proof.

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I won’t go over it with you. It has the same problems that Anselm’s and Leibniz’ proofs do, namely, the axioms are unprovable. Gödel hoist on his own petard !!!

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Gödel was a convinced theist, in the Christian tradition. He held the notion that God was personal, which differed from the religious views of his friend Albert Einstein. He believed firmly in an afterlife, stating: “Of course this supposes that there are many relationships which today’s science and received wisdom haven’t any inkling of. But I am convinced of this [the afterlife], independently of any theology.” It is “possible today to perceive, by pure reasoning” that it “is entirely consistent with known facts.” “If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife].”

In an (unmailed) answer to a questionnaire, Gödel described his religion as “baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza.” In describing religion in general, Gödel said: “Religions are, for the most part, bad—but religion is not”. According to his wife Adele, “Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning”, while of Islam, he said, “I like Islam: it is a consistent [or consequential] idea of religion and open-minded”. So much, again, for the smarty pants who think that if you are intelligent you can’t believe in God.

I am fully in accord with Gödel in this regard. It is certainly true that a great many religious statements are ridiculous, and a great many religious beliefs are idiotic, but that does not mean that basic religious beliefs are false. And we have Gödel to thank for showing that science and mathematics are every bit based on faith as religion is.

Later in his life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband’s food. In her absence, he refused to eat, eventually starving to death. He weighed 65 pounds (approximately 30 kg) when he died. I will admit that there is a fair degree of irrationality in starving yourself to death because you are afraid of being poisoned. His death certificate reported that he died of “malnutrition and inanition caused by personality disturbance” in Princeton Hospital on January 14, 1978. He was buried in Princeton Cemetery. Adele’s death followed in 1981.

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I don’t know what Adele made for him, but maybe she cooked sauerkraut soup once in a while. Sauerkraut is one of several dishes that Czechs and Germans agree upon as delectable, and you can find versions of sauerkraut soup in Germany, Austria, and the Czech Republic with the ingredients, other than sauerkraut varying somewhat. In this recipe you can use a German or Polish sausage depending on your tastes. You may also add a little pickle juice from the sauerkraut if you like. Be sure to have plenty of crusty bread on hand to dip into the broth.

Sauerkraut Soup

Ingredients:

7 oz sauerkraut
6 cups light broth
2 potatoes, diced
2 tbs flour
1 onion, diced
4 slices bacon, finely diced
1 large sausage, sliced thinly
5-10 white mushrooms, diced
1 tsp caraway seeds
1-2 tsp sweet paprika

Instructions

Sauté the finely diced bacon over medium-low heat in a large heavy pot. There is no need to add oil. When the fat is flowing, add the sliced Polish sausage, mushrooms and onion, and sauté for about 5 min.     Sprinkle with flour and stir everything around to mix. Add the drained sauerkraut, diced potatoes, caraway seeds and paprika and stir again.  Add the broth and salt to taste and cook until the potatoes are tender.

Serve in deep bowls with crusty bread. If you like you can pass around sour cream for guests to add.

Mar 032016
 

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Today is the birthday (1845) of Georg Ferdinand Ludwig Philipp Cantor a German mathematician of immense importance. He created set theory, which has become a fundamental theory in mathematics. Cantor’s work is of great philosophical interest, as well as being purely mathematical, a fact I want to dwell on after dribbling on a bit about his life.

Cantor’s theory of transfinite numbers (numbers larger than all finite numbers, yet not absolutely infinite) was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from his mathematical contemporaries and later from philosophers. Cantor, a devout Lutheran, believed the theory had been communicated to him by God. Some Christian theologians saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected.

The objections to Cantor’s work were occasionally fierce: Henri Poincaré referred to his ideas as a “grave disease infecting the discipline of mathematics,” and Leopold Kronecker’s public opposition and personal attacks included describing Cantor as a “scientific charlatan”, a “renegade” and a “corrupter of youth” (shades of Socrates !!). Writing decades after Cantor’s death, Wittgenstein lamented that mathematics is “ridden through and through with the pernicious idioms of set theory,” which he dismissed as “utter nonsense,” “laughable” and “wrong”. Cantor’s recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder.

Those whose eyes glaze over at the mere mention of mathematics might be amazed that mathematical propositions could engender such base emotions. But Cantor’s work had, and has, implications for some of the most basic, but enduring, human questions such as “what is real?” and “what is God?” I promise I won’t delve too deeply into mathematics, I’ll just use a few analogies, with apologies to those who know a bit more than the basics about number theory and set theory. I realize they are over-simplifications, as well as being vaguer than the underlying mathematics.

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Georg Cantor was born in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) was a well-known musician and soloist in a Russian imperial orchestra. Cantor’s father had been a member of the Saint Petersburg stock exchange but when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those in Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt. His exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the University of Zürich. After receiving a substantial inheritance upon his father’s death in 1863, Cantor shifted his studies to the University of Berlin, and then spent the summer of 1866 at the University of Göttingen, a major center for mathematical research.

Cantor submitted his dissertation on number theory to the University of Berlin in 1867. After teaching briefly in a Berlin girls’ school, Cantor took up a position at the University of Halle, where he spent his entire career. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible.

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Cantor suffered his first known bout of depression in 1884, which some scholars attribute to the constant criticism of his work by famous scholars that weighed heavily upon him. This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature thinking there might be evidence that Francis Bacon wrote the plays attributed to Shakespeare (in my opinion a fruitless quest, but endlessly simmering at the fringes of Shakespeare scholarship). It never ceases to annoy me that such investigation is based on the premise that a poor grammar school boy can’t be a genius.

After Cantor’s 1884 hospitalization his youngest son Rudolph died suddenly (while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König’s proof had failed, Cantor remained shaken, and momentarily questioning the existence of God. Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (I’ll get to it in a minute !!) to a meeting of the Deutsche Mathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

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Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Differential calculus was not developed until the 17th century, even though the basic building blocks had been available to ancient Arab and Greek mathematicians. The problem was that calculus requires use of infinity and ancient mathematicians could not accept the existence of infinity. It turns out that you don’t have to accept the existence of infinity to use the concept mathematically. The square root of -1 cannot logically exist, because there does not exist a pair of identical numbers that when multiplied together produce -1. But if you just give it a name (the letter “i”) and use it in equations without worrying about whether it exists or not, the equations often work out when you cancel out i. For example, x – i = y – i simplifies to x = y, so you don’t need to worry about whether i exists or not. Infinity can work in much the same way mathematically. But dealing with infinity is both tricky and counterintuitive.

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If you’re smart but not especially well versed in mathematics, I highly recommend George Gamow’s book, One, Two, Three . . . Infinity. It explores Cantor’s mathematics in simple terms. Gamow’s birthday is tomorrow and since I can’t very well celebrate a mathematician followed next day by a theoretical physicist (working in complex mathematics), I’ll tip my hat to Gamow today. He was an early advocate and developer of the Big Bang Theory. In the book he uses a thought experiment to help explain the weirdness of infinity. Imagine you have a hotel with finite rooms, all of which are full, and a new guest arrives. You have to send him away because there is no room. Now imagine you have a hotel with infinite rooms and a new guest arrives. “No problem,” you say. You move the person in room 1 to room 2, in room 2 to room 3, in room 3 to room 4 . . . and so on. The series of integers (whole numbers) is infinite, so you never run out at the upper end. Now room 1 is free for the new guest. Now suppose the infinity hotel is full and an infinite number of guests shows up. “No problem,” you say again. This time you put every guest in the room that is double the number of the room they are in now. The guest in room 1 goes to room 2, in room 2 goes to room 4, room 3 goes to room 6 . . . and so on. In this way you free up all the odd-numbered rooms (double ANY number is an even number). There is an infinite number of odd numbers, so you can accommodate an infinite number of new guests. Maybe now you are beginning to grasp the problem of the existence of infinity. This thought experiment is counter-intuitive, and, hence, why so many philosophers and mathematicians objected.

So . . . are mathematical objects (things) real or not? Mathematicians, philosophers, and theologians have been arguing about this question for millennia with no end in sight. Cantor believed that absolute infinity was God. I find this equation an admirable idea, but it does not answer the question as to whether absolute infinity exists any more than whether God exists. Even if infinity defines God it does not prove his existence. I can define unicorns, but they don’t exist.

Cantor’s set theory also can lead to paradoxes, and mathematicians don’t like paradoxes. The barber paradox is an informal version. Imagine a town in which all the men need to shave, but are clean shaven. Men either shave themselves OR are shaved by the barber. Who shaves the barber? The two sets “shaved by the barber” and “shave themselves” cannot logically be distinct sets.

 

If nothing else, I hope I have shown that mathematics prompts questions and puzzles that are more than of scientific or logical interest. They strike at the very heart of issues such as, “What is the meaning of life?” and “Is there a God?” or “What is existence?” Cantor died poor and believing he was a failure. How many giants have died likewise? We owe it to Cantor today to keep him in memory.

I’m going to give you a great Saint Petersburg recipe in Cantor’s honor, because he was born there, even though he spent most of his life in Germany. Germans will like this too. It is a version of stuffed cabbage, but not the kind that you are used to. Instead of peeling off the leaves and stuffing them individually, you stuff the whole cabbage intact. Herbs can be of your choosing.

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St Petersburg Stuffed Cabbage.

Ingredients:

1 small cabbage
1 carrot, peeled and sliced
3 onions, peeled and sliced
1 bouquet garni
1-2 bay leaves
2-3 cloves
5-8 peppercorns
salt

Filling:

14 oz veal
7 oz pork lard
2-3 slices stale white bread
1 cup milk
2 eggs, beaten
salt and pepper

Instructions:

To make the filling, grind the veal with the lard. Soak the bread in milk for several hours, then wring out the excess. Mix the bread thoroughly with the meat and lard, add the eggs and mix the whole filling uniformly. Season with salt and pepper to taste.

Remove the toughest outer leaves of the cabbage. Immerse fully in a large pot of water and simmer until the leaves are soft and pliant. Drain the cabbage and put the filling between the leaves without tearing them off. Tie the cabbage with strong (colorless) twine and simmer in water or stock. Add the carrot, onions and seasonings. Cook on low heat for around 30 minutes. Serve the cabbage whole with the vegetables to garnish, and with sour cream.