Aug 042016
 

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Today is the birthday (1834) of John Venn, English mathematician and logician, primarily remembered for his use of diagrams, which we now call Venn diagrams, to help explain concepts in set theory. Venn was not exactly a giant in his field, but I’d settle for having something reasonably commonplace named after me. Venn diagrams have served me very well in my own work.

Venn was born in Hull, and educated at private schools in London before studying mathematics at Cambridge University, at Gonville and Caius College where he subsequently became a fellow and then head of the college.

Venn’s father was an Anglican clergyman and Venn followed suit in the late 1850s, as was normal for fellows at Cambridge at the time. In fact, after receiving his degree in 1857 he did parish work for a few years before devoting himself full time to mathematics. Even after he left the clergy in the 1880s he continued being involved in the church, although he found strict Anglicanism incompatible with logic and mathematics.

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Venn’s first major publication was The Logic of Chance (1866), a significant accounting of the laws of probability. He then turned to George Boole’s work in logic and produced Symbolic Logic in 1881. It was in this work that he introduced Venn diagrams which he had been using as a teaching device for several years:

I began at once somewhat more steady work on the subjects and books which I should have to lecture on. I now first hit upon the diagrammatical device of representing propositions by inclusive and exclusive circles. Of course the device was not new then, but it was so obviously representative of the way in which any one, who approached the subject from the mathematical side, would attempt to visualise propositions, that it was forced upon me almost at once.

As Venn notes, other mathematicians, notably Gottfried Leibniz and Leonhard Euler, had used similar diagrams earlier, but Venn popularized them as well as extending their application to a wide variety of fields outside of mathematics and logic, and making their application more rigorous than previous attempts.

Venn diagrams don’t actually serve a technical function in mathematics or logic, but they do make certain concepts easier to grasp by displaying them visually. Here’s a simple example showing the Greek, Russian, and Roman alphabets each contained within circles which are drawn to overlap. Symbols in common between two of the alphabets are shown at the intersections of their circles, and symbols common to all three are shown in the central intersection.

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Venn diagrams also have the possibility of excluding items from any of the circles, such as in the example below of a diagram concerning “people I know” and the use of social media. Harry uses neither Facebook nor Twitter so fits inside the rectangle representing people I know but outside the circles representing social media.

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Venn was elected to the Royal Society in 1883 and continued to publish other works, including The Principles of Empirical or Inductive Logic (1889) and volumes on the history of Cambridge and a list of its alumni, compiled with the aid of his son, John Archibald Venn.

Venn died on April 4, 1923, in Cambridge at the age of 88. He is memorialized by a stained glass window at his old college.

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When dealing with mathematical subjects before I’ve focused on mathematical objects. The Venn diagram is not a mathematical object per se, but it does lend itself to cooking ideas. This site gives an idea for a Venn diagram pie. http://www.quirkbooks.com/post/happy-pi-day-make-venn-pie-agram  It was created for Pi Day (3/14 in countries that use month/day format), so it is more about being a pie than being an accurate Venn diagram. But you can take the original and modify it.

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The site shows you how to cut two disposable pie pans to make the Venn diagram shape, and the crust conforms to the general idea of sets – no crust and full crust intersect to make a lattice crust. The recipe fails with the fillings. It just suggests using three different ones. It shouldn’t be too difficult to come up with categories such as fruit and dairy, so that one side is fruit, no dairy, the other is dairy, no fruit, and the middle is fruit and dairy. I’ll leave it to you.

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.