8-2016 Topologists should not go swimming

Pavel Urysohn (1898-1924) was a Russian mathematician of Jewish origin who is best known for his contributions in dimension theory, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

Urysohn drowned while swimming in the sea, on vacation in France off the coast of Brittany, France, near Batz-sur-Mer, and is buried there. He was accompanied by his colleague and friend, Pavel Alexandrov (another famous mathematician). It is believed that Alexandrov was incredibly distressed by this tragedy, and deeply regretted his inability to save his friend.

MORAL OF THE STORY : Topologists should not go swimming.

7-2016 Algebraists are also human

Yutaka Taniyama (12 November 1927 – 17 November 1958) was a Japanese mathematician known for the Taniyama–Shimura-Weil conjecture. This conjecture inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem (FLT). The correct proof of FLT was published in May 1995. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians the Taniyama–Shimura-Weil conjecture was finally proven in 1999.

On 17 November 1958, Taniyama committed suicide. His enigmatic suicide note, which shows symptoms of stress and mental depression, mentions tiredness and a loss of confidence in his future. About a month later, Misako Suzuki, the woman whom he was planning to marry, also committed suicide, leaving a note reading: "We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him."

MORAL OF THE STORY : Algebraists are also human.

6-2016 Taxis are no good for game theorists

John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory, differential geometry, and the study of partial differential equations. Nash's work has provided insight into the factors that govern chance and decision making inside complex systems found in daily life

Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he was awarded the Abel Prize for his work on nonlinear partial differential equations. One would have expected him to be a leading contender, perhaps even a virtual certainty, for a 1962 Fields' Medal, but mental illness destroyed his career, long before those decisions were made.

Nash’s research into game theory and his long struggle with paranoid schizophrenia became well known to the general public because of the Academy Award-winning motion picture A Beautiful Mind (2001), which was based on Sylvia Nasar’s 1998 biography of the same name. The film opened in the United States cinemas on December 21, 2001. It went on to gross over $313 million worldwide and win four Academy Awards, for Best Picture, Best Director, Best Adapted Screenplay and Best Supporting Actress. It was also nominated for Best Actor, Best Film Editing, Best Makeup, and Best Original Score.

On May 23, 2015, John and his wife Alicia Nash were killed in a car crash while riding in a taxi on the New Jersey Turnpike near Monroe Township, New Jersey, USA. They were on their way home after a visit to Norway, where Nash had received the Abel Prize.

MORAL OF THE STORY: Game theorists should not hire taxis.

5-2016 Do not depend on your wife for food

Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, A. N. Whitehead, and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna.

He sadly succumbed to crippling paranoia later in life. In his sixties, he became convinced that his food was being poisoned, and would only trust the cooking of his wife Adele. When she was hospitalized for six months in 1977, Gödel refused to eat, and subsequently died of starvation.

MORAL OF THE STORY : If you are a logician, do not depend on your wife for food

4-2016 Might is not always right

Archimedes of Syracuse (287 BC –  212 BC) was an Ancient Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. He is generally considered the greatest mathematician of antiquity and one of the greatest of all time.

Archimedes was contemplating a mathematical diagram when the city of Syracuse was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were lethal weapons.

MORAL OF THE STORY: If an armed brute is talking to you, hide all the mathematical instruments you have.

3-2016 The revolutionary mathematician

Évariste Galois (French: (25 October 1811 – 31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel.

Galois was also a political firebrand. He was even jailed several times for his anti-royalty activities. On Bastille Day of 1831 (14 July), Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. He was again arrested. On 23 October he was sentenced to six months in prison for illegally wearing a uniform. He was released on 29 April 1832. Fortunately for all, during his imprisonment, he continued developing his mathematical ideas.

Tragedy was to strike Galois, for in July 1829 his father committed suicide. The priest of Bourg-la-Reine forged Mayor Galois' name on malicious forged epigrams directed at Galois' own relatives. Galois' father was a good natured man and the scandal that ensued was more than he could stand. He hanged himself in his Paris apartment only a few steps from Louis-le-Grand where his son was studying. Galois was deeply affected by his father's death and it greatly influenced the direction his life was to take.

Siméon Poisson, a French mathematician, geometer, and physicist. asked him to submit his work on the theory of equations, which he did on 17 January 1831. Around 4 July 1831, Poisson declared Galois's work "incomprehensible", declaring that "[Galois's] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion."Galois did not ignore Poisson's advice, as he began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832.

Galois's fatal duel took place on 30 May 1832. The true motives behind the duel are obscure. There has been much speculation, as to the reasons behind it. What is known is that five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.

In March 1832 a cholera epidemic swept Paris, and prisoners, including Galois, were transferred to the Sieur Faultrier hostel. There he apparently fell in love with Stephanie-Felice du Motel, the daughter of the resident physician. After he was released on 29 April Galois exchanged letters with Stephanie, and it is clear that she tried to distance herself from the affair. The name Stephanie appears several times as a marginal note in one of Galois' manuscripts. Galois fought a duel with Perscheux d'Herbinville on 30 May, the reason for the duel not being clear but certainly linked with Stephanie.

Galois was wounded in the duel and was abandoned by d'Herbinville and his own seconds and found by a peasant. He died in Cochin hospital on 31 May and his funeral was held on 2 June.

Detailed speculation, based on scant historical details, has been interpolated by many of Galois's biographers (most notably by Eric Temple Bell in Men of Mathematics), that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.

MORAL OF THE STORY: If you are an algebraist, do not indulge in fencing.

2-2016 How religious fanatism killed a woman mathematician

Hypatia of Alexandria was the first woman to make a substantial contribution to the development of mathematics. She is also considered to be the first woman mathematician. She was the daughter of the mathematician Theon, the last Professor at the University of Alexandria.

There is no evidence that Hypatia undertook original mathematical research. However she assisted her father Theon of Alexandria in writing his eleven part commentary on Ptolemy's Almagest. It is also thought that she also assisted her father in producing a new version of Euclid's Elements which has become the basis for all later editions of Euclid.

All Hypatia's work is lost except for its titles and some references to it. However no purely philosophical work is known, only work in mathematics and astronomy. Hypatia was an excellent compiler, editor, and preserver of earlier mathematical works. She is described by all commentators as a charismatic teacher.

Hypatia came to symbolise learning and science which the early Christians identified with paganism. This quality was enough for Christian fanatics of Alexaandria to antagonise her. She was also seen as a 'stumbling block' to those who would have accepted the 'truth' of Christianity were it not for her charisma, charm, and excellence in making difficult mathematical and philosophical concepts understandable to her students; concepts which contradicted the teachings of the relatively new church.

She became victim of the crossfire between Orestes and Cyril who spearheaded conflicts between Christians and non-Christians of Alexandria. In 412 Cyril, (later St Cyril) became patriarch of Alexandria. However the Roman prefect of Alexandria was Orestes (a pagan). Cyril and Orestes became bitter political rivals as church and state fought for control. Hypatia was a friend of Orestes and this, together with prejudice against her philosophical views which were seen by Christians to be pagan, led to Hypatia becoming the focal point of riots between Christians and non-Christians.

She was murdered in 415 CE by a Christian mob who attacked her on the streets of Alexandria. On her way home from delivering her daily lectures at the university, Hypatia was attacked by a mob of Christian monks, dragged from her chariot down the street into a church, and was there stripped naked, beaten to death, and burned. She was also severely tortured by scraping off her skin with clamshells (some say roofing tiles). Even those Christian writers who were hostile to her and claimed she was a witch, portray her as a woman who was widely known for her generosity, love of learning, and expertise in teaching in the subjects of Neo-Platonism, mathematics, science, and philosophy in general.

What certainly seems indisputable is that she was murdered by Christians who felt threatened by her scholarship, learning, and depth of scientific knowledge.

Whatever the precise motivation for the murder, the departure soon afterward of many scholars marked the beginning of the decline of Alexandria as a major centre of ancient learning.

MORAL OF THE STORY: If you teach maths, do not get involved with religious fanatics.

1-2016 Alan Turing, the man who knew too much.

Alan Mathison Turing ( 23 June 1912 – 7 June 1954) was a pioneering British computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. Alan Turing is widely considered to be the father of theoretical computer science and artificial intelligence. Turing's contribution to the theory of computation, and decision problems is legendary. In spite of all his genius, he was forced to lead a deplorable and humiliating life and die a mysterious death.

Much later after his death, British Prime Minister Gordon Brown officially apologized to Alan Turing on behalf of the people of his nation for "the appalling way he was treated." Parliament finally brought up a bill of "pardon", and on 24 December, 2013, Queen Elizabeth granted Turing pardon posthumously, 59 years AFTER his mysterious death (he died mysteriously on 7 June 1954).

As if to repent for the torture they inflicted on Turing, the world started naming things in his memory. The highest award in computer science (the Turing Award) "for contributions of lasting and major technical importance to the computer field", is named after him. The Turing Award is generally recognized as the highest distinction in computer science and the "Nobel Prize of computing".The first recipient, in 1966, was Alan Perlis, of Carnegie Mellon University. The fundamental model of a computer is called the Turing machine (much like Dalton's atomic theory which is basic to chemistry). The Turing test is an important contribution to artificial intelligence.

Turing had something of a reputation for eccentricity at Bletchley Park.While working at Bletchley, Turing, who was a talented long-distance runner, occasionally ran the 40 miles (64 km) to London when he was needed for high-level meetings, and he was capable of world-class marathon standards.

Turing played a pivotal role in cracking intercepted coded messages that enabled the Allies to defeat the Nazis in many crucial engagements, including the Battle of the Atlantic; His work, significantly shorted the war by at least two years, saving an estimated 17 million lives. Prime Minister Winston Churchill singled out Turning as the person whose work contributed the most to defeating the Germans. In 1945, Turing was awarded the OBE by King George VI for his wartime services, but his work remained secret for many years. However, many considered him at that time as "the man who knew too much", and hence a "security risk".

It may sound incredibly insane to record that Turing was arrested on 7 February 1952 for his affair with a young Manchester man. Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's cod ebreaking centre Turing's conviction led to the removal of his security clearance and barred him from continuing with his cryptographic consultancy for the Government Communications Headquarters (GCHQ), the British signals intelligence agency that had evolved from GC&CS in 1946 (though he kept his academic job at Manchester University). He was denied entry into the United States after his conviction in 1952, but was free to visit other European countries, even though this was viewed by some as a security risk.

He was obliged to undertake injections of female hormones intended to render him asexual. Rather than serving time in prison, Turing chose to undergo estrogen injections then considered in men as a form of “chemical castration” eliminating their sex drive.

For two years after his arrest, until his death, he suffered enormous amounts of humiliation, distrust, disgrace, and embarassment, both personally and professionally.

His housekeeper famously found the 41-year-old mathematician dead in his bed, with a half-eaten apple on his bedside table. The famous logo, a half-eaten apple, used by a famous computer company, reminds us of this incident (although there are other stories about this logo doing the rounds). Turing's habit was to take an apple at bedtime, and that it was quite usual for him not to finish it; the half-eaten remains found near his body cannot be seen as an indication of a deliberate act. Indeed, the police never tested the apple for the presence of cyanide.

This genius paid with his life, for doing so much for humanity (particularly the ungrateful British).

MORAL OF THE STORY :: If you are a cryptologist, do not eat apples. * * *

3-2015 When Satan meets Cantor -- A book review

Satan, Cantor, and Infinity, and Other Mind-Boggling Puzzles, Raymond M Smullyan (Pub.: Dover Publications, Mineola, New York)

Raymond Smullyan, the master story-teller, wears many feathers on his cap : mathematician, logician, concert pianist, stage magician, amateur astronomer, puzzle master, teacher. He often mixes his talents, and creates amazing books on logic and puzzles, which often call for profound reasoning and mathematical deduction. His book, “Satan, Cantor and Infinity: Mind-Boggling Puzzles (by Raymond M. Smullyan) ” is one such masterpiece.

In this book, Raymond Smullyan takes a puzzle-based perspective on the principles underlying the works of mathematician Georg Cantor, particularly on infinity. His fascinating riddles involve probability, certainty, time, and infinity, and they unfold amidst a landscape populated by honorable knights, lying knaves, quick-witted robots, and other fanciful characters. This 270+ pages tome, explains 25 puzzles, grouped into 7 parts. Each puzzle is as intriguing as the other, and demonstrates the inventive genius of the author. Using an imaginary sorcerer who is really a logician, Smullyan takes us on a tour of logic and mathematics, including Godel’s famous theorem. Which brings us to the pioneering discoveries of Georg Cantor. Each puzzle is not linked to any other in the book, and so the reader may peruse the contents in any order.

I chose to read the last part of the book (part 6) which had a tantalising title “A journey into infinity”. This part carries a puzzzle, or rather an introspection “What is infinity ?” A very lucid conversation involving the Sorcerer, talks about infinity and the famous “Hilbert’s hotel problem”. This lead me to my favourite concept – paradoxes. I also found in part 5 “The envelopes paradox”. I had written about this paradox, in this very journal “Gonit Sora”, some time ago.

The book has its fair share of puzzles involving truth-tellers (knights) and liars (knaves). It has therefore several interesting examples of the usage of Goodman’s principle.

Smullyan not only unveils his puzzles in his own fascinating style. He also gives clues/solutions to most of his puzzles. But wait ! You need to ponder well, before you can grasp the puzzle or its solution (my experience) . Finally, this book is not for the mathematically weak-minded. A good knowledge of Cantor’s set theory, and the works of Zermelo, Fraenkel, Russel, Godel, and Lewis Caroll, would make reading this book, a lot more enjoyable. My closing remark would be that rather than using this book for entertainment, it can also be a valuable and innovative tool for understanding mathematics and logic.

Smullyan beautifully sums up his thoughts as follows: “The moral of the story is that even fallen angels might benefit from a good course in mathematical logic.”

2-2015 My own corner on the web

Now, I have my own corner, on the web. The new URL for my website is ::
http://drpartha.org.in/index.htm

You can navigate down to all my pages starting from the above page.

Bookmark this page, and let all your friends know about this new site. If you own or administer a website, please add a link to the above page from your site.

Thanks, partha