Sunday, May 21, 2017

Mancala/Oware, a math game with a 5000-year-old history

Left: an African mancala board

To close out the math posts, here's an ancient board game from Africa. Today, we can learn to play online. It's usually played on a wooden board, but in a pinch, players can use an egg carton and forty-eight small stones, seeds or marbles. Mancala has a long history. Also called oware, it has many variants. Math began with the Ancient Sumerians, and this game is thought by many to have originated with them, five millennia ago.

Was its original function record-keeping, or did it have a ritual purpose? Historians aren't sure, but the presence of the game boards at African temples, laid out in alignment with the rising and setting sun, suggests some sort of symbolic significance.

It is thought that Arab traders brought the game from Sumeria (today Iraq and Kuwait) to ancient Egypt, from whence it spread over Africa and beyond. In its various forms, this mathematical game of skill and strategy is still played widely today.

Saturday, May 20, 2017

Julia Robinson, a rare mathematical woman

Much like Germany's David Hilbert, American mathematician Julia Robinson saw her breed as a single nation, "without distinction of geographical origins, race, creed, sex, age, or even time." (quoted in Marcus du Sautoy)

Born in Missouri, Julia Bowman lost her mother and was raised by her grandmother in the Arizona desert. A childhood illness afforded lots of time to think. After marrying a fellow mathematician, Rafael Robinson, she settled in California. Advised against having children by her physician, she devoted her energy to mathematics, tackling Hilbert's tenth problem of existential definability.

After spending much time on this problem, she collaborated with a twenty-year-old Russian mathematician, Yuri Matiyasevich, and visited him in Leningrad. Together they proved Hilbert's tenth problem unsolvable. Julia was delighted to meet the young man, opining that she must have been waiting for him to be born and grow up so they could work together on their proof.

Robinson was elected to the National Academy of Sciences in 1975, the first woman mathematician to receive this honour. She received a professorship at Berkeley the same year. In 1983, she was elected president of the American Mathematical Society, again the first woman to hold this post. Sadly, she died of leukemia before her term was complete.

Each year, in her honour, The American Institute of Mathematical Sciences holds the Julia Robinson Mathmatics Festival.

Friday, May 19, 2017

Bertrand Russell, mathematician and philosopher

Image from sapiengames

Bertrand Russell (1872-1970) was a well known 20th century mathematician and philosopher. His books include Principles of Mathematics (1903), Why I am not a Christian (1927), The Conquest of Happiness (1930) , and A History of Western Philosophy (1945).

He made new mathematical contributions to formal logic and discovered what came to be called Russell's paradox. He is ranked with Kurt Godel as one of his century's top logicians.

In 1950, he received the Nobel Prize for Literature, and in the 1960s, his anti-nuclear and anti-war protests inspired the youth of the day.

Russell was known for his bons mots, many of which are ironic or paradoxical. He said for instance, "...fools and fanatics are always so certain of themselves, and wiser people so full of doubts." Another of his statements: "It has been said that man is a rational animal. All my life I have been searching for evidence which could support this." He said that humans were born "ignorant, not stupid," and "made stupid by education," and that "To be without some things that you want is an indispensable part of happiness."

His famous personal essay, What I have lived for, is considered a model of its kind of writing.

Thursday, May 18, 2017

Alan Turing, mathematician, marathon runner, and father of computing

Alan Turing was an avid cyclist and marathon runner, and the mathematician who conceived of artificial intelligence. He built the world's first computer at the WWII code-breaking centre of Bletchley Park, enabling the break of the German naval code, Ultra. Many believe this breakthrough shortened the war by a couple of years. Math and computing science were not his only areas of knowledge. For his groundbreaking work on morphogenesis, he was made a Fellow of the Royal Society. 
Sadly, although Turing's nation used his skills in war time, the government treated him shockingly afterwards. In 1948, he'd been named Deputy Director of the computer laboratory at Manchester University, where he became the first person to use a computer for mathematical research. 

The trouble began in Manchester in 1952. While investigating a theft of money from Turing, the police learned of a homosexual affair which he did not attempt to hide, though homosexual activity was illegal. He was tried for gross indecency and found guilty. To avoid a prison sentence, he agreed to take estrogen injections. 

In the post-war period, Alan Turing was still working for GCHQ, with Hugh Alexander, whom he'd known at Bletchley Park. However, under the cold war alliance with the Americans, who considered homosexuals ineligible for security clearance, his own government stripped him of the clearance he'd had since the war. 

In June of 1954, he was found by his housekeeper, dead of cyanide poisoning. The presence of a half-eaten apple and the presence of cyanide on his fingers led to speculation that he'd died accidentally, while carrying out an experiment. However, the coroner found the death to be suicide. Yet it seemed strange; for one thing, he'd just enrolled in an upcoming conference. At the time of his death, this talented thinker was only forty-two years old.

The Turing Award for Computing was established in 1966, and later a Turing monument was put up in Manchester. In 1998, his birthplace in London was marked with a blue plaque, according to the custom. In 2013, Queen Elizabeth granted him a full pardon for his former "crime." 

Tuesday, May 16, 2017

Hardy, Littlewood and Ramanujan, a mathematical trio










After an exchange of letters with Srinivasa Ramanujan revealed his amazing mathematical insights, GH Hardy and John E. Littlewood, two Cambridge dons, encouraged the talented but untrained young mathematical genius to come to Cambridge. There they arranged to get him a position and collaborated with him on mathematical work. 

Sadly, Ramanujan did not adapt well to the climate of England, and found it difficult as a vegetarian to stomach the food in college. When his health began to fail, Hardy encouraged him to return to India for a holiday. He lived to see his home and family, but died in Madras of a parasitic infection. He was only thirty-three years old.
At the request of a friend, Hardy penned a memoir, A Mathematician's Apology, during his final illness. This is a seen by many as a study of the creative mind and process. Published in 1940, a year into the war, the book reveals a somewhat gloomy outlook.

Sunday, May 14, 2017

David Hilbert, German mathematician

Image from britannica

"Mathematics knows no races or geographic boundaries;...the cultural world is one country." So thought David Hilbert. He wrote Mathematics and the Imagination, and Nature and Mathematics, among other books.

Born in 1862 in Konigsberg, where Euler solved the problem of the Seven Bridges of Konigsberg, Hilbert died in 1943 in Gottingen, which had been the European centre of mathematics before the war. He contributed to geometry, invented "Hilbert space" in calculus, established a formalist school of mathematics, and expanded his work into mathematical physics.

At the Paris International Congress of Mathematicians in 1900, he gave a key lecture in which he assigned his fellow mathematicians "homework" for the new century. Among his twenty-three problems, several remain unsolved today. At the top of Hilbert's list is the Riemann Hypothesis, about which he once said, "If I were to awaken after a thousand years, my first question would be, 'Has the Reimann Hypothesis been proven?'"

In a blog post seven years ago, I recorded an anecdote about something strange Hilbert did. It's right here.

Thursday, May 11, 2017

Fiction with Heart: Elinor Florence at Canadian Authors

Bestselling novelist Elinor Florence carries Canadian history in her DNA. Wednesday she talked to Canadian Authors about salting fiction with nuggets from real life. The protagonist of Bird's Eye View is a prairie farm girl who works as a map interpreter at RAF Medmenham.

Born and raised on a former airport in North Battleford, Florence spoke of going to sleep thinking about the ghosts of airmen on her family's farm. One pilot crashed and died across the road on a training flight. Her father bought the farm land from the air force, and until he built the house, the family lived in a former airport building.

This airport, along with many more across the prairies, was built for the Commonwealth Air Training Plan, to ensure that pilots from Britain and the Commonwealth could learn to fly in safety before entering the Battle of Britain.

A journalist who worked on newspapers in BC and across the prairies, Florence talked about how she unearthed the telling details that place readers in the heart of the story.

The book was a long time in the making, and over a period of years, she interviewed people who remembered and served in the war. She read memoirs, including one by Constance Babington-Smith, the map interpreter who discovered the first V1 rocket on a German aerial map. Known today as cruise missiles, these flying bombs were a serious threat to London, and potentially to Halifax and New York as well, before the Normandy invasion.

To ensure she described the setting accurately, Florence visited Danesfield House, now a hotel. At this former site of RAF Medmenham, she met and interviewed a woman who had worked there during the war. By an amazing coincidence, this was the former map reader's first one day visit back to see her old workplace.

Florence also tracked down letters and pictures from obscure corners of museums. Astonishingly, a photo of the pilot who crashed in North Battleford led her, via the internet, to his relatives in Tasmania. She learned that the mother knew the young man and had written to his mother when he died. The Tasmanian cousin was delighted to learn where he was buried, and arranged to have the RCAF lay a wreath in remembrance.

A warm audience of Canadian Authors members and guests enjoyed Elinor Florence's fascinating presentation. Another book, My Favourite Veterans, is based on interviews she posts regularly on her blog, Wartime Wednesdays.

Wednesday, May 10, 2017

Everyone Brave is Forgiven by Chris Cleave

Chris Cleave begins this war novel with page after page of lighthearted and relentless repartee. At the outbreak of war, Tom thinks himself possibly "the only man in London who did not think the war was an unmissable parade lap." Raised to the post of running a school district, he reflects that he "might celebrate the promotion if he didn't have the gift for noticing that the schools were empty."

Tom's friend and flatmate, Alistair, restored paintings for the Tate before the museum's treasured works were trucked to Wales and hidden in a mine shaft for the duration. When he decides to sign up, Tom gives him a jar of blackberry jam he's just made with wild fruit and the last of their sugar ration. When his friend vows to "lay it down" to "open together at war's end," the reader knows that jam will return, sadly not shared by the two pals in peacetime.

The author then introduces Mary and Hilda. Unlike their male counterparts, the women friends are from wealthy families. They talk and laugh and argue and fight and have known each other for a lifetime. The war tests this friendship and throws their different attitudes into sharp relief. While Hilda proclaims herself content to find a handsome man in a uniform, Mary volunteers for war service and stumbles upon her gift and calling for teaching children.

When her class is evacuated, she prevails on Tom to give her a job teaching the children who've been left behind, or sent back unwanted from the countryside. Some are physically disabled, and one child, eleven-year-old Zachary, is a black American with whom Mary forms an instant bond.

Mary and Tom fall in love, and she prepares her classroom, complete with a basement air raid shelter. In the midst of the Christmas pageant, the sirens sound and they go down, along with spectators Tom and Alistair. Due to Mary's pre-planning, they're able to continue the pageant below stairs. Knowing that Alistair has now been accepted by the army, Tom broods beside his friend, wondering if their friendship is over. Perhaps "it is not possible, after all, for a man who had gone to war to abide a man who had stayed."

Cleave alters his language as the situation darkens. The light witticisms that initially mark the conversations between Tom and Alistair and Tom and Mary give way to images of ruthless realism. The reader is party to Alistair's grim thoughts about "the crushing fatigue and the fear, and the constant mental strain of saying to others, We shall prevail, and to oneself, I am defeated." Under the air raid, Tom speaks seriously to Mary, admitting that now he really hates the Germans, though "he never thought he had it in him." Her witty reply is filled with brittle bitterness, "'That's why we call them the enemy. See how it works now, darling?'"

Tom reveals that he too volunteered, and was refused, adding that he thought she might be proud of him in uniform. "'Do you really think so little of me?'" she responds. When in a funk, he tries to break it off with her, Mary speaks of life and fate and choices in this simple speech. '''But it wouldn't be my life, don't you see? You're the one I've chosen, and I love you even more for being good enough to ask me not to choose you.'"

Meanwhile, on the detested rocky starving island of Malta, a gloomy Alistair remembers London, where one views history "as a reworkable legend, a great entertainment of doubtful veracity and liable in any case to revision whenever the next mudlark waded into the Thames at low tide and pulled out some iconoclastic sherd." His irony deepens with his gloom, and he bitterly supposes "the War Office had established a vast cache of polishes" to make boots glow "as if with an inner light" and the band instruments blaze "like the armor of Achilles." Since the arrival of letters might leave his men "upbeat or homesick, or a queer mix of the two" he drops by them to "project a soothing equanimity." Then he goes to Christmas dinner made of "bread crumbs and canned malevolence."

In London, Mary watches "the devastation roll by, each bomb "a breach in the carapace, laying bare the living nerve." When she writes to Alistair about her guilt and regret over the air raid that struck her classroom, she tells him she was "brought up to believe that everyone brave is forgiven, but in wartime courage is cheap and clemency out of season." For his part, Alistair feels unequipped to reply to her letter, believing that in all of history "there was not one example of a man ever having written a satisfactory letter to a woman who mattered to him."

When they do meet, he suggests she drive an ambulance, asking, with a flash of his old gaiety, "'Why wander through your thoughts when you could drive through them quite recklessly, with sirens?'" After he returns to duty, they begin to exchange humorous letters. One day, he looks up after reading one of hers, "surprised to find the war." The mood is lifting, and with it, Cleave's fresh and incisive language.

We see more of this upshift in the language as the novel moves toward a hesitant hope. In the Ritz with friends, Mary reflects that "Here they honored one's name in that generous way the Ritz knew, which was to remember it only when one was sober."

Yet even with the promise of light, there are dark moments, as Mary feels herself stepping "into the dark, even though she knew that each step took her no further from who she was," and foresees that when the zoo animals are "returned to their old labeled cages," the world will remain unable to "wake from its pattern."

At the end of this unputdownable story, the reader is left with the hope of "an air one might still breathe, if everyone forgiven was brave." Cleave's grandfather was in the war, and from the intimacy of this portrayal of the effects of war on individuals, one is tempted to believe that he too might have witnessed it at first hand. Perhaps in another life.

Tuesday, May 9, 2017

Canadian Nonfiction Collective conference at Green College inspires

Peaceful Green College, surrounded by lush gardens and ocean views, was a great setting to hear the wise words of CNFC guest speakers, who made the recent gathering of nonfiction writers uplifting and inspiring. One big topic was the effect of new media on thought and language. Andreas Schroeder sadly commented that while people rely on the speed and accept the shallowness of internet research, you could "drive a truck through the new Koerner library." Blogging may have its place, but "I defy anyone to be able to produce anything worth reading every single day."

Fellow writers learned much from the thoughtful journalist Deborah Campbell, who has just won the BC Book Prize for Nonfiction and the Hillary Weston Prize. A Disappearance in Damascus took years to write, and many times she was tempted to quit. The insight that kept her going was this. "The point of writing is not to change the world, but to keep the truth alive." The challenge of keeping the truth alive in a "post-truth world," when journalists are told that "Iraq is over," and "Refugees are over," Campbell pointed out that this problem is not new. Like Jonathan Swift in "A Modest Proposal," the writer must engage the reader using the ancient techniques of storytelling. She offered words from Berthold Brecht, written in 1956, as an aid to comfort and hope. Writers in all places and all eras, said Brecht, need the courage, keenness, skill, judgment and cunning, but it is possible to get our words in front of the audience that needs them.

Joy Kogawa recently published a memoir, Gently to Nagasaki. She spoke about the paradoxes that attend the writing life, and the struggle to discover how to live. She acknowledges the evil in the world; it has challenged her mightily in her own life. Her family were evicted from their home as enemy aliens in 1942, and sent to internment camps. Yet now that re-purposed house has become a beacon for diverse writers and a historical reminder of our past. Kogawa loved an ancient and scarred cherry tree in the garden of that house, and wanted to take cuttings of it and plant elsewhere as symbols of love and reconciliation. Nevertheless, somebody deliberately killed it in a displaced expression of rage against her deceased father, a minister who had been exposed as a pedophile.

Kogawa also shared some bizarrely ironic history about the bombing of Nagasaki. This place was not the initial target, and was hit when changing weather conditions and low fuel in the bomber made it urgent to drop the deadly payload. The people the bomb fell on were a hidden community of Japanese Christians, long persecuted by their own government for their religious difference.

"Getting your own back is sweet, and we call it justice," said Kogawa, "but it doesn't satisfy for long." The wisdom I took from this wise elder was more nourishing. "If you can't forgive yet, you can still intend to." If we live with two parts mercy to one part abundance, we can "be with people who have different truths, even Donald Trump." Humans are a "sense-making species, and we make our stories by struggling with them" (a message also shared by Campbell).

But the line that makes my scalp prickle with resonance as I retype it from my notes: "Have confidence in this: The long arc bends toward the good."

Monday, May 8, 2017

Mary Walsh left them laughing

Last Thursday evening at Frederic Wood Theatre at UBC, Hal Wake and The Vancouver Writers' Fest hosted another wonderful evening with a writer: the hilarious actor and comedian Mary Walsh is now the author of a novel. When CBC's Lisa Christiansen interviewed Mary about her latest artistic accomplishment, the answers to the journalist's questions were true, evocative and hilarious.

Crying for the Moon portrays a Newfoundland Catholic girl, inspired in part by Mary's growing-up years in St. John's. Maureen makes it to Expo 67, but when she actually meets Leonard Cohen in a bar, she can't think of a thing to say. What a lost opportunity. The novel has been called gritty, and it involves poisoning a wife beater. If what Mary read to us was any indication, it also has humour and heart. I can't wait to read it.

By chance, I sat in the front row beside Lisa Christiansen's mother, and we chatted before the show. Hearing Mary explain to Lisa how to pronounce NewfoundLAND, I thought about Mompy, my own late mother, also from St. John's. For a moment, I heard her speak in the Newfy brogue she kept in spite of more than half a lifetime living "out west." This is what she said in my mind's ear: "Don't be crying for the moon, my dear." No, Mompy. Indeed I won't.

Saturday, May 6, 2017

Tartaglia of Bologna - a sharp mathematical mind behind a scarred face


In 1535, a young Venetian called Nicolo Fontana won the Mathematics Competition at Bologna University. His achievement, formerly considered impossible, involved using the square roots of negative numbers. Ten years later, this work was claimed and published by Cardano, along with work by Cardano's student, Ferrari, whose formula was inspired by Fontana's work.

A bookkeeper as well as an engineer in the Venetian army, Fontana was a topographical surveyor who designed fortifications. He was wounded in an attack by the French army, and the resulting facial scar stuttering gave rise to the epithet he became known by, Tartaglia, the Stammerer.

Friday, May 5, 2017

Ferrari - not the car designer, but the Renaissance mathematician

Image from wikipedia

Lodovico Ferrari of Bologna (1522-1565) was the first to discover an alegebraic solution to the quartic or biquadratic equation.

Taken in and trained by Gerolamo Cardano, he learned from his patron's lectures and eventually succeeded him as a lecturer in mathematics in Milan. Ferrari publicly challenged fellow mathematician Nicolo Tartaglia over how to solve a cubic equation, and won a math contest against him in 1548. This success brought him a chance at the job of tax assessor in Mantua, where he gained considerable wealth. Later he quarreled with the Cardinal and had to leave this post.

Thursday, May 4, 2017

Waiting for the rain to let up

Yesterday I spent a luxurious afternoon at Van Dusen Gardens. After lunching at Truffles, I spent most of the afternoon sitting on a sofa on the empty outdoor deck under a wide overhang, reading and dozing. When the weather cleared, I took in the rhododendron walk, sadly damaged by the recent harsh winter, but still lovely. Umbrellas are a necessity in Vancouver this spring.

Leonardo Fibbonaci, observant mathematician of Pisa

Image from Leaning Tower of Pisa

Leonardo Pisano, often called Fibonnaci (son of Bonacci) was born in Pisa in 1150 CE. He travelled through North Africa in childhood with his diplomat father before returning to Pisa to write math books including Practica Geometriae and Liber Quadratorum.

Another book, the Liber abaci, introduced what came to be known as the Fibonnaci sequence. Beginning 1,1,2,3,5,8, this sequence predicts the numbers of seeds on a sunflower head, and numbers of rabbits descending from a pair. It is also foundational work predating Pascal's Triangle.

This sequence in important enough to be the subject of a modern journal called the Fibonnaci Quarterly, which delves into the mathematics of Fibonnaci numbers.

Wednesday, May 3, 2017

Omar Khayyam, poet and mathematician

Image from St. Andrew's University

Today Omar Khayyam is well-known as a poet. This 11th century native of the learned city of Nishapur (now in Iran) was an expert in music and geography, and also an astronomer, philosopher, and  mathematician. His math books included Problems of Arithmetic and Treatise on Demonstration of Problems of Algebra. 

Omar Khayyam contributed to calendar reform, and accurately calculated the length of a year to 11 decimal places. He also solved a cubic equation using the intersection of a circle with a rectangular hyperbola, a new method. He said this equation could be solved, not by the ruler and compass method, but using cubic sections. Indeed, this claim was proved 750 years later.

The following are probably his most famous poetic lines:
"The moving finger writes, and having writ,
Moves on: nor all thy piety nor wit,
Shall lure it back to cancel half a line,
Nor all thy tears wash out a word of it."

Tuesday, May 2, 2017

Algebra with Al Khwarizmi in the 8th century

Image from yesIknowthat

The great mathematician Muhammad Al-Khwarizmi was born about 780 CE and became one of the first directors of the House of Wisdom in Baghdad, where he had the work of earlier mathematicians translated into Arabic.

An advocate of the Hindu number system, and advocated their use. Indeed, now known as the Hindu-Arabic numerals, they remain in worldwide use today. His name is remembered through the Arabic-derived words algebra, from one of his book titles, and algorithm, his Latinized name.

Monday, May 1, 2017

Bhaskara the Teacher, 12th century Indian mathematician

Image from india.com

This great Indian mathematician was given the epithet Bhaskaracharya, meaning the Teacher. Son of an astrologer, he headed the observatory at Ujjain, the centre of math and astronomy.

Like Brahmagupta before him, he understood zero and negative numbers. He also devised
four special methods for squaring numbers.