Saturday, April 30, 2016

Evariste Galois

The death of General Jean Maximilien Lamarque on June 1, 1832 was the catalyst for the fighting to break out in the Parisian June Rebellion of 1832.  This event is captured in Les Miserables by Victor Hugo.  Lemarque's death came one day after the fatal duel of young mathematician Evariste Galois (1811-1832) - whose death may have been purposeful and intended to trigger the raising of the barricades.  If so, so much the worse for Galois, and so much the worse for us.

There is a great deal of legend that has grown up around Galois's life, and it was easy for biographers such as E. T. Bell and Leopold Infeld to romanticize his life.  He died at 20 in a duel, and by that young age he had already revolutionized mathematics - though that wasn't recognized for at least a decade after his death.  He wrote out a great deal of mathematics the night before he died - as if he knew for sure he was going to die - almost as if he planned to die - and in this writing we find the phrase, "Je n'ai pas le temps!"  ("I have no time!")  My understanding is that he had no second in the duel and that only one shot was fired.  Galois was shot in the stomach and left for dead in the field of Glacier Pond in Gentilly, 13th Arrondisement, Paris.

I spent a great deal of time trying to find the location of the duel - wanting to visit, pay my respects, take photographs, but as I assumed and was later told much has changed in Paris since 1832.  Apparently the field was near swamps, and, as I was told by a mathematical correspondent in Germany who is familiar with Paris, "Definitely the old swamps ("Glacière") don't exist, they used to be swamps that froze in winter, and whose ice was stored on the outskirts of Paris, but this part of Paris has been entirely built, and even the river is underground now.  There is a street ("rue de la Glacière") which corresponds roughly to the place."

The following picture is of  "rue de la Glacière" - a very short street with no sign of a field or swamp on or near it!

A passing peasant found Galois a couple of hours after the duel and brought him to La Chochin Hospital - which, in my pilgrimage I also sought out.  Obviously this too has changed since 1832 - remodeling, new wings and buildings added - as one would hope for a hospital over the course of 184 years!  But, though updated, here it is, the hospital where Galois was taken.  It's only two blocks away from rue de la Glacière.

Galois died at the Cochin Hospital at 10:00am the following morning.  He had refused the offices of a priest, but his younger brother Alfred was there with him, and Galois's last words were to his brother, “Ne pleure pas, Alfred! J'ai besoin de tout mon courage pour mourir à vingt ans!” ("Don't cry, Alfred!  I need all my courage to die at twenty!")

Galois was buried in a common grave at Montparnasse Cemetery, and no trace of it is left.  A cenotaph memorializing him has been raised in the Bourg-la-Reine Cemetery near where family members are buried.  The following are pictures of Montparnasse Cemetery - his remains being somewhere inside:

The reason I began this post with a picture of the flag is that Galois was very political.  He was part of the Artillery of the National Guard, of which 19 officers were arrested late in 1830 for conspiracy to overthrow the government.  They were acquitted in May 9, 1831, and 200 republicans gathered for a dinner to celebrate the acquittal.  During the dinner Galois appeared to make threats against King Louis Phillipe - standing and raising both a glass and an unsheathed dagger.  It was said that he stated, "To Louis-Phillipe, if he betrays."  But the reality is that his words were drowned out in the surrounding noise.  Galois was arrested but, surprisingly, acquitted, however on July 14, Bastille Day, he was arrested again because he was wearing the uniform of the Artillery of the National Guard, which was illegal; he was also carrying loaded guns and a dagger.

While in prison he tried to commit suicide by stabbing himself with a dagger, but the other prisoners stopped him.  In 1832 all the prisoners were moved to another location due to a cholera epidemic.  It seems that it was here that he fell in love with Stephanie-Felice du Motel, daughter of the attending physician.  She later spurned Galois, and this is involved at least in some way with his death.  In letters he wrote to friends not long before he died he says, "I was provoked by two patriots  .  .  .  it was impossible to refuse" and "I die a victim of an infamous coquette."

We don't know for certain what the cause of the duel was.  Was it purely a duel of love and honor?  Was it a set-up by other political forces to get rid of him (using Stephanie as a convenient ruse)?  Was it a way of "honorably" committing suicide, which he had been unsuccessful with in prison?  Was it self-sacrifice - him laying his life down for the cause he believed in, to be a trigger for barricades to go up and fighting to begin?

What we do know for certain is that his mathematical work - all done by the age of 20 - has altered the face of mathematics.  As mathematician Arthur Cayley said of Galois's work, "The idea of a group as applied to permutations or substitutions is due to Galois, and the introduction of it may be considered as marking an epoch in the progress of the theory of algebraic equations."  Biographer John Derbyshire also views the work of Galois as the beginning of a new epoch in algebra - the first being when people when from asking, "This plus this equals what?" to "This plus what equals that?" - the second being the development of "literal notation" (i.e. using letters in algebra in order to "relieve the imagination") in the 1500s and 1600s (if you think algebra is hard because they threw the alphabet in with the numbers, try doing it without the letters!) - and the third being Galois's move to higher levels of abstraction (this making up a huge part of what mathematics is today, but it's not at all the "algebra" you see in middle school, high school or junior college - it has to do with such things as fields, rings, ideals, groups - both abelian and non-abelian, as well as kernels and so on - and that's just the beginning of what his revolution has brought forth!).

As to biographies, I find it hard to let go of the romanticized versions of his life that I'd always heard.  It seems E. T. Bell wanted to inspire young people to go into mathematics (as Sophie Germain had been inspired by the story of Archimedes) and so played fast and loose with the facts in biographies of mathematicians in his book Men of Mathematics, of which there is a chapter on Galois.  (I don't know if I can fault him TOO much as this book did inspire such phenomenons as John Forbes Nash and Freeman Dyson to go into mathematics, and his book The Last Problem inspired Andrew Wiles to pursue and finally solve Fermat's Last Theorem, but still  .  .  .).  I first fell in love with the story of Galois when I read Leopold Infeld's biography Whom the Gods Love twenty-five years ago or so.  So if you want a lovely, romanticized version of Galois's life, these are the places to go.

If you want the cold, clarifying water of well-researched reality splashed in your face, check out Tony Rothman's essay on the facts surrounding the events at the end of Galois's life.

If you'd like a brief, accurate, and interesting biography of Galois read either the chapter "Pistols at Dawn" in John Derbyshire's book Unknown Quantity, or the chapter "The Romantic Mathematician" in Mario Livio's book The Equation that Couldn't be Solved.

Though as I've researched further into his life and have begun to feel that "ignorance is bliss," I don't know that we need all the romanticizing that has taken place.  The bare facts are intriguing enough in my mind - a genius whose mathematics changes the discipline entirely, who engages in questionable political activities, dies in a mysterious duel at age 20 the day before the death of General Lemarque which triggered the fighting portrayed in Les Miserables, is left for dead on the field, and dies in his brothers arm's the next day asking his brother not to cry since he needs all of his courage to die at 20 - seems dramatic enough to me!

Friday, April 29, 2016

Le Jardin des Tuileries

 The majority of my "recording" time goes to my blog, but I try to save at least a bit of time for written journaling as well - mostly it's just when I can steal a rare moment, but today's journaling time was set in place before I left the United States.  Part of my planning for Paris was to spend time relaxing in Le Jardin des Tuileries.  The reason for that is that this is where mathematician Mary Somerville came to relax when she was in Paris in 1817.
The family traveled here because Mary's health was poor - partly from having been working too hard in England.  Here are her own words:

"My health was never good at Chelsea [London], and as I had been working too hard, I became so ill, that change of air and scene were thought absolutely necessary for me. We went accordingly to Paris; partly, because it was near home, as Somerville could not remain long with us at a time, and, partly, because we thought it a good opportunity to give masters to the girls, which we could not afford to do in London. When we arrived, I was so weak, that I always remained in bed writing till one o'clock, and then, either went to sit in the Tuileries gardens, or else received visits"

I can see why she found the Tuileries Gardens relaxing and restorative.  The rest of this post will simply be pictures and a video.  Sadly I have no means of video editing, so it's rough - but hopefully make the place more alive anyway.

I wonder what Mary Somerville might make of it had she known that very nearly 200 years later someone else would come here to reflect and relax simply because she had done so once upon a time.

Wednesday, April 27, 2016

Sophie Germain

She's buried here at Pere Lachaise Cemetery, Paris.

She died here, in her home at 13 Rue de Savoie, Paris

But Sophie Germain (1776-1831) should be here - listed among the 72 French scientists, engineers and mathematicians recognized for their contributions.  In particular she should be here as her work on the elasticity of metals was crucial in the building of the Eiffel Tower.

As I mentioned in my post about Ada Byron Lovelace, prior to the 20th century women were discouraged - a better word might be disallowed - from going into mathematics.  The only exceptions that I am aware of are Ada Lovelace (1815-1852) and Hypatia (c. 360-415) - and things didn't really end well for either one of them.  Ada's work was forgotten until Alan Turing rediscovered it in the 1940s, and Hypatia was murdered by a mob.

Sophie Germain was born in Paris 13 years before the beginning of the French Revolution.  During her youth it was necessary for her to remain indoors due to the bedlam in the streets relating to the revolution, so she spent a great deal of time in her father's library.  It was here that she came across the story of the death of Archimedes in a book on the history of mathematics by J. E. Montucla.  As the story goes, Archimedes was working on a geometrical problem when his hometown of Syracuse was invaded by the Romans.  Archimedes was well-known and was credited with devising inventions that had held off the Roman invasion.  Soldiers were given orders to bring Archimedes back alive, but Archimedes incensed the soldier who came for him by saying, when that soldier's shadow obscuered his drawing in the sand, "Do not disturb my circles."  The soldier ran him through with a sword.

Sophie felt that if mathematics was THAT engaging then it was a topic that she needed to learn.

Sophie's parents were concerned about her studying mathematics and did their best to put a stop to it.  When they didn't let her do mathematics, she began to work on math in her bedroom at night.  When her parents realized what she was doing they took away her candles and her clothing so that it would be too cold and dark for her to do mathematics.  Sophie snuck candles into her room and wrapped up in her blankets for warmth.  Her parents finally relented when they found her asleep at her desk one morning with the ink frozen in the ink-horn.  It is said that after some time her mother secretly supported her.  Sophie never married, and her father supported her financially throughout her life.

ASIDE - As one small example of the bedlam going on in the streets of Paris, the row of kings on the facade of Notre Dame Cathedral was beheaded by the mob, thinking they represented kings of France, when actually they represented biblical kings of Judah.  Thankfully it was later restored.

West Facade Notre Dame
When Sophie was 18, the now-famous Paris university Ecole Polytechnique was opened, but, of course, as a woman, she could not attend.  Here too she found a way.  She used the identity of a young man who was no longer attending, Monsieur Antoine-August Le Blanc.  Under this pseudonym she was able to receive lecture notes and to submit work.  The person she began submitting her work to is one of those whose name can be found on the Eiffel Tower, Joseph-Louis Lagrange.

Sophie's intellect stood out, so much so that the great mathematician Lagrange took notice and asked for a meeting.  Thus Sophie had to give up her secret identity.  Thankfully, Lagrange did not mind that she was a woman, and he became her mentor, visited her home and also provided moral support.

Germain later became interested in number theory after reading work of Legendre, and she initiated a correspondence with him about number theory and elasticity.  He included some of her work in a supplement to one of his books.  Her interest in number theory was heightened when she read Carl Friedrich Gauss's famous work Disquisitiones Arithmeticae.  She also began a correspondence with him using her old pseudonym Monsieur Le Blanc.  With Gauss as well the secret was finally revealed - this time by way of her protecting him during a time of war.  It was about 1807, and the French were occupying the area near Braunschwieg where Gauss lived.  Sophie was concerned that Gauss might meet the same fate as Archimedes, so she wrote to a general in the army who was a family friend asking him to ensure Gauss's safety.  The general went to meet Gauss to make sure that he was safe, which was certainly a welcome gesture, but Gauss was puzzled when the general said he owed the visit to Sophie Germain - whom Gauss had never heard of!

Three months later Sophie admitted her identity to Gauss.  Here is his response:

"How can I describe my astonishment and admiration on seeing my esteemed correspondent M leBlanc metamorphosed into this celebrated person. . . when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius"

Sophie is one of many mathematicians who worked on Fermat's Last Theorem, and as a result of the specifics of that work a particular type of prime number bears her name.  A Sophie Germain prime is a prime number p where 2p+1 is also prime.

In spite of her achievements she never received a degree and even on her death certificate she is not named a mathematician or scientist but only as a property holder.  When the matter of honorary degrees came up as a topic at Gottingen it was six years after her death, and Gauss (the most prominent mathematician in the world at that time) lamented that she who had proved to the world that even a woman could contribute to that most rigorous and abstract of sciences, mathematics, could not receive the degree.

Here's to you Sophie!  Thinking of you and the intellectual heights you achieved from the height of the tower on which your name should appear!

Sunday, April 24, 2016

Lady Ada Augusta Byron King, Countess of Lovelace

Babbage's Difference Engine
Ada, Countess of Lovelace (1815-1852) was born Augusta Byron, daughter of the poet George Gordon, Lord Byron and his wife Anne Isabelle Milbanke.  They were married in January of 1815 but separated in January 1816 when Ada (as she was nick-named by her father) was one month old.  Four months later he left the country and never saw her again.  Despite this, and his reputation as a womanizer (one woman with whom he had a publicly well-known affair called him "mad, bad and dangerous to know"), Ada requested to be buried next to her father in the family vault at the Church of St. Mary Magdalene, Hucknall, England.
A poet of Byron's stature would normally have been buried in Poet's Corner at Westminster Abbey, and his remains were sent there, but the Abbey refused to bury him there for the reason of "questionable morality."  (The Abbey finally put in a memorial stone to him in Poet's Corner in 1969; he died in 1824.)  Following are more pictures of Lady Ada and Lord Byron's resting place at Hucknall, which seems quite proud to have them both there.

 The light you see shining up out of the grave is for a wreath that was given at the time of his death that has been well preserved and proudly displayed.
There's also an opening that allows for viewing into the vault.

Along with the grave, there are also wall plaques and a stone in the floor.  The marble stone in the floor was sent by the King of Greece.  Lord Byron had died in Greece - having given money to the Greeks for their war of independence from the Ottoman Turks.  His money went to refit the Greek fleet.  Byron had planned to be part of the attack on the Turkish-held fortress of Lepanto, but he fell ill and died of his illness.  The Greeks saw him as a hero, and the marble slab in the floor of the church in Hucknall was given by the King of Greece.

Though Ada never knew her father, she was fascinated by him and came to call herself a "poetical scientist."
Portrait of Lord Byron - Charlotte Bronte Bicentennial Celebration - National Portrait Gallery - London
Ada's mother, on the other hand, did everything she could to make sure that Ada did not grow up to be a crazy poet like her father.  Anne Isabelle (Annabella, as she was known) made sure that Ada's instruction was focused on mathematics and science in order to make sure she developed a disciplined and balanced mind - rather than an insane, poetic mind.  This is in strong contrast to other female mathematicians prior to the twentieth century, all of whom I know had a tremendous struggle to gain mathematical education due to societal or familial concerns that studying mathematics would be too much for a woman's mind to handle and would perhaps drive her mad.

One of these other female mathematicians who had had to struggle in order to study mathematics and whose family feared she would go mad (her father said they would have her in a strait-jacket if she didn't stop studying mathematics) was Mary Fairfax Somerville of whom I posted earlier.  She was 35 years Ada's senior and became one of Ada's math tutors.  Another of Ada's math tutors was Augustus de Morgan, but though he was her tutor, even he felt that mathematics was too much for the mind of a woman in general.  He wrote to her mother that women should avoid doing hard mathematics - "the reason is obvious - the very great tension of mind which mathematics requires is beyond the strength of a woman's physical power of application."  In the same letter he stated, however, that Ada unquestionably had as much power as would require all the strength of a man's constitution.

Mary Somerville and Ada developed a close friendship, as did their families.  Mary mentored Ada, and it was Mary who introduced Ada to Charles Babbage, he who developed ideas for early computing devices - the Difference Engine and the Analytical Engine - and is considered by some to be a father of modern computing.  He is described as a pre-eminent polymath among the many polymaths of his day - involved in mathematics, inventing, philosophy and mechanical engineering.

The pictures below and the picture at the very top of this post are of the uncompleted Difference Engine on display at the National Science Museum, London.

Babbage was not the only well-known personage with whom Ada was in contact. Among other acquaintances were Michael Faraday and Charles Dickens. Ada's life was one of privilege, and at one point she nearly became a lady-in-waiting to Queen Victoria.

I had opportunity as part of these studies to read the correspondence between the Byron/Lovelace and Somerville/Grieg families, and it was amazing to see this relationship unfold.  Early-on Ada's mother writes to Mary Somerville of her appreciation for the affect she has on balancing Ada's mind.  Later Ada herself writes to Mary and sweetly asks her if she will be her chaperone to go to parties put on by Babbage so that she could interact with him and learn more about his computing machines.  Ada threw herself into understanding the Analytical Engine, and saw, even more than Babbage did, the full extent of what it could do.  She saw that the hardware was only half of the story - and that the computer needed software if it were to be able to calculate any type of equation (Babbage's vision having been mostly about number-crunching and creating tables of logarithms).
Ada Byron Lovelace - from a display in St. Mary Magdalene Church, Hucknall

Display at St. Mary Magdalene Church, Hucknall

From a display about Ada in St. Mary Magdalene Church Hucknall
In her twenties Ada said to her mother that she wanted to compensate for her father's misguided genius, expressing that if he had transmitted any portion of his genius to her that she wanted to use it to bring out great truths and principles.  She is indisputably the world's first published computer programmer.  She saw the possibility of interchanging numbers and symbols, and one example of her vision is that of an engine that might compose elaborate, scientific music.  After her death her works seems to have been forgotten for a while, but when Alan Turing was working on building The Bombe to help break the German Enigma Code during WWII he came across Ada's work, and it is thanks to him that her work has come fully to light.  A new programming language developed in 1986 and used by UK air traffic control is called ADA, which is very fitting since Ada had invented a mechanical bird when she was young; now Ada has finally gotten to fly!

There's so much I wanted to share about Ada that I fear this post is very disjoint!  I think that any commentary about her life needs a book, or maybe an epic poem, rather than a blog post!  But this is the best I can do for now.
Lord Byron's memorial outside St. Mary Magdalene Church, Hucknall, England
Ada died at age 36, the same age at which her father had died - he of fever, she of cancer.  Because of the closeness of their families, Mary Somerville's son Woronzow Grieg wrote to Ada's doctor asking him to send on any news.  I know this from the reading I did of the correspondence in the Bodleian at Oxford, and when I read the letter from the doctor I was in tears.  The doctor let Woronzow know that it was cancer and that there was no hope.  He ended his letter by saying that while a doctor is necessary in such circumstances, the best a doctor can ever hope for is to RESTORE health, but in this case the best he could hope for was euthanasia - in other words a "good death."

On her father's memorial outside the Hucknall church is a quote from his work Childe Harold's Pilgrimage, "But there is that within me  .  .  .  that shall breathe when I expire."  I'd certainly say that is true of Ada!