“Proof:” Carl Friedrich Gauss & Sophie Germain

By Barry Honold

Author Elizabeth Klaver asks, why does “Auburn choose the prime numbers and not some other area of mathematics?” Her answer: Sophie Germain, Catherine’s character, and Germain Primes provide a rather “nice connection.” Catherine is working on a proof involving prime numbers. Germain was one of the first professionally-accepted female mathematicians, and her story ties directly into that of her mentor: Carl Friedrich Gauss (1777 – 1855).

Gauss is arguably the greatest mathematician who ever lived. His skill first manifested itself at the age of nine. He was clearly meant to enter academia. Under Gauss, Germany would become the “mathematical Mecca of Europe.” A colleague thought him the perfect man to finally solve Fermat’s Last Theorem. Gauss declined, replying that one could easily “lay down a multitude of such provisions, which one could neither prove nor disprove.” He made a number of advancements in optics, map-making, magnetism, non-Euclidean geometry, and invented an early telegraph. He was a master of the Romance languages and spent his sunset years learning Sanskrit and mastering Russian to correspond with colleagues there in their native tongue. He was, to quote one biographer, “a master in the deepest and most abstract questions of knowledge.”

He corresponded with many mathematicians on the Continent, though he never left his native Germany. One in particular fascinated him: an inquisitive and bright young Frenchman by the name of Monsieur Leblanc. When the French occupied his native Brunswick, an officer paid a visit, and said that he had been instructed by a certain Mademoiselle Germain to inquire about Gauss’ health. Gauss was confused; the name “Germain” meant nothing to him. His colleagues began to needle him about keeping a Frenchwoman as a lover.

Three months later, the mystery was solved: in a letter, one Sophie Germain admitted to having “previously taken the name of Leblanc” in writing to Gauss. Gauss was an open-minded individual, and pleasantly surprised at her little trick. He replied:

“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 familiarizing 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.”

Germain (1776 – 1831) defied all expectations of her time. She received a Grand Prix from Napoleon for her work in acoustics and prefigured Comte’s positivism. When she was thirteen, she was in her father’s study reading a copy of History of Mathematics. Upon reading the legend of Archimedes’ death, she developed an interest in mathematics. Her parents were fiercely opposed to her studying math, so she studied at night. When they discovered this, they took her candles. She responded by hiding candles, and using them after bedtime. This escalation continued until her parents gave up in exasperation. Later, she realized that she needed to know Latin to read Newton and Euler. Having no instructors, she went about it “alone and unaided.” 

Germain realized that this bias against her would be the norm, and adopted the name of “M. Leblanc” to correspond with noted mathematicians. One in particular, Joseph Louis Lagrange, took note of her skill and the two routinely wrote.  As Jess Fernandez-Martinez wrote, she “procured for herself students’ notebooks,” particularly those in Lagrange’s classes. She then went over them thoroughly, and began sending in notebooks of her own. Lagrange was immediately intrigued by this student’s exceptional skill and began touting the pupil to anyone who would listen. Lagrange arranged to meet “him,” and met her. So delighted was he that he lent his prestigious name to her cause, and she was soon meeting with some of the foremost minds in France.

The French government had a call to solve a problem regarding acoustics, which she did on her third try, while none of her male colleagues would touch the problem. Gauss recommended that she be granted an honorary degree. Unlike him, she did try her hand at proving FLT.  As a result, we have the Germain Primes, without which FLT would have taken even longer to prove. Unfortunately, she died of breast cancer before receiving her honorary degree.

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“Proof:” Fermat, Wiles, & Auburn

By Barry Honold

David Auburn, in deciding how much math to include in Proof, had to strike a delicate balance: though math is an integral part of the story, it is not a “math play.” However, he did try “to get in as much kind of math lore as possible” One central theme is that of the lone genius. Catherine, the lone genius in Proof, has many real-life counterparts in math and science. However, none may be more fitting than Princeton University’s Andrew Wiles.

In 1993, Wiles flew to a conference in England. There was a steady drip of rumor regarding what he had been secretly working on. This, writes Amir Aczel, took seven years and kept Wiles “a virtual prisoner in his own attic.” He was allotted an unusual three hours of lecture time. When he arrived, he kept to himself and discussed nothing. This uncharacteristic secrecy from a colleague sparked curiosity, and attendance to his lectures soared.

As he spoke, it became obvious what problem he was about to solve. For over 300 years, it had gone unsolved by the best and the brightest in math. History was made with the utterance of two simple sentences: “And this proves Fermat’s Last Theorem. I think I’ll stop here.” The lecture hall exploded in a standing ovation. Wiles had done it. The cameras snapped and the global press was soon in contact.

Calvin Clawson writes that Pierre de Fermat, after whom Fermat’s Last Theorem was named, may be “the greatest mathematician of the seventeenth century.” Only after his son posthumously published his work did he receive the acclaim he so rightly deserved. Fermat was a lawyer by education, who spent his career performing nominal legal duties that allowed him to fervently pursue his hobby: math.

If Fermat the amateur is judged on both quality and sheer volume of his work, he outshines many professional mathematicians. What he became famous for was a margin note. After his posthumous publication, a scribble was discovered in the margin of a Fermat manuscript. Per Aczel, it stated that “ has no whole number solution if (n) is greater than 2.”

Fermat wrote that he had a proof for this formula, but “the margin is not large enough to contain it.” The equation went on to bedevil mathematicians for over 300 years. In the play, Hal says that Catherine’s proof regards “a mathematical theorem about prime numbers, something mathematicians have been trying to prove since…there were mathematicians. Most people thought it couldn’t be done.” That was most certainly the consensus in the math community about Fermat’s Last Theorem for three centuries. Until Wiles.

Richard Hornsby writes that Proof is about “the obsessive, fascinating, all-consuming process of doing mathematics” and that Auburn is “second to none in depicting all three mathematicians’ enthusiasm for their work.” The play was never meant to be based on the events of 1993 – 1995. But the mathematicians all share Wiles’ enthusiasm and obsession. He described proving FLT (its shortened name) as akin to “entering a dark mansion. After some scientific, trial-and-error fumbling about, you learn the location of the furniture. Then you locate that room’s light switch. Then, it’s on to the next room to repeat the process.”

 Catherine and Wiles both built upon centuries of work for their achievements. The completion of their work would have been impossible if not for their predecessors. According to Hal, Catherine’s proof was very “hip,” and utilized a “lot of newer techniques” that were developed after her father’s time. Wiles’ final proof relied heavily upon post-WWII advances in math. Wiles, in essence, merely completed the process. Both lone geniuses merely completed a process started long before they were even born. They were the final, elegant piece of the mathematical puzzle.

Published in: on January 20, 2010 at 5:52 pm  Comments (1)  
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“Proof:” Men, Women, & Math Skills

By Barry Honold

In Proof, Catherine is depicted as a math prodigy. Playwright David Auburn has said that he was fascinated with the idea of a young genius toiling away in the small hours of the night and producing a work that would revolutionize an entire field. Likewise, Tom Stoppard’s hit play Arcadia features a young and exceptionally talented female protagonist.  Both deal with advanced mathematical concepts: chaos theory in Arcadia and Fermat’s Last Theorem in Proof.

The sex of the protagonist is significant, largely because it is not considered the norm. Most stories about the hard sciences involve men. And, in scientific and mathematical jobs around the country, men outnumber women. Auburn wrote about “a panel of women mathematicians [that] used [Proof] to discuss questions of sexism and bias in their profession” at New York University. Bias against women in math is common.

In 2005, former Clinton Administration official and Harvard President Larry Summers spoke at a diversity conference on campus. He began musing on the disparity between men and women in tenured positions. As it was reported, he had a three-part theory: high-power, high-prestige jobs require sacrifices most women seem unwilling to make; it is possible that men have more “intrinsic aptitude” for advanced math and science; and the normal glass ceiling problems. “In my own view, their importance probably ranks in exactly the order that I just described,” he finished.

The reaction was immediate. MIT biologist Nancy Hopkins walked out, saying that if she did not leave, she would have “blacked out or thrown up…this kind of bias makes me physically ill.” Feminist provocateur Camille Paglia immediately pounced on this remark, expressing sympathy that Hopkins’ doctoral degree students were unfortunate enough to have a professor prone to “swooning” when confronted with an idea she found disagreeable. The left, right, and center talking heads on TV began blasting away at each other.

Time ran an entire issue devoted to the “gender gap” shortly after the flap, asking “[i]s it true, even a little bit, that men are better equipped for scientific genius?

As usual, there is no simple “yes” or “no” answer. One of Summers’ cited sources was the University of Michigan’s Yu Xie, who said that “I didn’t exclude biology as an explanation. But I know biological factors would not play a role unless they interacted with social conditions.”

There are certainly role models for women in math: Hypatia, probably the first prominent female mathematician. Her father taught at the Library of Alexandria. Or Maria Agnesi, noted for differential and integral calculus. Or one of Voltaire’s associates, Emilie du Chatelet, the French Enlightenment mathematician who translated Sir Isaac Newton’s Principia Mathematica.  

In fact, if there are any intrinsic differences, recent research suggests they can be accounted for and bypassed. In “The Gender Gap in Math: Its Possible Origins in Neighborhood Effects” the researchers devised a contextual study to look at the disparity. Essentially, the normal socialization of boys encourages them to be outside of the house more than girls. Boys are more likely to be encouraged to play sports that involve math in some form and explore their community (and thus access libraries, after-school centers, and museums). However, girls who have access to these resources do equally well. Education and affluence tend to trump biology.

Another article, “The Myth of the Gender Gap,” pointed out recent studies that have shown girls’ math scores and participation to be pulling even with the boys. This “may reflect the simple fact that more female students are now taking math courses.” It furthermore discovered that “girls are increasingly sticking with math classes through school…girls and boys take advanced math in high schools in equal numbers, and women receive nearly half of all bachelor degrees in the US – and their scores are closing the gap.” The stereotype about boys doing better in math persists, though.

 Allanah Thomas, who teaches adult women math skills, explains that “[w]hat often holds girls back is self-confidence; it drops sharply in middle school.” One unlikely ally is the video game “Tetris.” Time writes that when girls played the game after math tests, their spatial reasoning skills shot up dramatically. High schools and colleges can make their math departments more “women friendly” by hiring more female math faculty. The most important thing is for parents and teachers to encourage their girls to work at math, and refuse to give up.

Published in: on January 20, 2010 at 5:50 pm  Leave a Comment  
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DARPP-32: Genius & Madness in “Proof”

By Barry Honold

David Auburn’s Proof deals with a recurrent theme in literature – the alleged link between genius and madness. Catherine’s father, Robert, almost certainly suffers from schizophrenia. According to the National Institute for Mental Health, it is a “chronic, severe, and disabling brain disorder that affects about 1.1 percent of the U.S. population age 18 and older ….People with schizophrenia sometimes hear voices others don’t hear, believe that others are broadcasting their thoughts to the world, or become convinced that others are plotting to harm them. These experiences can make them fearful and withdrawn and cause difficulties when they try to have relationships with others.” Robert is also a mathematical genius who did some of his best work before his mid-twenties.

In Proof, Catherine may have inherited some of her father’s illness: she has a perfectly lucid conversation with him wherein he reminds her that he is, in fact, dead. We are never told whether this is a dream or a delusion. In a flashback, she discovers him working in the cold on the front porch. He convinces her that his illness is in remission and that he has resumed real work again. She is cautiously optimistic, but when she looks over his work, realizes that it is gibberish. Interestingly enough, Auburn wrote of meeting with one woman at a book signing: she told him that her mathematician father had a nervous breakdown and she had “spent her whole life caring for him.” Then, she said, “This is the story of my life. How did you know?”

The idea of some link between genius and madness is not new: it stretches back from the Roman orator Seneca, through the Middle Ages with Paracelsus, to Van Gogh’s self-mutilation and the description of Lord Byron as “mad, bad, and dangerous to know.” Not just Van Gogh, but Edvard Munch, Paul Gaugin, and Jackson Pollock also suffered from mental illness. Derek Hillard, writing in “The Rhetoric of Originality: Paul Lolan and the Disentanglement of Illness and Creativity,” writes that the notion of madness sharing “ground with originality is at first not evident: both the products of genius and madness may seem to be fancy; they neither ground themselves on a universally accessible logic nor define themselves based on previous grounds.” Auburn discovered a similar idea while reading A Mathematician’s Apology, by G.H. Hardy, while researching the play. Hardy had written that, “in a good proof, there is a very high degree of unexpectedness, combined with inevitability and economy.”

And here we have the difference between Robert and Catherine’s toil: his, a work of insanity which could never hope to survive peer review. Hers is a work of genius that will cement her name in the annals of brilliant mathematicians.

Vanderbilt psychologists Brad Folley and Sohee Park did a study that showed creative people to be more likely to suffer from depression and schizophrenia. The most well-known case involving a mathematician is John Nash, Jr. (portrayed by Russell Crowe in A Beautiful Mind).

Nash is a seminal figure in the field of Game Theory, and formulated the Nash Equilibrium. Game Theory analyzes how people, states, and organizations behave, usually in competition. His work was first applied to Cold War strategy and later to economics. After he received his master’s degree in math, his advisor wrote a one- sentence recommendation letter: “This man is a genius.” He began having auditory hallucinations and believed himself to be persecuted by the Pope and President Eisenhower. He was diagnosed with paranoid schizophrenia, hospitalized, and treated with Thorazine and psychoanalysis. His hallucinations lasted, in varying degrees, over the years without ever really ceasing. In 1994, he was awarded the Nobel Prize in Economics.

As science has advanced over time, madness has gone from something viewed as quasi-spiritual to a simple medical condition. As such, madness has been de-mystified, is diagnosed by doctors, treated with pharmaceuticals, and its storied history as a link to the divine severed.

Now we have a more prosaic explanation: DARPP-32. A UK Daily Mail article explains that DARPP-32 is a gene that controls thought and action, but is also partially responsible for schizophrenia. When “genes and environmental events” create an information overload for the brain, everything backfires, and may result in schizophrenia.

The “antic disposition” of Hamlet and Lear screaming into the storm gave Romantic writers an iconic image to emulate: the tortured soul. They took the step of rooting the cause of that torture in over-reliance on cold reason and the rejection of nature and the spiritual. It is certainly poetic that “cold reason,” if anything, will be what finally ends or ameliorates madness.

The Foundations of Modern Math: Pythagoras, Euclid, and Archimedes

Barry Honold, Dramaturg

According to Calvin C. Clawson’s Mathematical Sorcery: Revealing the Secrets of Numbers, the earliest archaeological evidence for counting is “two animal bones which show clear grouped marks.” These are also called tally sticks. The first is a baboon’s thigh bone discovered in the Lebembo Mountains in Africa and dates back to 35,000 years ago. The second is a wolf bone that is 33,000 years old. It is, he notes, “especially intriguing” because it is “notched with fifty-five marks, grouped in eleven sets of five marks each.” Other early counting methods were developed by Egyptian and Babylonian farmers and bureaucrats for record-keepin and determining crop size.

Pythagoras (ca. 570 – 501 BC) – Pythagoras was born on the island of Samos in the Aegean Sea. He was the son of a prosperous merchant, In “Pythagoras, Son of Mnesarchos,” author Nancy Demand surmised that his extensive travels were business trips made with and for his father. In 529, he went to Sicily, then Tarentum, before finally settling in Croton. Amir D. Eczel explained that he was in an ideal locale to have seen all Seven Wonders of the Ancient World. His philosophical order was founded was the assent of the local government, allowed women to join, and was very much a secret society. It was also strict: one initiate who was overly garrulous was thrown into the sea and drowned. Eczel writes that they “followed an ascetic lifestyle, and were strict vegetarians.” He saw that many aspects of nature were cyclical (the tides, day and night, lunar phases, etc), and extrapolated that everything was understandable via numbers. He discovered the concept of odd/even integers. The Pythagoreans believed that everything in nature was essentially mathematical, and that even concepts like justice and friendship had correlating numbers. He is recognized as one of the greatest mathematicians of all time, with Carl Friedrich Gauss, and is most known for the Pythagorean Theorem and Pythagorean Triples.

Euclid (365 – 300 BC) – He was invited by Ptolemy to head the Academy and Library of Alexandria after the death of Alexander the Great. Aside from that, we know very little of his life. His greatest work is the Elements, and Eczel writes that the first two volumes of this work are believed to be entirely influenced by the work of Pythagoras and his society. Euclid is viewed as more of a compiler of Greek mathematics than an innovator. Clawson writes that, using five postulates and axioms, Euclid “logically deduced all of his theorems of geometry. This was a monumental achievement and served as a model for all of mathematics right up to the twentieth century,” selling second only to the Bible. “The Elements was used as a geometry text for two thousand years.”

Archimedes (287 – 212 BC) – Archimedes is considered by some to be the pre-eminent mathematician of antiquity. Marshall Clagett writes that Galileo revered him and described him “in almost Homeric hyperbole.” Eczel writes that his work pre-figured integral calculus and differential calculus. When he suspected that he was being plagiarized, he would intentionally fudge an equation, and later point out the error to the embarrassment of the thief. He invented the Archimedes’ screw, which is a drill that pulls water out of the ground and is still used today. He discovered the first law of hydrostatics, which says that “a submerged body loses from its weight the weight of the liquid it displaces,” via an apocryphal tale: the king was concerned that one of the local gold merchants was dishonest. He asked Archimedes to find out the truth. Archimedes began using his own body to determine weight. He ran some numbers, and arrived at the truth. “When he discovered the law, he jumped out of the bath and ran naked through the streets of Syracuse shouting ‘Eureka! Eureka!’ (I found it!).”

He used his mathematical skill to help defend Syracuse against the Romans via catapults. This gives us another story: while he was doing geometry in the sand, a Roman soldier approached him and ordered him to leave. Archimedes was too caught up in his work, and didn’t hear the soldier. The soldier thought he was being ignored and angrily drew his sword. He walked too close to the figures, and Archimedes ordered him away. The soldier killed him. This was reported by Roman general Marcellus in the Second Punic War. According to Mary Jaegar, “Archimedes’ death was a sore point for the Romans – their commander Marcellus is said to have been both angered and aggrieved. Cicero later sought out and found his grave. Then, the grave disappeared to history. In 1963, construction workers found his tomb while beginning work on a hotel.

Published in: on January 20, 2010 at 5:42 pm  Leave a Comment  
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