Dafne at Palazzo Corsi

The splendid Palazzo Corsi Salviatti in Via Tornabuoni conceals a series of curious historical facts that are still relatively unknown to the general public. These include the fact of having been built around a piazza, the apartments of the “Flash Pope” and the room where the very first opera in the world was held, making this one of the most original buildings ever to be erected in Florentine Renaissance style.

Today the structure, which sits at the intersection of Via degli Strozzi and Via de’ Tornabuoni in the heart of Florence, is part of the Palazzo Tornabuoni, a Four Seasons–managed private residence club. The space where Dafne was performed more than 400 years ago is now one of 37 residences — each dramatic in its own right — available for whole or fractional ownership at Tornabuoni.

Library in present-day Palazzo Corsi Tornabuoni

The Brunelleschi suite sits on the third floor of the palazzo, above the library, and overlooks Renaissance architect Filippo Brunelleschi’s Duomo.

Other residences include the Strozzi suite and the Galileo, with its own private rooftop terrace with a 360-degree view of Florence.

Galileo suite roof top terrace

The private residence club has 10 club suites (available for share ownership) and 27 private ownership suites (at prices from about $2.5 million to $6.8 million – 2009 dollars).

Turning the palazzo into a residential property required extensive renovation that took five years,led by Florence-based architect and interior designer Michele Bonan. The renovation involved maintaining the architectural integrity of the structure, updating it with modern systems, and restoring its sculptures, frescoes, and other objets d’art. The Department of Fine Arts in Florence supervised the restoration of the palazzo’s artworks, which took 14 artisans three years to complete.

Upon entering, the huntress Diana welcomes you. Resplendent salons and suites follow, each a timeless repository of successive generations of frescoes and sculptures.

The palazzo dates to about 1450, when the architect Michelozzo constructed it as a private residence for the Tornabuoni family. Since then the building has changed hands several times. Alessandro Ottaviano de’ Medici, who would become Pope Leo XI, acquired the property in 1574. Then, in the 1590s, Corsi took ownership of the property, which remained in his family for more than three centuries. In 1901, the building became a bank, and it served in that capacity until the Florence-based Fingen Group purchased it in 2004.

The palazzo rises up over the foundations of three pre-existing medieval buildings that once belonged to the Consorteria of the Tornabuoni and Tornaquinci families; and in one of these Lucrezia Tornabuoni, Lorenzo il Magnifico’s mother, was born and spent her childhood.

The three houses faced onto an internal piazza connected by two lanes to the adjacent main streets. Commissioned by Giovanni Tornabuoni, famous architect Michelozzo Michelozzi took inspiration from this piazza to build the present day palazzo all around it, creating the splendid courtyard that can still be seen today.

When Alessandro de Medici, Archbishop of Florence, became the owner of this palazzo, he transferred the city archbishopric here temporarily seeing that the old headquarters in Piazza San Giovanni had been devastated by fire and required extensive restructuring and renovation. The Archbishop surrounded himself with renowned artists like Agostino Ciampelli and Lodovico Cardi, known as “Il Cigoli” for these building and redecorating operations. Ciampelli was famous for having frescoed various halls to the theme of the Old Testament, whereas Il Cigoli, in his role as architect, had previously restructured a loggia located at the corner of the building facing onto Via de Ferravecchi (old irons) and Via de Belli Sporti (beautiful architectural projections), now via Strozzi and Via Tornabuoni, known as “Canto a Tornaquinci”. Alessandro de Medici’s rapid ecclesiastic ascent soon took him far from Florence. On becoming cardinal he moved to Rome taking with him his faithful Ciampelli, and at the age of eighty, he was nominated Pope with the name of Leone XI. However this was an extremely short-lived pontificate that lasted a mere 27 days (1-27 April 1605), so short in fact that he was called the “Flash Pope” by the people of that era.

In the meantime the palazzo had passed into the hands of the Corsi, an ancient Florentine family which boasted a history of priors and gonfaloniers of the republic. Jacopo Corsi was certainly their most outstanding member. A refined benefactor and great music lover, he was in the habit of holding gatherings in the palazzo with a selection of the greatest poets and musicians of that time, like Claudio Monteverdi, Torquato Tasso, Ottavio Rinuccini, Jacopo Peri and Giovanbattista Marino, who called themselves the Academy of Music.

Jacopo Peri was born in Rome but relocated to Florence to study music. In the 1590s, he met Jacopo Corsi, the leading patron of music in Florence, and they decided to recreate a form of Greek tragedy, following in the footsteps of the Florentine Camerata, which had produced the first experiments in monody.

Jacopo Peri

Based on an account by Jacopo Peri it is commonly thought that the first performance of Dafne took place in 1594. However Peri’s account is misleading. He might have meant that Jacopo Corsi and Ottavio Rinuccini requested him to compose Dafne in 1594 or that he composed Dafne in 1594 at the request of Corsi and Rinuccini. He definitely did not state that the first performance was held in 1594.

The only definite date of performance of Dafne is given by Marco da Gagliano. He says that Jacopo Corsi had Peri’s Dafne performed in the presence of Giovani Medici and some of the principal gentlemen of Florence during the carnival in 1597. He does not specifically state that the performance took place at Palazzo Corsi nor that it was the first performance but it is reasonable to interpret his remarks to that effect.

Ottavio Rinuccini, who composed the libretto to Dafne, did not a date of performance. He merely said that Dafne was performed before a few enthusiastic listeners and later in an improved form of the text at Palazzo Corsi before a large audience of Florentine noblemen, the Grand Duchess and the cardinals Del Monte and Montalto. That date could have been around 18 January 1599 when the household accounts of Palazzo Corsi show expenses incurred for a performance of Dafne. Sala delle Muse at Palazzo Corsi seems like the obvious choice of venue.

On 21 January 1599 Dafne was performed again, this time at Palazzo Pitti, before the same cardinals and a large audience of Florentine nobility.

From the household accounts of Palazzo Corsi, we know that Dafne was again performed at the palace in late August 1600 and it was perhaps for this performance that the libretto was printed.

Dafne was revived on 26 October 1604 at Palazzo Pitti, in Sala Bianca, in honour of the Duke of Parma. A libretto of this performance exists.

Cover page of Dafne libretto

The libretto by Ottavio Rinuccini survives complete, but Peri’s score has been lost. The surviving music fragments are by Jacopo Corsi, who was the first to compose parts of Rinuccini’s text.

Peri’s later composition, Euridice, written in 1600 based on a libretto by Ottavio Rinuccini, is the earliest surviving opera and was initially performed as part of the wedding festivities of Maria de’ Medici and Henry IV of France, thereby catapulting opera into the mainstream of court entertainment. Some of the music used in the first performance of L’Euridice was composed by Peri’s rival at court, Giulio Caccini.

Cover page of Euridice libretto

Vienna opted for 1598 as the year of first performance of Dafne and in 1998 it celebrated opera’s 400th anniversary with the event Universe of Opera held over three days and showcasing 53 internationally known singers offering solo arias and duets.

WA Opera Mezzo-soprano Fleuranne Brockway at the Florentine Festival – the opening of the Corsini Collection exhibition at AGWA

Fleuranne was selected in 2017 to be one of West Australian Opera’s Wesfarmers Young Artists and she is the winner of the 2018 Royal College of Music Scholarship, offered jointly by Australian International Opera Awards with the Royal College of Music in London.

Mathematician to know: Maria Gaetana Agnesi

One more sleep, one more sleep!

How about one more story? From Italy?

Renaissance Europe, for all its splendour, did not offer many scholarly opportunities for women, unless they chose to join nunneries. A notable exception was Italy, which espoused a more enlightened view that allowed a few women to flourish in the arts, medicine, literature, and mathematics. Among the most notable of mathematically minded women of the era was Maria Gaetana Agnesi.

The eldest of 21 children – her father married three times – Agnesi was born in 1718. (On May 16 this year we celebrate the 300th anniversary of her birth.) She was very much a child prodigy, known in her family as “the Walking Polyglot” because she could speak French, Italian, Greek, Hebrew, Spanish, German, and Latin by the time she was 13. By her late teens, she had also mastered mathematics.

Agnesi had the advantage of a wealthy upbringing; the family fortune came from the silk trade. She also had a highly supportive father, who hired the very best tutors for his talented elder daughter. Unfortunately for the shy, retiring Agnesi, he also insisted she participate in regular intellectual “salons” he hosted for great thinkers hailing from all over Europe. She was expected to recite long speeches from memory in Latin or participate in discussions about philosophy or science with men who made it their life’s work. Her younger sister, a brilliant harpsichordist and composer, was also tapped to impress visitors with her extraordinary abilities.

In the summer of 1727, a particularly noble gathering was held in the garden of the Palazzo Agnesi. Maria Gaetana, aged nine, declaimed from memory a long Latin oration against the rooted prejudice that women should not be allowed to study and practice the fine arts and the sciences. Among those in attendance were senators and magistrates. The child’s remarkable performance caused much enthusiasm among the guests, who decided to publish in her honour a pamphlet that included the oration and a series of poetic compositions in various meters and languages. The latter are, in general, as nebulous and pompous as most of the Arcadian poetry of the time; in contrast, the oration stands out as a clear and effective defense of the right of women to the pursuit of any kind of knowledge.

During the 1730s, Maria Gaetana debated topics in natural philosophy and mathematics in a series of disputes with her father’s guests. Manuscript material held at the Biblioteca Ambrosiana tells us about Agnesi’s cursus studorium in those years: lists of Latin terms and their Greek and Hebrew translations; a Latin pamphlet on mythology and its Greek translation; a Latin text on the life of Alexander the Greate translated into Italian, French, German and Greek.

In 1738, at the age of twenty, Maria Agnesi completed her studies by publishing a list of philosophical theses, Propositiones Philosophicae, most of which she had defended in the disputes held at her father’s palazzo.

Maria Gaetana Agnesi

By the time Agnesi’s Propositiones Philosophicae went to press in 1738, Milanese salon culture had entered a period of stagnation that would last for nearly two decades. This culture relied essentially on a few families and on small and ephemeral private academies. From 1734, when Lombardy became involved in the war of the Polish succession, and throughout the war over the Austrian succession (1740-1748), many salons and academies interrupted their scientific activities.

Yet in 1739 the Palazzo Agnesi was still at the centre of Milanese social life, thanks to the brilliant performances of the filosofessa. Agnesi was requested by her “most loving father” to attend an increasing number of “salons”. One of these was particularly remarkable. The heir to the throne of Poland had been visiting Milan and was invited to attend events at the palaces of the great patrician families: the Borromeo, the Simonetta and the Pallavicini. On a December evening, the prince, “followed by a number of the most qualified and erudite nobles”, visited the Palazzo Agnesi. Pietro received them “with great joy”; the palace was adorned with plentiful decorations and lights. The structure of the gathering was familiar: Maria Gaetana debated the guests on topics in natural philosophy (including the explanation of the tides, for which she referred to Newton). A report of the evening appeared in the pages of the Gazzetta di Milano.

A few months earlier, in July 1739, Charles de Brosses had attended a similar meeting at Palazzo Agnesi. There he found some thirty people from across Europe in a circle around Agnesi, who sat on a sofa awaiting questions and challenges. In his rusty Latin, the Frenchman debated with Agnesi on subjects of his own choosing – the relation between soul and body, perception, the propagation of light and the nature of colours.

De Brosses admired her intellectual prowess greatly, he described Agnesi as “something more stupendous than the cathedral of Milan”, and expressed his horror upon learning that she wished to become a nun. Antonio Frisi effectively describes Pietro Agnesi’s reaction to this request: “It was as if he had been struck by lightning.” Frisi refers to long discussions and negotiations between father and daughter. Eventually, Agnesi declared herself convinced that “God had destined her to live in the world” and to assist and relieve “suffering humanity”. She agreed to maintain her lay status, but only on certain conditions that would make her life an unusually private one. Occasionally, to please her father, she would still participate in the “salons” at the palazzo. But her glittering public career was at an end.

During the same period, Agnesi increasingly turned toward mathematics, “the only province of the literary world where peace reigns.” As with so many other academic pursuits, she took to it immediately. She studied amid globes and mathematical instruments, ploughing through calculus before anyone else in Milan was studying it.

Largely self-educated, Agnesi had the good fortune to find a mathematical mentor in a monk named Ramiro Rampinelli, a frequent visitor to the Agnesi home, who directed her study of calculus. He also encouraged her to write a seminal mathematics textbook, Analytical Institutions, and through his influence, she was able to gain the input of Jacopo Riccati, one of the leading Italian mathematicians of the day, while writing her seminal manuscript. She revised the draft text to incorporate Riccati’s comments.

Perhaps Agnesi began her book project as a way to pass her knowledge on to her younger siblings. Or maybe she realized how annoying it was to have mathematics instruction siloed into individual branches and one-off books so that getting an education required hunting down a whole collection of resources and hiring a tutor to fill in the gaps. Whatever the case, Agnesi saw a need for a unified textbook covering algebra, geometry, and calculus, so she wrote one.

Agnesi wrote the book in Tuscan, the dialect that would become modern Italian, instead of her own Milanese. Because she chose Italian over Latin — the language of scholars and one she knew well — it appears the text was aimed at a school-age population from the very beginning. Analytical Institutions would provide generations of Italian students with a solid and well-rounded mathematics education.

As was Agnesi’s style, when she decided to take on a project, she went big. In 1748, Agnesi published a two-volume, 1,020-page text called Instituzioni Analitiche (Analytical Institutions), believed to be the first mathematics book published by a woman. Thanks to her father’s wealth, Agnesi arranged for a private printing of the book, ensuring she could oversee the book’s typesetting and verify that her formulas were accurately represented. If a particularly unwieldy equation ran past the bottom of the page, it was printed on a long sheet of paper that was folded up and tucked into the regular-size pages.

Volume one covered arithmetic, geometry, trigonometry, analytic geometry and calculus. Volume two included discussions of infinite series and differential equations. In the preface, Agnesi paid tribute to her monkish mentor, declaring that without Rampinelli’s help, “I should have become altogether entangled in the great labyrinth of insuperable difficulty… to him I owe all advances that my small talent has sufficed to make.”

In England, John Colson, a professor at Cambridge, heard about the book and the impact it was making abroad, and felt that British students urgently needed access to the same information. Colson was getting on in age, so he scrambled to bone up on his Italian in order to translate Agnesi’s text. He hadn’t yet published the translated manuscript when he died in 1760. The work was finally released in 1801 in English, thanks to a vicar who edited and shepherded it through the publication process.

More than 250 years later, Agnesi’s name continues to appear in calculus textbooks: she lends it to a curve that rolls over a sphere like a gentle hill. She wasn’t the first to discover the curve, although it was assumed she was at the time; mathematics historians found someone who had claimed it earlier. The “witch of Agnesi”, as the curve is called, is actually the product of a mistranslation. In Instituzioni Analitiche, Agnesi calls her cubic curve versiera, which meant “turning in every direction”. Colson translated it as versicra, or “witch”.

Among other phenomena, this curve describes a driven oscillator near resonance; the spectral line distribution of optical lines and x-rays; and the amount of power that is dissipated in resonant circuits. Today, Agnesi’s curve is used primarily as a modelling and statistical tool. Some computer models for weather and atmospheric conditions, for example, use Agnesi’s curve to model topographic peaks of terrain. It can also be used as a distribution model, substituting for the standard bell curve model in statistics. It can be difficult to integrate over a specified range using the bell curve; the Witch of Agnesi’s algebraic expression is relatively straightforward in contrast, and thus easier to integrate.

Most biographies, while admiring, feel compelled to note that Agnesi’s seminal tome contained “no original mathematics”. Her accomplishment was noteworthy in part because Agnesi’s gift for languages enabled her to read mathematical papers from around the world and synthesize those works in a single text. Notably, Analytical Institutions was the first tome discussing calculus that included the very different methods developed by co-inventors Isaac Newton and Gottfried von Leibniz.

Even before publishing her textbook, Agnesi had been invited to join a number of learned academies in Italy, including the Instituto delle Scienze in Bologna. After its publication, in 1748, she became famous. Letters of congratulations were sent by numerous personalities, including Laura Bassi (also a member of the Academy of Sciences of Bologna), Jacopo Riccati, Giovanni Poleni, Etienne de Montigny (who read and commented very favourably on the book on behalf of the Académie Royale des Sciences) and the plenipotentiary minister Gian Luca Pallavicini, writing on behalf of the Empress Maria Theresa, who also sent her a diamond ring and jewel-encrusted box.

“For if at any time there can be an excuse for the rashness of a Woman who ventures to aspire to the subtleties of a science, which knows no bounds, not even those of infinity itself, it certainly should be at this glorious period, in which a Woman reigns…” — Maria Gaetana Agnesi’s dedication of her book Analytical Institutions to Maria Theresa, Holy Roman Empress, in 1748.

Father François Jacquier – a protégé of Benedict XIV, a professor of physics at La Sapienza University, and co-author of the translation of the Principia that legitimated Newton’s natural philosophy in official Catholic culture – wrote from Rome. The pope himself sent Agnesi a personal letter of congratulations that showed some knowledge of the contents of her textbook. he also recommended that the University of Bologna appoint her lecturer in mathematics. Agnesi was then approached by the president of the Academy of Bologna and three other professors of the Academy and invited to accept the Chair of mathematics at the University of Bologna. While her name remained on the rolls of the university for 45 years, she never went to Bologna.

In 1752, when Agnesi was thirty-four, her father died and she was finally able to claim her freedom. She gave up mathematics and her other scholarly pursuits in order to spend the rest of her life serving the poor, donating her entire inheritance to the cause. She died a pauper in 1799, in one of the houses for the poor and sick she managed, having given away everything she owned.

Story from Headstrong – 52 women who changed science and the world, by Rachel Swaby, American Physical Society and The World of Maria Gaetana Agnesi, Mathematician of God by historian Massimo Mazzotti.

Mathematician to know: Ada Lovelace

What story are you going to tell us today?

Lucky I know lots of stories! 🙂 How about a story about Ada Lovelace?

Ada Lovelace (née Augusta Byron) was given a famous name before she made her own. Her father was Lord Byron, the bad boy of English Romantic poetry, whose epic mood swings could be topped only by his string of scandalous affairs — affairs with women, men, and his half-sister.

True to character, he was hardly an exemplary father. The first words he spoke to his newly born daughter were, “Oh! What an implement of torture have I acquired in you!”

According to the book Lady Byron and Her Daughters by Julia Markus, less than a month after the birth of their daughter, Lord Byron informed his wife of his intention to continue an affair with a stage actress and three days later wrote Lady Byron telling her to find a convenient day to leave their home. “The child will of course accompany you,” he added. Soon after, Lord Byron left England and never saw his daughter again. He died when Ada Lovelace was 8.

However brief their time in each other’s company, Lord Byron was ever-present in Lovelace’s upbringing — as a model of what not to be. Lady Byron, herself a mathematical wiz called “Princess of Parallelograms” by Lord Byron, believed a rigorous course of study rooted in logic and reason would enable her daughter to avoid the romantic ideals and moody nature of her father. From the age of 4, Ada Lovelace was tutored in mathematics and science, an unusual course of study for a woman in 19th century England. When Ada became sick with the measles, she was bedridden, only permitted to rise to a sitting position thirty minutes a day. Any impulsive behaviour was systematically ironed out.

Lady Byron (Anne Isabella Milbanke, 11th Baroness Wentworth)

It may have been a strict upbringing, but Lady Byron did provide her daughter with a solid education — one that would pay off when Lovelace was introduced to the mathematician Charles Babbage. The meeting occurred in the middle of her “season” in London, that time when noblewomen of a certain age were paraded around to attract potential suitors. Babbage was forty-one when he made Lovelace’s acquaintance in 1833. They hit it off. And then he extended the same offer to her that he had to so many: come by to see my Difference Engine.

Ada Byron at 17 (~1932)

Babbage’s Difference Engine was a two-ton, hand-cranked calculator with four thousand separate parts designed to expedite time-consuming mathematical tasks. Lovelace was immediately drawn to the machine and its creator. She would find a way to work with Babbage. She would.

Her first attempt was in the context of education. Lovelace wanted tutoring in math, and in 1839, she asked Babbage to take her on as his student. The two corresponded, but Babbage didn’t bite. He was too busy with his own projects. He was, after all, dreaming up machines capable of streamlining industry, automating manual processes, and freeing up workers tied to mindless tasks.

Lovelace’s mother may have tried to purge her of her father’s influence, but as she reached adulthood, her Byron side started to emerge. Lovelace experienced stretches of depression and then fits of elation. She would fly between frenzied hours of harp practice to the concentrated study of biquadratic equations. Over time, she shook off the behavioural constraints imposed by her mother, and gave herself over to whatever pleased her. All the while, she produced a steady stream of letters. A playfulness emerged. To Babbage, she signed her letters, “Your Fairy”.

Meanwhile, Babbage began spreading the word of his Analytical Engine, another project of his—a programmable beast of a machine, rigged with thousands of stacked and rotating cogwheels. It was just theoretical, but the plans for it were to far exceed the capabilities of any existing calculators, including Babbage’s own Difference Engine. In a series of lectures delivered to an audience of prominent philosophers and scientists in Turin, Italy, Babbage unveiled his visionary idea. He convinced an Italian engineer in attendance to document the talks. In 1842, the resulting article came out in a Swiss journal, published in French.

A decade since their first meeting, Lovelace remained a believer in Babbage’s ideas. With this Swiss publication, she saw her opening to offer support. Babbage’s Analytical Engine deserved a massive audience, and Lovelace knew she could get it in front of more eyeballs by translating the article into English.

Charles Babbage

Lovelace’s next step was her most significant. She took the base text from the article — some eight thousand words — and annotated it, gracefully comparing the Analytical Engine to its antecedents and explaining its place in the future. If other machines could calculate, reflecting the intelligence of their owners, the Analytical Engine would amplify its owner’s knowledge, able to store data and programs that could process it. Lovelace pointed out that getting the most out of the Analytical Engine meant designing instructions tailored to the owner’s interests. Programming the thing would go a long way. She also saw the possibility for it to process more than numbers, suggesting “the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.”

Reining in easily excitable imaginations, Lovelace also explained the Engine’s limitations (“It can follow analysis; but it has no power of anticipating any analytical relations or truths”) and illustrated its strengths (“the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves”).

The most extraordinary of her annotations was Lovelace’s so-called Note G. In it, she explained how a punch-card-based algorithm could return a scrolling sequence of special rational numbers, called Bernoulli numbers. Lovelace’s explanation of how to tell the machine to return Bernoulli numbers is considered the world’s first computer program. What began as a simple translation, as one Babbage scholar points out, became “the most important paper in the history of digital computing before modern times.”

Babbage corresponded with Lovelace throughout the annotation process. Lovelace sent Babbage her commentary for feedback, and where she needed help and clarification, he offered it. Scholars differ on the degree of influence they believe Babbage had on Lovelace’s notes. Some believe that his mind was behind her words. Others, like journalist Suw Charman-Anderson, call her “[not] the first woman [computer programmer]. The first person.”

Ada Lovelace

Lovelace guarded her work, and sometimes fiercely. To one of Babbage’s edits, she replied firmly, “I am much annoyed at your having altered my Note… I cannot endure another person to meddle with my sentences.” She also possessed a strong confidence in the range of her own abilities. In one letter, she confided, “That brain of mine is something more than merely mortal… Before ten years are out, the Devil’s in it if I haven’t sucked out some of the lifeblood from the mysteries of the universe, in a way that no purely mortal lips or brains could do.”

For what it’s worth, Babbage himself was effusive about her contributions. “All this was impossible for you to know by intuition and the more I read your notes the more surprised I am at them and regret not having earlier explored so rich a vein of the noblest metal.”

Lovelace’s ideas about computing were so far ahead of their time that it took nearly a century for technology to catch up. While Lovelace’s notes on Babbage’s analytical engine gained little attention at the time they were originally published in 1843, they found a much wider audience when republished in B.V. Bowden’s 1953 book Faster Than Thought: A Symposium on Digital Computing Machines. As the field of computer science dawned in the 1950s, Lovelace gained a new following in the digital age.

During the 1970s the US Department of Defense developed a high-order computer programming language to supersede the hundreds of different ones then in use by the military. When US Navy Commander Jack Cooper suggested naming the new language “Ada” in honour of Lovelace in 1979, the proposal was unanimously approved. Ada is still used around the world today in the operation of real-time systems in the aviation, health care, transportation, financial, infrastructure and space industries.

Ada Lovelace Day (second Tuesday of October) celebrates the extraordinary achievements of women in science, technology, engineering, and math. The “Ada Lovelace Edit-a-thon” is an annual event aimed at beefing up online entries for women in science whose accomplishments are unsung or misattributed. When her name is mentioned today, it’s more than a tip of the hat; it’s a call to arms.

Story from Headstrong – 52 women who changed science and the world, by Rachel Swaby.

Mathematician to know: Emmy Noether

On any list of history’s great mathematicians who were ignored or underappreciated simply because they were women, you’ll find the name of Emmy Noether. Despite the barricades erected by 19th century antediluvian attitudes, she managed to establish herself as one of Germany’s premier mathematicians. She made significant contributions to various math specialties, including advanced forms of algebra. And in 1918, she published a theorem that provided the foundation for 20th century physicists’ understanding of reality. She showed that symmetries in nature implied the conservation laws that physicists had discovered without really understanding.

Joule’s conservation of energy, it turns out, is a requirement of time symmetry — the fact that no point in time differs from any other. Similarly, conservation of momentum is required if space is symmetric, that is, moving to a different point in space changes nothing about anything else. And if all directions in space are similarly equivalent — rotational symmetry — then the law of conservation of angular momentum is assured and figure skating remains a legitimate Olympic sport. Decades after she died in 1935, physicists are still attempting to exploit Noether’s insight to gain a deeper understanding of the symmetries underlying the laws of the cosmos.

Yay, it’s story time!

Albert Einstein was in over his head. He had worked out his general theory of relativity, but he was having problems with the mathematics that would have to correspond. So Einstein pulled in a team of experts from the University of Göttingen to help him formulate the concepts. The team was led by David Hilbert and Felix Klein, who were held in extremely high regard for their contributions to mathematical invariants. But their legacy, in part, is the community of scholars they fostered at Göttingen, who helped the university grow into one of the world’s most respected mathematics institutions. They scouted talent. For the Einstein project, Emmy Noether was their draft pick.

Noether had been making a name for herself steadily. In the eight years prior, she worked at the University of Erlangen without a salary or a job title. By the time she left for Göttingen, she had published half a dozen or so papers, lectured abroad, taken on PhD students, and filled in as a lecturer for her father, Max Noether, who was an Erlangen mathematics professor suffering from deteriorating health.

At the time, Noether’s specialty was invariants, or the unchangeable elements that remain constant throughout transformations like rotation or reflection. For the general theory of relativity, her knowledge base was crucial. Those interlinked equations that Einstein needed? Noether helped create them. Her formulas were elegant, and her thought process and imagination enlightening. Einstein thought highly of her work, writing, “Frl. Noether is continually advising me in my projects and…it is really through her that I have become competent in the subject.”

It didn’t take long for Noether’s closest colleagues to realize that she was a mathematical force, someone of extraordinary value who should be kept around with a faculty position. However, Noether faced sharp opposition. Many of the people who supported the push to make her a lecturer also believed that she was a special case and that, in general, women shouldn’t be allowed to teach in universities. The Prussian ministry of religion and education, whose approval the university needed, shut down her appointment: “She won’t be allowed to become a lecturer at Göttingen, Frankfurt, or anywhere else.”

The shifting political landscape finally cracked open the stubborn set of regulations governing women in academia. When Germany was defeated in World War I, socialists took over and gave women the right to vote. There was still a movement internally to get Noether on staff, and Einstein offered to advocate for her. “On receiving the new work from Fräulein Noether, I again find it a great injustice that she cannot lecture officially,” he wrote. Though Noether had been teaching, on paper her classes were David Hilbert’s. Finally, Noether was allowed a real position at the university with a title that sounded like fiction. As the “unofficial, extraordinary professor,” Emmy Noether would receive no pay. (Her colleagues joked about the title, saying “an extraordinary professor knows nothing ordinary, and an ordinary professor knows nothing extraordinary.”) When she finally did receive a salary, she was Göttingen’s lowest-paid faculty member.

Pay or no pay, at Göttingen she thrived. Here’s how deeply one line of study, now called Noether’s theory, influenced physics, according to a physicist quoted in the New York Times: “You can make a strong case that her theorem is the backbone on which all of modern physics is built.” And the dent she made in mathematics? She was a founder of abstract algebra. In one paper, published in 1921 and titled “Theory of Ideals in Rings,” Noether dusted her work free of numbers, formulas, and concrete examples. Instead she compared concepts, which, the science writer Sharon Bertsch McGrayne, explains, “is as if she were describing and comparing the characteristics of buildings—tallness, solidarity, usefulness, size—without ever mentioning buildings themselves.” By zooming way, way out, Noether noticed connections between concepts that scientists and mathematicians hadn’t previously realized were related, like time and conservation of energy.

Noether would get so excited discussing math that neither a dropped piece of food at lunch nor a tress of hair sprung from her bun would slow her down for a second. She spoke loudly and exuberantly, and like Einstein was interested in appearance only as it related to comfort. Einstein loved his gray cotton sweatshirts when wool ones were the fashion; Noether wore long, loose dresses, and cut her hair short before it was in style. For Einstein, we call these the traits of an absentminded genius. For Noether, there was a double standard—her weight and appearance became the subject of persistent teasing and chatter behind her back. Like the trivial annoyances of title, pay, and politics, the comments didn’t bother Noether. When students tried to replace hairpins that had come loose and to straighten her blouse during a break in a particularly passionate lecture, she shooed them away. Hairstyles and clothes would change, but for Noether, math was her invariant.

With a mind working as rapidly as hers, it was a challenge for even Noether to keep up with her own thoughts. As she worked out an idea in front of the class, the blackboard would be filled up and cleared and filled up and cleared in rapid succession. When she got stuck on a new idea, students recalled her hurling the chalk to the floor and stomping on it, particles rising around her like dust at a demolition. Effortlessly, she could redo the problem in a more traditional way.

Both social and generous with sharing ideas, many, many important papers were sparked by Noether’s brainpower and published without her byline but with her blessing. In fact, whole chunks of the second edition of the textbook Modern Algebra can be traced back to her influence.

Politics in Germany affected her career again. Though Noether had established herself as one of the greatest mathematical minds of the twentieth century, the Nazis judged only her left political leanings and her Jewish ancestry. In May 1933, Noether was one of the first Jewish professors fired at Göttingen. Even in the face of blatant discrimination, perhaps naively, the math came first. When she could no longer teach at the university, Noether tutored students illegally from her modest apartment, including Nazis who showed up in full military gear. It wasn’t that she agreed with what was happening, but she brushed it aside for the dedicated student. “Her heart knew no malice,” remembered a friend and colleague. “She did not believe in evil — indeed it never entered her mind that it could play a role among men.”

For her generosity, Noether’s friends were wholly dedicated to her. Understanding that staying in Germany would put her in serious danger, in 1933 her friends arranged for Noether to take a position at Bryn Mawr College in the United States. It was meant to be a temporary post until she could land somewhere more prestigious. But just two years after she arrived, Noether died while recovering from a surgery on an ovarian cyst. She was fifty-three. Following her death, Einstein wrote a letter to the New York Times. “Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.” Today, some scientists believe her contributions, long hidden beneath the bylines and titles of others, outshine even the accomplishments of the ode’s writer.

Physicists tend to know Noether’s work primarily through her 1918 theorem. Because their work relies on symmetry and conservation laws, nearly every modern physicist uses Noether’s theorem. It’s a thread woven into the fabric of the science, part of the whole cloth. Every time scientists use a symmetry or a conservation law, from the quantum physics of atoms to the flow of matter on the scale of the cosmos, Noether’s theorem is present. Noetherian symmetries answer questions like these: If you perform an experiment at different times or in different places, what changes and what stays the same? Can you rotate your experimental setup? Which properties of particles can change, and which are inviolable?

Conservation of energy comes from time-shift symmetry: You can repeat an experiment at different times, and the result is the same. Conservation of momentum comes from space-shift symmetry: You can perform the same experiment in different places, and it comes out with the same results. Conservation of angular momentum, which when combined with the conservation of energy under the force of gravity explains the Earth’s motion around the sun, comes from symmetry under rotations. And the list goes on.

The greatest success of Noether’s theorem came with quantum physics, and especially the particle physics revolution that rose after Noether’s death. Many physicists, inspired by Noether’s theorem and the success of Einstein’s general theory of relativity, looked at geometrical descriptions and mathematical symmetries to describe the new types of particles they were discovering.

Emmy Noether’s theorem is so vital to physics that she deserves to be as well known as Einstein. – Brian Greene

Noether’s theorem to me is as important a theorem in our understanding of the world as the Pythagorean theorem. – Christopher Hill

Mathematicians are familiar with a variety of Noether theorems, Noetherian rings, Noether groups, Noether equations, Noether modules and many more. Over the course of her career, Noether developed much of modern abstract algebra: the grammar and the syntax of math, letting us say what we need to in math and science. She also contributed to the theory of groups, which is another way to treat symmetries; this work has influenced mathematical side of quantum mechanics and superstring theory.

Story from Headstrong – 52 women who changed science and the world, by Rachel Swaby, and Fermilab/SLAC National Accelerator Laboratory Symmetry Magazine.

Tuesday Comedy Club

Three statisticians go out hunting together. After a while they spot a rabbit. The first statistician takes aim and overshoots. The second aims and undershoots. The third shouts: “We got him!”

After a talking sheepdog gets all the sheep in the pen, he reports back to the farmer: “All 40 accounted for.” “But I only count 36 sheep,” says the farmer. “I know,” says the sheepdog, “But you asked me to round them up.”

I hired an odd-job person to do 8 jobs for me, but when I got back, she’d only done half of them.

Last night I dreamed that I was weightless… I was like, 0mg!

Did you hear about the mathematician who was afraid of negative numbers? He’d stop at nothing to avoid them.

Did you hear about the Improper Fractions shop? It’s open 24/7.

What’s the difference between a pupil studying exponentials and a lumberjack? Nothing, they both involve moving logs around.

What’s the difference between an angle measurer and the President of the Agriculturists’ Union? Nothing, they’re both pro-tractors.

What’s the difference between 0.9 recurring and 1?

My turn!

Why did the Viking fail the graph question?
He forgot to label his axes.

How many numbers are there between 1 and 10 inclusive?
Five, because 1, 3, 5, 7 and 9 aren’t even numbers.

What do you say to a mathematical cat who’s stuck in a (geome-)tree?

Why did you divide sin by tan?
Just cos.

Why is 6 afraid of 7?
Because 7 8 9.

How does a ghost solve a quadratic equation?
By completing the scare.

I win!

Thank you Cambridge Mathematics for the math puns 🙂