Tag Archives: Hypatia

Hypatia of Alexandria – The Unsung Mathematician

In our studies of math, we learn from hundreds of individuals who’ve greatly influenced our perspectives of mathematics. These famous individuals created various theorems and proofs, and left us with questions to resolve. Individuals such as Euclid brought us marvels that were passed on for generations. His book The Elements created a foundation for many mathematicians, and helped them find breakthroughs in science and math. Unquestionably, our society and technology may not be where it is if it weren’t for these individuals. As we look at our own development of math, we learn names such as Pythagorean and use it repeatedly throughout our math education. Although the works of these persons are marvelous, there are many mathematicians and scientists who are unknown. They may not have found a theorem that completely changed the world around us, but they contributed towards something equally great. Truly these are our unsung mathematicians that are unknown to the world but known to the few who’ve learned or specialized in the field of math or science. One individual that I’ve found and never heard of before was Hypatia. Her biography may not be found in a history book about Rome, but she was an influential person to those around her. She is considered to be the first female mathematician known. Personally, reading about her impresses me because the math profession at the time was highly male dominated and lacked women. I can imagine that she had to go through many challenging experiences to prove herself as a respected mathematician. Her life could be seen as a tragedy and it is even romanticized in the 2009 film Agora, but those who knew her revered her greatly.

Depiction of Hypatia. Image: Public domain, via Wikimedia Commons.

Who was she?
There is little information about where Hypatia was born or even a birth date. Scholars believe that she was born between 351 AD and 370 AD. Around this time period, Rome was in a slow decline, and there were civil disputes because of the influence of Christianity. Her father Theon was a well-known mathematician and astronomer who taught Hypatia the foundations of her knowledge in those fields. It’s significant to note that Theon was the last known survivor of the Museum of Alexandria and played a part in the preservation of Euclid’s The Elements. Hypatia and her father Theon would collaborate on commentaries on classical mathematical works, and she would eventually have students of her own. As she followed her father’s footsteps, she also developed knowledge in philosophy and became head of the Platonist school of Alexandria. She primarily taught the works of Plato and Aristotle and it was recorded that many people would come from different places to learn from her. She would wear the robes of a scholar and nobody questioned that she didn’t wear the traditional female clothes. Her work was well respected everywhere and it simply didn’t matter how she appeared to her audiences. The people of Alexandria revered her as a virtuous figure and an intelligent scholar. She believed in Neo-Platonism, a belief system that states everything emanates from the One. This belief system states that the One is considered God or the Good of all things. But what’s interesting is that the One is neither existent nor a being. This abstract thought provoked the teachings of Christianity and most likely made Hypatia a target for zealots. Although Hypatia’s beliefs conflicted with others of the Christian faith, she tutored Synesius of Cyrene who later became a bishop and incorporated Neo-Platonism into Christianity.

Depiction of Hypatia’s Death. Image: Public domain, via WIkimedia Commons.

Hypatia’s popularity to those around her eventually led to her own death. There are a couple of different accounts of her death but it basically resulted from a dispute between her friend Orestes, the Governor of Alexandria, and Cyril, the Bishop of Alexandria. Whatever conflict or feud these two individuals had, this led to Hypatia being brutally murdered by a mob of Chrstian zealots. There are two different documentations of this event and little is known how old she was when she died. Although she was murdered, her students fled to Athens and the school she taught in Alexandria continued her teachings.

Her Works and Legacy
Many of Hypatia’s works were destroyed around 652 AD when the Arabs invaded Alexandria. However, a couple of letters that she exchanged with her students were still available. One of her students mentioned before, Synesius, exchanged a letter that talked about a hydrometer and astrolabe. These writings expanded on the concepts and structures of these two objects. Some other works that she has done, in collaboration with her father, were commentaries on Arthimetica by Diophantus and The Elements by Euclid.

Hypatia leaves a legacy that has been seen as an inspiration or tragedy. Many politicians, poets, and writers used her death as part of a cause for them and to influence others. I see her as one of our history’s unsung mathematicians who had so much more to give if she had lived longer. As time has progressed, we live in a day and age where there are more opportunities for unsung heroes such as Hypatia. Because of technology and access to colleges and universities, anyone can study mathematics. Although you and I may not be the next Euclid or even Hypatia, our small contributions and interest may eventually lead to something great and we can be known as the unsung heroes of our generation.

References:
http://www.smithsonianmag.com/womens-history/hypatia-ancient-alexandrias-great-female-scholar-10942888/?no-ist
http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/
http://en.wikipedia.org/wiki/Hypatia#Works
http://www.britannica.com/EBchecked/topic/279463/Hypatia
http://www.sheisanastronomer.org/index.php/history/hypatia-of-alexandria

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A Glimpse into Female Mathematicians of the Past

After our discussion in class about the work of Sophie Germain, I was interested in learning more about other prominent women in mathematics. I’m sure we will go over some of them in class, but here is what I discovered about some very smart women.

One of the earliest known female mathematicians was Hypatia. She lived in the time period of approximately 350-416 C.E. She was excellent at mathematics, astronomy and philosophy. No doubt this is because her father was Theon, one of the last members of the library of Alexandria. Unfortunately for us, we do not know many of her contributions to science. She is more well known for her brutal death. She was riding in her carriage, when she was forcefully removed, stripped, beaten to death, and then her body was burned. Not a nice way to go. Regardless, of that cruelty, she is one of the first well known women mathematicians, and in her time that was quite an accomplishment.

Another leading lady in mathematics was Ada Lovelace. She lived from 1815-1852 as the daughter of well known writer, Lord Byron. She never met her father, and her mother advocated her to study fields that were different from language and poems. Essentially, anything different from what her father was well known for. It must have been a bad break up. Thus, math and science it was. Turns out, she is credited with being the world’s first programmer. But before that achievement, she demonstrated ingenuity as a child. She set her mind toward the daunting task of flying, at the young age of twelve. She researched materials, how to build wings, and even wanted to incorporated steam! Being curious from a young age really inspired her to continue her study of the sciences.

Because of the strict laws against the education of women she had to study mathematics with a tutor, she could not technically enroll in university.  She met Charles Babbage later in life and their friendship encouraged her studies. They continued their correspondence even after her marriage to the Earl of Lovelace. At the time Babbage was working on a theoretical machine called the Analytical Engine. The idea was that the Engine could store numbers, and it could do long cycles and loops without the help of people. She wrote to Babbage about including Bernoulli numbers and how such implicit functions could be solved by the Engine. According to Wolfram Alpha, “The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function. x/(e^x-1)=sum_(n=0)^infty(B_nx^n)/(n!).  These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.” In order to calculate Bernoulli numbers, there must be a lot of operations involved. To top it off, they anticipated that the Analytical Engine could perform this task. Below I have pictured one of Ada’s tables on how she envisioned the Engine could compute this. Remarkably enough, Lady Lovelace once said, “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with.” She understood that the machine is only as good as the people who are using it. It cannot come up with new ideas, or understand why it is doing computation, it can only do said computation. If this machine were to have been made, it would have been an incredible invention. However, the fact that it was never brought to production, does not in any way reduce all of the work both Ada and Charles did.

Ada Lovelace's plan to generate Bernoulli numbers.

Ada Lovelace’s plan to generate Bernoulli numbers. Image: Betty Toole.

Unfortunately, there has been speculation that Ada did not contribute in the mathematical sense, but was merely a notetaker for Babbage. This is baffling because in his autobiography, Babbage gives her credit for all of the theoretical math she did for his Analytical Engine. I could continue this post with a commentary about women in science even today, but I’d better move onto the final female mathematician I wish to recognize.

The final female mathematician I wish to discuss is Emmy Noether. Emmy was born in Germany in the late 1800’s. She was denied a lot of formal education because she was a woman. She began her studies with piano and languages, but soon discovered a passion for math, like her father, and her brother.  Universities in Germany were hesitant to let her become a professor, although, she did get the status of Associate Professor eventually. This title was taken away however, when the Nazi’s came to power because she was Jewish. Despite all of this, she had many notable accomplishments. So much so, that Albert Einstein once referred to her as “the most significant creative mathematical genius thus far produced since the higher education of women began.” This is high praise, especially coming from a man our society reveres as the most intelligent man ever known.

She was behind a revolutionary theorem, called Noether’s Theorem. This theorem states that: “Each symmetry of a system leads to a physically conserved quantity. Symmetry under translation corresponds to conservation of momentum, symmetry under rotation to conservation of angular momentum, symmetry in time to conservation of energy, etc.” And when I first read this, I was quite confused. However, with some help from my sources, I was able to wrap my mind around it to a certain extent. Noether is telling us that when we find symmetrical things, in nature or otherwise, there is some sort of conservation force that goes with it. One example of this, that is referenced in the New York Times article, is the relationship between time and energy. To paraphrase, if a person throws a ball up in the air right now, or throws it the same way sometime in the future, the time does not affect the trajectory of the ball. This means that the symmetry of time is related to the conservation of energy. This is crucial to how we think about physics today, and I could definitely relate this to my old physics teacher being like a broken record and telling us energy cannot be created or destroyed, it only changes form. Emmy clearly made an impact on not only math, but the way we think about certain concepts today. She even developed some of the mathematical formulas that Einstein used for his Theory of Relativity.

It seems to me that Emmy deserves much more recognition than she is receiving. Truthfully, I had not even heard of her until I began research for this blog post. I know this is not a class about how our society can improve, but one way would be to get more women in math and science. It is interesting to think about how limited women once were. I am optimistic about the progress we have made in that regard, but just think about how much further along we could possibly be in terms of figuring out the mysteries of the world if we had help from every person, from every demographic, and every gender. I do not know if this is possible, but inclusion is a nice thought. These ladies kicked butt in their time, and I hope that the women of the present and the future follow their example and continue to do the same.

I have recently learned that October 14th was Ada Lovelace Day! Ada Lovelace Day celebrates women in all areas of science. And because of that, I would like to dedicate this post to all the amazing ladies out there making leaps and bounds in the sciences. You are an inspiration to me, but all young women of the world.

Works Cited:

  • Angier, Natalie. “The Mighty Mathematician You’ve Never Heard Of.” The New York Times. N.p., 26 Mar. 2012. Web. 29 Sept. 2014. <www.nytimes.com%2F2012%2F03%2F27%2Fscience%2Femmy-noether-the-most-significant-mathematician-youve-never-heard-of.html%3Fpagewanted%3D1%26_r%3D0>.
  • “Bernoulli Number.” — from Wolfram MathWorld. N.p., n.d. Web. 10 Oct. 2014.
  • Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Women in Math

Cora Ratto de Sadosky (1912-1981). Image credit: Courtesy Cora Sadosky. This photo and biography was featured on MAA’s Women of Mathematics poster, via flickr.

As I have gone through the process of gaining a higher education in mathematics, I have made a startling realization that I am alone. Sure, there are other women in my math classes, but the majority of the students are men. I have had to rely on my own strength and diligence to get through the challenging courses. When I started working on my degree, I had many counselors and professors that discouraged me from entering into such a field due to the fact that it was challenging, and the odds were I would not succeed.  Whether this opinion was developed from me being a female or not, I have a hard time believing that a male would receive that same type of consolation. Also, it is a popular belief in our culture here in Utah that most women should not enter into the fields of science and mathematics, and are better off obtaining degrees that will benefit them as   homemakers. Hence, most women do not pursue a degree in mathematics or science. It troubled me to know that there are no women that I could turn to for help in my field of choice. In my History of Mathematics class, we have been learning about the great minds of mathematics which have mostly been men. However, last week in class, we learned about Sophie Germain, a woman mathematician. This got me thinking that I have never before heard, or learned about other women in the field of mathematics. I’ve been asking myself, why don’t I know more about these women? So I decided to do some research and find other women who have contributed to the field of mathematics and made it possible for other women, like myself, to gain a higher education.

One of the first known female mathematicians was Hypatia (370-415 A.D.). Her father was a well-educated man, and Hypatia spent a lot of time in the world of education learning from her father. From her father’s teachings, Hypatia become very educated in math, science, and astronomy and would impart this knowledge to students in her home. Large crowds would also come and listen to her teach in the streets. Her fame and popularity, however, turned to be her downfall as she was killed by Christian zealots.

Sophie Germain (1776-1831) was born in a time of revolution, which was shown in her character. During this time, it wasn’t socially acceptable for women to have access to the same education as men. This didn’t stop Sophie from becoming a great mathematician, and being the first woman to win a prize from the French Academy of Sciences for her work on the theory of elasticity. It should be noted that during her life she often worked under a false name to avoid persecution for breaking social boundaries of women in education.

Sofia Kovalevskaya (1850-1891) was born in Russia, where women were not allowed to attend universities. In order for her to pursue some type of higher education, she decided to get married so she could travel to Germany, and was able to be privately tutored by a professor. Sofia was granted a PhD, and went on to produce wonderful works in the fields of mathematics and science, but was always faced with adversity. Despite her hardships, her contributions were vast, and she expanded the opportunities for women in education and women’s rights.

Emmy Noether (1882-1935) grew up in Germany, where she wasn’t allowed to receive a university education. Growing up, she was educated in language, and the common tasks expected from women. At age eighteen, she decided to take courses in mathematics, and was able to become a university student. She received a PhD, and became an unofficial associate professor at the University of Göttingen. However, in 1933 she lost that title because she was Jewish. She decided to move to America and became a lecturer and researcher. There she developed many of the mathematical foundations for Einstein’s general theory of relativity. Einstein later wrote of her that she was “the most significant creative mathematical genius thus far produced since the higher education of women began”(Zielinski).

Ingrid Daubechies (1954-Present) is the first female president of the International Mathematical Union, and is a strong advocate for women in science and mathematics. As a girl, she studied physics and eventually received her PhD, along with other awards. Her most important discovery was in the field of wavelets, which are “mathematical functions useful in digital signal processing and image compression as well as in many other branches of applied and pure mathematics”(Riddle). In a recent interview Daubechies was asked why there is the assumption that men are better at mathematics than women. Her response to this question, “I disagree with this view – completely. There is a highly variable percentage of women in academia and in departments of mathematics across Europe. Differences are so enormous that it becomes obvious that it has something to do with cultural habits, which differ from one nation to another, and not with intelligence”(TWAS).

In conclusion, there have been many women who have made significant contributions to the fields of math and science, and have influenced the works of other male scholars. As a woman in higher education and mathematics, I admire the hardships and work these women accomplished, and wish that more was said about them. In doing this research, I’ve realized I am not alone, and I have many great examples of women who have worked hard and overcame societal obstacles.  As a future teacher, I aspire to influence more girls to pursue college degrees and not be intimidated by the “male dominated” subjects, and realize that women are just as intelligent as men.

References:

“Math Is (also) for Women.” TWAS. July 29, 2014. Accessed September 30, 2014. http://www.twas.org/article/math-also-women.

Riddle, Larry. “Women Mathematicians Alphabetical Index.” Women Mathematicians Alphabetical Index. Accessed September 28, 2014.

Zielinski, Sarah. “Five Historic Female Mathematicians You Should Know.” Smithsonian.com. October 7, 2011. Accessed September 28, 2014.

I thought we had him, Sophie

Portrait of Sophie Germain by Auguste Eugene Leray. Image: public domain, via Wikimedia Commons.

Portrait of Sophie Germain by Auguste Eugene Leray. Image: public domain, via Wikimedia Commons.

Like most people, I end up having dinner with my family every so often. Unlike most people, our family conversations always seem to include math or physics. Generally we end up trying to stump each other with various questions. It’s like a game to us.

A few nights ago we had one of those dinners. I found a lull in the conversation and I fired off, “Hey dad, do you know any significant female mathematicians?” His jaw went a little loose and he gave me a blank stare. It was the same look people give me when they find out I talk about math at the dinner table.

My crippling question had developed from a classroom discussion about an influential female mathematician, Sophie Germain. I’d realized in that discussion that I could name several influential female scientists. For whatever reason, I’d never heard of any women known for their mathematics.

As the feeling of triumph settled in, a smile developed across my face. Sophie and I were about to taste victory. I watched confidently as my father’s eyes slowly rolled back into his head. Gradually his eyes came back down and his smile met mine. I knew I was in trouble when he said, “Well, the oldest one I can think of is Hypatia”.

He went on to tell us the story of the daughter of Theon of Alexandria. Theon was a known scholar and professor of mathematics at the University of Alexandria. His daughter, Hypatia, was given all the best. In particular, her education was second to none. With such great influences, many historians believe she was able to eclipse her father’s knowledge at an early age. In time, people would come from distant cities to learn from her.

Sadly, none of her original work has survived to this day. As a mathematician we remember her largely for her editing and insightful comments on other great works of the time. Some of the more important works included the Arithmetica by Diophantus and Conics of Apollonius. The book on conics was particularly significant. It contained progressive information about cutting cones with planes that helped develop ellipses, hyperbolas and parabolas.

Unfortunately, Hypatia is possibly better known for the way she died than the way she lived. During her lifetime, the quality of life in Alexandria was on the decline. Fighting had developed among the different religious factions and it threatened to destroy the city. At one point, the Roman emperor ordered the destruction of all pagan temples. As an educated pagan that often spoke about non-Christian philosophy, Hypatia was a likely target. Ultimately her end came when a group of Christians pulled her from her carriage, drug her into a church, stripped her, beat her to death, tore her to pieces, burned the pieces and disgracefully scattered her remains.

At this point my mother had made it to the other side of the table. She began slapping my father’s arm while muttering something through clenched teeth. “Ok, Ok”, he conceded and continued, “My favorite was Emmy Noether anyway”. “She died of natural causes!” he taunted my mother as she wandered into the kitchen. By this time my smile was long gone. He’d already won the game and was just showing off.

Much like Hypatia, Noether was the daughter of a successful professor of mathematics. Her German family was quite wealthy and provided all her needs. Unfortunately, sometimes society isn’t as helpful.

Noether found herself suffering from restricted access to the University of Erlangen. Her father was on faculty there and I expect it felt like home. Fortunately, she was able to audit classes until they started accepting female students. I’m just speculating here, but I’d like to think she helped drive the decision. A brief four years later she was awarded her Ph.D. summa cum laude.

In spite of her obvious achievement, society still wasn’t quite ready for Noether. She spent the next several years working with David Hilbert and other prominent mathematicians at Göttingen University. Due to a lack of insight, much of the faculty refused to allow a female member. As a result, she worked for free.

As the years went by at Göttingen she worked on incredible theories and taught classes in Hilbert’s name. Eventually her situation improved when she was allowed to become a licensed lecturer. The university wasn’t paying her yet, but at least she could collect student fees. Years later she was finally granted a position as an adjunct teacher.

It’s worth mentioning she was considered a remarkable instructor. More than a few of her students went on to make significant contributions to mathematics. If all things were equal, it would be hard to imagine her without tenure.

While Noether may not be well known to the general public, great minds have given her enormous respect. Einstein referred to her as a “significant creative mathematical genius”. This may well be due to her contributions to general relativity.

Her major mathematical contribution was in a 1921. It was a groundbreaking paper on the theory of ideal rings. She was able to think of things in an extremely abstract way. Many consider her the mother of modern algebra.

One of the more easily understandable ideas she developed involved creating ascending or descending chains. Imagine a set A1 is contained in a set A2, which is contained in a set A3, etc. Sometimes you can see that after a certain point, the rest are the same. For example we might notice that A10, A11, A12, etc are all the same. On the surface it may not seem like a big deal. But techniques like these are frequently exploited in proofs.

Quite honestly, the scope of her contribution is breathtaking. It warms my heart to know people have been making such impressive advances in the last hundred years. It’s even more rewarding to see remarkable individuals like Emmy Noether push through an unjust barrier. In spite of never being treated equally, she emerge a champion with the landscape changed behind her.

By the time my father had finished his glowing praise of Emmy Noether, my face was in my palm. I had clearly lost my challenge. I thought we had him, Sophie.

http://www.smithsonianmag.com/womens-history/hypatia-ancient-alexandrias-great-female-scholar-10942888/

http://www-history.mcs.st-and.ac.uk/Biographies/Hypatia.html

http://www.agnesscott.edu/lriddle/women/hypatia.htm

http://www.bu.edu/lernet/pathways/bios.html

http://www-history.mcs.st-andrews.ac.uk/Obits2/Noether_Emmy_Einstein.html

http://www.encyclopedia.com/topic/Emmy_Noether.aspx

http://en.wikipedia.org/wiki/Emmy_Noether#Ascending_and_descending_chain_conditions