(via etsu.edu)

On today’s episode Samuel Hansen has a conversation with noted mathematical journalist, Director of Publications and Communications for the Mathematical Association of America, and, most importantly, the Mathematical Tourist Ivars Peterson. You can find out more about Ivars Peterson on his website and can find his columns at the MAA.

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• 37 is the smallest irregular prime (submitted by Andy Baker and John Kiehl)
• 37 is the smallest left and right truncatable prime having more than one digit
• 37 is the only prime with period length three: 1/37 = 0.027 027 027 …
• 37 is the prime you get if a three digit number having the same digits is divided by its digit sum

Children learn from this that numbers are wonderful shortcuts. Instead of saying the word “fish” exactly as many times as there are penguins, Humphrey could use the more powerful concept of “six.”

As adults, however, we might notice a potential downside to numbers. Sure, they are great time savers, but at a serious cost in abstraction. Six is more ethereal than six fish, precisely because it’s more general. It applies to six of anything: six plates, six penguins, six utterances of the word “fish.” It’s the ineffable thing they all have in common.

No matter who you are Strogatz’s exposition is plenty good enough to hold your attention, and the content is parse-able by anyone. If you are a mathematician go and read this to help reground yourself in the most basic contents, then go tell all the non-mathematicians you know to go read this so they know what the hell you have been talking about for all these years. (Strogatz Opionionator)

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(via aimmath.org)

For today’s episode of Strongly Connected Components Samuel Hansen called up director of the American Institute of Mathematics Brian Conrey. Together they talked about the AIM library, whymath circles and square mathematicians are a good thing, and just what the math castle is. To find out more about Brian Conrey visit his website and to find out more about AIM just click here.

Video of the Castle

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This resonated with me as I spent my weekend in the seat of United States of America’s federal power, Washington DC. I was there to participate in the Students for Free Culture, http://freeculture.org, annual Free Culture Conference. This conference is in the words of the creators: “A convening of the international free culture community for two days of networking, learning and acting. The vision is to bring together student activists and free culture luminaries to discuss free software and open standards, open access scholarship, open educational resources, network neutrality, and university patent policy, especially in the context of higher education.” The conference itself was a shot in the arm for me in particular, as it has pushed me towards really starting work on some projects that I have had on the back burner for a long time.

The conference, while concentrating a lot on education, spent a decent amount of time on politics, a subject that I have only allocated the minimal amount of interest to since I joined up and became on the few, the proud, the graduate students. It required that I open my mind and start thinking less like a mathematician, i.e. in closed logical fashion, where the strongest of formal arguments is obviously the correct one, and start cogitating in the way that normal people, and more specifically politicians, do on a daily basis. It was not the easiest thing for me to do, I would listen to some of the panelists talking about Net Neutrality or Open Educational Resources and immediately wonder why every does not just do things in the way that one of the panelists puts forth because it was obviously the best way. As a mathematician I often forget that most people do not think about the work in such a clean and dry way. One thing that became clear to me at this conference was that if I were the government I would spend as much money as I possibly could to make mathematics more open.

Lack of openness is a huge problem in academics. Most people in the United States of America, as well as the rest of the world, have no easy way to gain access to the scholarly results that are obtained on a daily basis from academics, most of whom work in state sponsored facilities. One of the scariest statistics I learned at the Free Culture Conference is that Scholarly Journals are an 8 billion dollar industry, and to put that in perspective so is the National Football League. Scientific Journals are consistently the most expensive type of journal too. Since all academics have to publish in order to gain promotion and, the ultimate goal, tenure they have no choice but to turn the the Cretaceous era behemoth that is the Scholarly Journal therefore cutting off all those who are not currently enrolled or employed by an institution large enough to afford a subscription to said journal. This does not just exclude people who are not currently enrolled at a college though, as most community and technical colleges do not have substantial budgets and therefore are unable to gain access to scholarly works for their students.

To put it simply this is not acceptable. It hurts enough to think that the average person is unable to access new results, but the thought that students can not get access is simply beyond the pale. It all gets worse when one factors in that Journals rarely, if ever, actually compensate authors for the articles the use to make their 8 billion dollars.

There are a couple of ways that the closed Journal format is being challenged today. The first way is governmental and that is a bill before Congress that would give open access, after six months, to all federally funded, non-classified so National Security Agency funded research could very well remained a walled off field in mathematics, research. Since governmental funding is such a huge part of how academics pay for their research this bill, if passed, will go a really long way to opening up citizens ability to access research. The other way that the closed yard that is academic publication is being challenged is through Open Journals. These are journals that allow anyone to read and link to articles that are published through them, for free. The environment of open journals is growing very rapidly, in fact one of the newest ones is a mathematical journal: Journal of Computational Geometry, http://jocg.org.

While these movements are very important, and are slowly chipping away at the edifice of the deeply entrenched Scholarly Journal system, if I were in charge of governmental funding I would start sinking a lot of money into ways that would allow every person access to this knowledge that has, for way too long, been walled away from people. The most basic way of becoming educated is having access to information, and if we as a country are serious about educating our populace we can not afford to have such a small, and arguably so elite, a group being the only ones who can see and use the newest results in mathematics, science, and all the rest of academia.

Another topic that we covered was the history of mathematics in Japan. One of the most famous of the historical mathematicians of Japan was “The Arithmetical Sage” Seki Kowa. Born in 1642 to the samurai family of Uchiyama Shichibei, Seki Kowa brought to Japanese mathematics a sense of analysis and his prodigious teaching skills. In his infancy Kowa was adopted by Seki Gorozayemon, hence the surname Seki and not Uchiyama. As his life progressed he served as the examiner of accounts for the Lord of Koshu, who later became Shogun at which point Kowa took the exalted position of Master of Ceremonies in the Shogun’s household. One of the most interesting things about Kowa are the stories that are told of his life. At five it was said, very similar to the story of Gauss and the summation of the integers 1 to 100, that he corrected his elder’s calculations, a perilous action to take for a culture that revolved around the Bushido code, but instead of recriminations the elders apparently gave him the name of the Divine Child. The stories do not end with childhood though, they say that while traveling in the typical samurai palanquin he noted distances and elevations of all that he passed and used those notes to make a high quality map of the region through which he passed. One of the main reasons for the telling of this story was the blending of the Samurai, physical, world that Kowa was born into and the mental world in which he lived. The final, and my favorite, story was that of how, when all the clock-makers and artisans failed to repair the spring in a clock that the Shogun had gifted to the Emperor of China Kowa came in and used his knowledge of mathematics to fix the clock in a couple of days.

While all of these stories are apocryphal, there are plenty of concrete examples of the work of Kowa. He did much work on the Yeuri, Japanese Calculus, and while he may not have invented it, he is often credited with that achievement, he definitely broadened its scope and depth. He surely did invent the Tenzan algebraic system though, but the secrecy inherent in the Japanese educational, and European for the matter, system meant that he only revealed the Tenzan to his students after they swore a blood oath to not reveal the method. Kowa also did work on determinants that predated Leibniz’s by 10 years, discovered positive and negative, not complex, Bernoulli numbers, beat Newton to both his method for solving equations and interpolation formula. Other important work that Kowa had included a method for approximating pi, work on magic squares, and the indeterminate equation ax-by=1 where the coefficients are constant.

Even more important than his mathematical discoveries though, was probably Seki Kowa’s Yendan Jutsu, roughly translating to explanation process. When Kowa published the Hatsui Sampo he did something that no previous mathematician in Japan had attempted, he explained the steps he had taken in his work. His predecessor had, while working on a problem, on stated the rules which they followed and the results which these rules allowed them to gain. It was Kowa’s work with this type of analysis that explains his success in teaching, and the large amount of students that came to him to learn the mathematics. In fact his student Takebe Katahiro, himself a very well known Japanese mathematician, had the following to say about his teacher Seki Kowa’s clear style of analysis: “It is one of the brilliant products of my master’s school and it must be agreed that it surpasses all other mathematical achievements ancient or modern.” Considering that it was clear analysis that also explains the explosion of mathematics in Europe around the same time I do not think Takebe Katahiro is that far off. Kowa passed away in 1708 and has his title “The Arithmetical Sage” is inscribed on his tombstone.

The next class we started to delve into the topic of women in mathematics and touched on some of the bigger names, such as: Sophie Germain and Sofia Vasilyevna Kovalevskaya. One name that did not come up, and I have to say that I am sad that it did not appear in this section of our text, is the Enchantress of Numbers, as Charles Babbage named her, Ada Lovelace. The child of Anne Isabella Byron and the Lord Byron of poetry fame, Ada never met had the chance to know her father as her parents split a few months after her birth, December 10th 1815, in the most unamicable of circumstances leading Lord Byron into self-imposed exile from Britain. Lady Byron was terrified that Ada would become like her infamous and scandalous father that she decreed to herself that Ada would not learn poetry but instead teach her the ways mathematics and music. Lord Byron had nicknamed his wife “princess of parallelograms”, a name that had negative connotation to the poet, and as an amateur mathematician Anne Byron was well prepared to help lead her daughter down the path she had chosen. It did not always go well though, Ada once showed more interest in geography than mathematics and Anne did not just impose punishments, which for her included solitary for her daughter, but also deleted geography from Ada’s curriculum and fired the tutor. Ada did show aptitude for mathematics, even with the strict nature of her studies, and enjoyed the life of the academic enough to continue her studies through a near disastrous bought of the measles. Her friendship with Mary Somerville, the Queen of 19th C science, could not have hurt though.

It was at a party thrown by Somerville in 1834 that Ada learns of Charles Babbage’s, founder of the Analytical Society who sought to introduce European mathematical developments to England and creator of the first reliable Actuarial Tables, Analytic Engine which was his follow up to the Difference Engine. She began a correspondence with him around that time, and much of the rest of her life revolved around this decision. At around the same time Ada married the 8th Baron King, who was later elevated to the 1st Earl of Lovelace, which made Ada the Countess of Lovelace. Even after her marriage Ada’s life is controlled by the Lady Byron, whom apparently became good friends with King much to Ada’s chagrin. Her studies do not stop and the Countess began to surround herself with such scientific luminaries as Demorgan and Faraday, as well as other well known people such as Charles Dickens.

The Countess’s major contribution to the field of mathematics came in 1843 when she published a translated and annotated edition of Luigi’s Menabreu’s memoir on Babbage’s Analytic Engine. Ada’s notes are significantly longer than the translation itself and include  such brilliant observations of the Engine’s strengths beyond calculations themselves, which was Babbage’s main driving motivation, such as the production of graphics and music proving quite prescient giving what computer eventually became. In her notes she said, “The Analytical Engine weaves algebraic patterns just as the Jacquard Loom weaves flowers and leaves.” Her notes also include what was the first computer program, a method to compute Bernoulli numbers.

While her life was not always happy, she wrote in a letter to her mother, “If you can not give me poetry, can you not give me poetical science?” Her death death on the 27th of November 1852 is not the end of her story. Her notes were one of the early models for software when the digital computer age came upon the world, the US Department of Defense named the Ada programming language for her, and the British Computer Society awards a medal in her name. Even more important through is Ada Lovelace Day, 24th of March, which is a day of internet celebration and proselytization about women in technology and science that had over 2000 participants last year.