• 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.

Joseph Gallian is a Mathematics Professor at the university of Minnesota-Duluth. Samuel Hanse spoke with him at the Joint Mathematics Meeting in San Francisco about his text book Contemporary Abstract Algebra, the mathematics movie Hard Problems he executive produced, how he cracked the Driver’s License code, and more about the Beatles than you could think would be on a math podcast. You can find out more about Professor Gallian at his website.

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One interesting way to look at the idea of a paradigm shift is through the lens of the Singularity that Venor Vinge was kind enough to give us. The term singularity is one that mathematicians are very comfortable as they are a term for where some mathematical object happens to not be defined.  Say we are talking about a function that is not defined at zero, then the function tends to behave oddly near this singularity. The same can be said for matter near black holes, which are called gravitational singularities. It was from these ideas that Vinge came up with the term Singularity, in this case referring to some point in the future when technology will stop increasing in speed at an algebraic level and start progressing at essentially infinite speed; usually this refers to AI or self-replicating machines. The reason that I feel this lens could be useful to look at paradigm shifts through is because of something that Singularity Science Fiction, the Singularity does have it own sub-genre of Science Fiction literature, author Cory Doctorow once said, that for all essential purposes the Singularity is the point at which human beings that were raised under the conditions caused by the Singularity are incapable of meaningfully communication with those born before the event. He went on to posit that human beings have already gone through multiple points of Singularity, the greatest of which would be the invention of spoken language. After which it is quite clear that meaningful communication with those who do not have spoken language by those who do would be, for any practical purposes, impossible.

While I do not believe that any of these paradigm shifts qualify as full Singularity events, I do appreciate the problems that those who learned mathematics after we had the calculus would have communicating the mathematics of, say a thrown projectile to those who came before the calculus was know. It is in this way that I wish to discuss Alan Turing and the beginning of computing.

Computers for a long time did not refer to the machines that make the soft white noise hum that constantly surrounds us wherever we go, or who’s flickering LED’s let us know that the world is continuing to function. No, computers for a long time referred to the human beings who computed the values of arithmetic operations. Whether they did it on paper, with an abacus, or some early predecessor of the digital computing, such as Leibniz’s Stepped Reckoner, the computers were people. It was Alan Turing, 1912-1954, who changed this paradigm on the 28th of May, 1936 with the publication of one of the most important papers of all time, “On Computable Numbers, with an Application to the Entscheidungsproblem”. The paper was a response to another on the list of all-time important work, that of Kurt Godel and his incompleteness theorem. Turing introduced the idea of Turing machines, and through manipulations of these incredibly powerful and simple abstract machines showed that any computation that can be done can be done using a Turing machine.

This basic meaning of this, also known as the Church-Turing Thesis, is that instead of having to build machines, or write an algorithm that is computable by hand, all one has to do to show that any computation is possible is show that it can be done using a Turing Machine.  This was not the only important thing in that paper though, since it also contained a proof that the Halting Problem is undecidable proving there is no solution to the Entscheidungsproblem; not the first such proof but by far the most intuitive and easy to understand. It is this intuitive nature of Turing’s work that is so important. All of the theoretical work of people like Godel, Church, and Turing was fine and dandy, but in order for it to have a real impact on human being’s lives, set theoretic work such as this is quite often the most abstract and hardest to apply, some application was going to have to occur. This application turned out to first be the analog computing devices, such as those used by Bletchly Park to crack the Nazi Enigma code in WWII, and then the digital computer, upon which I am currently writing this sentence. The type of people who tend to build things like computers from the ground up, engineers primarily, are usually quite different from a mathematician who spends his life at a blackboard abstracting the world away. Therefore the equivalent definitions of Godel and Church which are nearly impossible to get a good handle on without decades of training, which is an unlikely  time investment for the engineer who just wants to build a machine, but, this is the the most important component of Turing’s work, the Turing Machine would be easy for the engineer to grasp at an intuitive level and begin to apply.

We are now nearly 74 years past the submission of Turing’s paper and our entire world is run by the physical manifestations of his machines. I try to imagine going back 80 years and discussing research with a mathematician of the time. No matter what problem we decided that we were going to research for I would inevitably just say that we should Google it as an initial step, then use things like JSTOR or MathSciNet to find more scholarly results. The mathematician of the 1920s would look at me as if I were a loony and then I would quickly become very happy I did not go farther back in time as if I had I probably would have been burnt at the stake as a warlock for such insane sounding statements. I would then have to join this mathematician at some library, walking through the stuffed shelves looking for some forgotten tome, and, sparing that, writing with a quill on paper to some mathematician elsewhere asking for their help in procuring the material that we needed to continue. The differences caused by this 80 year gap are nearly impossible for me to gain any purchase on, and I think would make it very hard for two people of the eras to work together. This is not simply conjecture though, as I have had the experience of dealing with those who refuse to adapt to the computer age and I find it very hard to find any sort of shared ground upon which we can share a conversation since computers are such an integral part of my life and being. It is really this inability to communicate with those before, this mini-Singularity, that really characterizes this paradigm shift.

I am not fully satisfied just talking about paradigm shifts that have already happened though, I want to speak a bit about one that I feel is coming in mathematics and that harnesses that which is so powerful, crowds. Crowds have become the poster child of internet, and more specifically how to Get Things Done using the internet. Want to make a repository of all the world knowledge, well crowds can do that and have on Wikipedia. Want to have an algorithm that gives out meaningful search results, Google leveraged the knowledge of crowds to do this using Page Rank. Want to prove a mathematical theorem, well this is the question that Field’s medalist Timothy Gowers asked on a pot to his blog, gowers.wordpress.com, on the 27th of January last year. In the post “Is massively collaborative mathematics possible?” Gowers outlined a problem, Density Hales-Jewett, and a method, using the comment sections on his blog, and then let the crowd go. A couple of months later he had his answer, which was: YES! The first polymath paper, “A new proof of the density Hales-Jewett theorem, is available over at cs.cmu.edu.

This is a can of worms that will not be able to be closed; in fact fellow Field’s medalist Terrance Tao jumped on board with his own polymath projects, as has Gil Kalai and Gowers himself has a couple new ones running at this time with many more on the horizon. I truly believe that this is not simply something new, it is something great which will lead to some incredible, interesting, and unexpected results. In fact it was even listed by the New York Times as on the Big Ideas of 2009. While it leverages many of the same people that already did mathematical research, professors, graduate students, and industry mathematicians, the real breakthrough is the way that this method of research can bring in the mathematically inclined amateur who up till this point had no easy path into mathematical research. These people bring such a different perspective, one that has not been tainted by the methods and biases present in high level mathematics education, that they will be able to see and attack problems in ways that the current mathematical elite would never dream of using. I will go on record right now that with in a decade a major result will be proven using a version of the polymath method and that within a few decades the students coming into their own in the field of mathematics will wonder how work was ever done before their predecessors started using the wisdom of the crowd.

Other than the obvious intelligence of deciding to use crowds to help prove theorems, the shrewdest thing that Gowers did was realizing that mathematics is not a lonely pursuit. There are of course very foundational examples that would prove me wrong on this, most recently the Poincare Conjecture proof by noted recluse Grigori Perelman, but if one looks through the mathematical publications in any journal it is an incredibly rare event to find one that does not have at least two authors. By understanding that most work is collaborative and the importance of an active community within mathematics, and the strengthening of this community by working on theorems together may have an even stronger impact than the polymath results themselves, Gowers was able to bring together many people who would not have worked together under other circumstances and knit them together into a functioning mathematics producing machine. I am just glad that I will be able to look back and say I remember when this started.

The young Isaac Newton’s “debtor’s ledger of sins”:

• Stealing cherry cobs from Eduard Storer
• Denying that I did so
• Robbing my mothers box of plums and sugar
• Calling Derothy Rose a jade
• Punching my sister
• Striking many
• Wishing death and hoping it to some
• Threating my father and mother Smith to burne them and the house over them
• Striving to cheat with a brass halfe crowne
• Making pies on Sunday night
• Squirting water on Thy day
• Not turning nearer to Thee according to my belief
• Setting my heart on money learning pleasures more than Thee
• having uncleane thoughts words and actions and dreamses

(Gravity and Levity via Aisha)