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.

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.

We started this week with a reading of a few section from chapter two of our book, The History of Mathematics by Roger Cooke, specifically those dealing with the mathematical history of India and the Maya. The mathematical history of India is itself, removed from any external context such as the over-all study of the history of mathematics, incredibly interesting.

One of the oldest cultures in the world the history of Indian mathematics reaches, of course, well into BCE. As is common with mathematics of that time, the math seems to be mostly concerned with geometry and artimetics. In fact, according to Cooke, sometime between 800 to 500 BCE the Sulva Sutras, who’s root words come from measure and cord, a collection of mathematically based verses were inserted into the Vedas. These verses, and the idea that the content probably springing from the maintenance of  altars, are intimately tied to a conversation that we had in class on Tuesday: The importance that culture and religion have on studies, and mathematics in particular.

Professor Bhatnagar brought up the semester he spent at the University of Nizwa in Oman, and the perspective the students there brought to their education, specifically that they came into classes expecting to be able to memorize their way through instead of learning basic concepts and then extrapolating from there to solve here to fore unseen problems. Professor Bhatnagar then posited that there was a good reason for this and someone else from the class spoke up that it could have something to do with the practice of memorizing large section of the Qur’an for recitation, a hypothesis that was quickly seconded by many in the class and was agreed with by our Professor. Of course it does simply end there because, as our Professor quite rightly pointed out, it is also a great honor to be one chosen to do the recitation and because of that the students were not only well practiced in memorization but have a large respect for the method.

There is no reason to stop the speculation on the effect that religion and culture have on mathematics there though, let me spend a second talking about mathematics in the United States. As we spoke about on Thursday after the USA declared its independence from the Untied Kingdom way back in 1774 it was not only in governing that we decided to break away from the British model. We also changed our education system quite a but as well, so much so in fact that there is very little in common with the two systems now only 236 years since independence. The United States university system tends to function on the idea of: If more than one person wants to study it, it is probably worth studying as opposed to a more track based system such as that in the United Kingdom. While I can not say I agree completely with this idea, I am proud to say that I am a product of a system that does, for some reason that eludes even my radically liberalized mind, offer underwater basket weaving as a for-credit course in more than one university.

The important part of offering all of these courses, and all the different ways to take them, is that having such an open system allows and encourages innovation. There are universities such as Evergreen in Washington state that do not seem to even have grades in classes, Redd where you get to design your own degree from the bottom up, and technical colleges where regimentation is the name of the game. This culture of openness has also nurtured special minds like Terrance Tao, a man more brilliant than even his Fields Medal gives him credit for as he managed to use Analytic techniques in Number Theoretic fields a marriage that many great mathematicians had thought impossible, to their greatest possible potential where it is possible a more discretely structured system could have stunted the creativity in Tao’s intellect which would have never the less produced a great mathematician but not one who could have generated the same results. It is not all greener grass in the United States though, we also have a very strong fundamentalist Christian population and I would be remiss if I did not mention the effect I feel that that religion has on mathematics.

One of the biggest philosophies of the fundamentalist Evangelical sects in the USA is that science is much more than simply wrong, it is the work of the devil. That science makes people question the truth of the world being 5000 years old, the truth that dinosaur bones are just a test of faith, and the truth that humans, and everything else, were created by the One True God. This has become a very large problem in the education system in the United States, from schools being forced to put stickers on their science books stating that evolution is just a THEORY to teaching creationism in biology class. There are much more insidious and non-obvious effects though and the one that I feel most dangerous is a distrust that all of these preachings cause in science and, by extension, mathematics. Students will go into a class room having been told that everything they will learn there are falsehoods and therefore they see no reason to pay any mind to what is being taught. Mathematics really challenges such dogmatic beliefs with the idea of proof and mathematical truths. If there can only be one truth, the truth of God, where does the Pythagorean Theorem fit?

This conversations quickly spun into a discussion of the conditions that are needed in order to have a flourishing mathematical culture. Going from the reading it seems that one possible way is to have a lot of questions that needed answered. Look at say the Mayans, who had a very strong culture of Mathematics; a huge amount of it centered around astronomy and the calender, such as the Dresden Codex that Cooke wrote about, questions very central to the Mayan culture and religion. I also feel that one of the sufficient conditions for mathematics is having a liberal and available education system, as most of the Western World have accompanied by a huge amount of new work being published in mathematics, because as long as people have access to a subject at least some of the people will work on it something. That is obviously not true if they can not receive the education necessary to understand said subject. One final thing on this subject we covered in class was: what happens to mathematics in a despotic system?

The two examples that were mentioned in class I believe are quite good illustrations, but I do not know if they are inclusive. The first was Napoleon. While Napoleon took control in France he was of the mind that pure mathematical research was not of that much import, he was much more worried about applied mathematics and of course how it could help him win his war with the rest of the world. This was at least part of the reason he sank so much money into the Ecole Polytechnique in France where mathematicians were trained in the applied side of the science. While this very well could mean that we lost out on some very good pure mathematicians but it did not completely backfire for mathematics as a whole as  Poisson and Poinsot were proud products of the Ecole Polytechnique. Also left out of the Polytechnique though were the women. Sophie Germain chief among them. A correspondent of both Lagrange and Gauss, Germain was not allowed at attend the Ecole Polytechnique and was therefore forced to teach herself mathematics eventually winning an award from the French Academy of Sciences. Germain was due to finally receive the deserved doctorate the Ecole Polytechnique would not give her from Gottingen University when she perished to breast cancer before she was able to meet with Gauss, who was to award her the degree.

The other despot that was mentioned in class was the one that almost all conversation of despots and dictators end up, Adolph Hitler. Nazi Germany was of course well known for the suppression of any sort of creative art that did not coincide with the philosophies of the National Socialist Party, but the Nazis did invest a lot of money in the sciences. The reasons were not that different than the reasons that Napoleon had, they had a war they needed to win. Once again it was applied mathematics that was to benefit over the pure as the Nazis sank more and more money into the creation of more accurate guns, fighter jets, bombs, and, of course, missiles. One important thing that can be seen from this more modern example though is the effect that a despot can have on the mathematics and sciences of other countries. Without the shadow of the Nazis, and the Japanese, there is a good chance that the breakthroughs in cryptanalysis at Bletchly Park and the amazing work of the Manhattan Project, no matter your thoughts on the Atomic Bomb still an amazing feat, would have happened. In fact, while Turing had already written his seminal paper in Computational Theory when the war broke out, would we have seen a working digital computer as quickly as we had without World War Two? I personally do not think that we would have had the knowledge without the computational machines built to help defeat the Axis powers.

As I mentioned before this is not a sufficient example to make any causative claims about the effect that despots have on mathematics, and the sciences, but I would be willing to wager, and living in Las Vegas I now know the true strength of those words, that looking at a much larger samples the same results would continue to appear. The reasoning is quite trivial: despots crave power above all, in order to gain and maintain said power they need more and better weapons than all of their foes, to gain this advantage they have to leverage all of their options, and the best of those options are the mathematicians and scientists whose mind are singularly skilled to tackle such problems.

We finished off the week with a discussion of the projects that we will be tackling as our main research task during the semester. The projects are all tied to the task of giving the 50 year history UNLV Mathematics Department. The basics are good listings of faculty, graduates, and class histories. I personally decided to do something a bit different and volunteered to put together an oral history of the UNLV Mathematics Department. This is of course because of my experience working with audio in the podcast world, but also because in history I think the story is perhaps the most important, and definitely the most captivating, thing. Oral histories have also become incredibly popular, especially as a way to consume history. I can only conjecture that this is because most raw data, facts, figures, and the like, is now not only available, but easily searchable on the internet. As a matter of fact Google has really killed the old practice of sitting in libraries until all hours, coughing from the dust shot into the air by opening old tomes to skim and losing one’s eyesight scanning microfiche for some tiny piece of information. The one thing that can not be found on the internet a lot of times though is the story of what happened by someone that was there. As a mathematician I am quite aware that numbers rarely tell the whole story, and analyzing a topic from a fact only perspective will lose all the nuance and color that a real history should have.

I found myself in the odd position of missing the second class of the semester and therefore missing the first real lecture of the year, as well as losing my change to gain an insight into how the class was going to be approached. Not that I wasted the time I should have been spending in class adsorbing the material. Instead I found myself in San Francisco at the Joint Mathematics Meetings. The second conference that I have attended and, thankfully, the second at which I presented. It was a radically different experience though as the first was a Graph Theory and Combinatorics conference with approximately 300 attendees, rather a smaller amount than the, at least, 5,500 people who made it to the JMM this year. I would be lying if I said it was not surreal scooting past Ron Rivest in a lecture hall or rubbing shoulders with Donald Knuth, sharing the same air with such luminaries of mathematics reminded once again the importance and gravity of our chosen subject. With my presentation, and the ones that I attended, I remembered the mutating and growing nature that is mathematics which really helps to put the challenge of studying its history in perspective. Also speaking to, and interviewing for my podcast, people like Richard Stanley from M.I.T., Steve Strogatz from Cornell, and Joseph Gallian from Duluth I was able to learn just how much mathematics can change in a short period of time. In the end though it was not all work for me in San Francisco, as I was able to spend a lot of time just talking to other mathematicians near my age. Therefore I was able to revel in the companionship that only the shared knowledge of such an exalted subject can bring.

All of this made the first chapter of the book slightly surreal. I had spent four days immersed in a sea mathematicians wearing name tags so then reading about people of whom we only have the vaguest of grasps bordered on spooky. To then leave the realm of certainty in class to talk about whether numbers were created or discovered was an even greater departure but not an unwelcome one. It is that kind of question, along with how does a child perceive mathematics and what is the intersection of mathematics and art that I feel that most mathematicians tend to avoid because they are so called soft questions. Just as what the History of Mathematics is seen to be. To not ask these questions though is rather obviously a mistake in my mind. As we learn more about the mathematics of the ancients through archeology or finding something someone else missed, we can get a more precise image of the mistakes they made and, more vitally, how they succeeded. Through these stories of achievement and failure we will come to gaze on the story of our discipline and better see where and how to move forward.