Ramanujan

I have now gone through all the different types of blog posts that were assigned in our course. I am looping back to looking at the history of mathematics in this blog since I feel that my last post on the history of mathematics was a tad recent.

Ramanujan interested me right away when we talked about him in class. I didn’t know exactly what drew me to him at first, though by the end of my research I was a little clearer. I watched a BBC documentary on him for one of my class daily’s and decided that writing a brief history on him would be an enlightening experience for why I was interested. The first part of this blog will be a brief history of his life, followed by a section on his work, and finishing with a critical view of his life and legacy.

Ramanujan is one of those mathematicians that come around once in a hundred years. He had an innate ability to understand math in ways that were far from traditional. Growing up in a small town in India, his ability in math could not carry him forward in traditional academics since his other subjects were far from proficient. He taught himself mathematics from limited sources and began communicating with GH Hardy in his early twenties. In the early 1900’s Ramanujan left India and came to Cambridge to work with Hardy. His life was far from happy here. He was a strict vegetarian because of his Hinduism. But because of the intensity of his vegetarianism (nothing could be cooked in a pan that had once held meat) he often didn’t eat. He wrote home to his wife to send food “the cheapest route possible.” Because of the weather and his lack of nutrition Ramanujan was frequently sick. He left after six years in England, but was so sick that he lived barely a year in India.

According to his widow, Ramanujan was always doing “sums.” His mathematics was focused on infinite series, continual fractions, and other areas of mathematics. His development of a theorem that can predict the number of divisions for any given number is revolutionary. It combines areas of mathematics that many wouldn’t equate to each other. His way of doing math was simple and not like that of traditional math that requires rigorous proofs. Because of this he wasn’t well received in an academic environment that was more traditional. It is only because of Hardy that his work was known at all. Even though Hardy acknowledges the brilliance of Ramanujan, much of Ramanujan’s work is written with Hardy instead of alone.

The last part of this blog will be a more critical view of his life and legacy. Ramanujan is viewed as this brilliant mind that left us with so much that we don’t understand. Rightfully so, there is a critique of how he was treated in England. But more than anything else the story is just sad. In the entirety of my research there seems to be a prevalence of sadness. The unforgiving school system in India didn’t support him, the racist undertones in England didn’t respect his work unless he conformed to their norms, the unsupported restrictions of his religion and diet, and more led to a life that was sad in England. But all of this sadness can be blamed on others. My thoughts are that he partly chose this sadness as well.

There is a story from his widow that talks about how he would be so focused on his sums that he wouldn’t stop to eat. His wife was forced to put rice in his hand so that he wouldn’t forget to eat. This makes me think the guy was a jerk. What kind of person doesn’t care enough about his wife to stop and eat dinner with her? I understand that I can’t judge a culture by my standards, but I don’t like this. Ramanujan also leaves his wife alone for years while he pursues his math in England. I don’t care how brilliant one is and how important one’s work is, there has to be a level of humanity in oneself.

What I would argue is that there needs to be a broader scope when looking at the life and legacy of Ramanujan. There needs to be a continuation of understanding that England’s academic system was not as receiving as it should have been. But there needs to be an affirmation of the support he did get from Hardy. There needs to be a disdain for a culture that was less welcoming than it should have been for someone who came with a different way of doing things, but through the proper hermeneutical and forgiving lens. There needs to be honesty when looking a Ramanujan’s personal life that allows one to be critical while again looking through the correct lens. Ramanujan is a man who will not be forgotten but a story that is laden with sadness. 

No Controversy Post on The Eight Queens Problem

This blog is going to take the place of our third blog post. The one topic that I haven’t covered yet is communicating mathematics. I doubt this will be as controversial of a piece as the previous post was since it will be more informative and less opinion based. So if one is reading this looking for something similar, I apologize.

In looking back through our course page on different thinkers around the time of the third blog post I was drawn to Thabit ibn Qurra since I knew so little about him. I wanted to delve into his mathematics for my blog. Looking at his Wikipedia page, I noticed a link to a chess solution. From that page I found a link to the eight queens puzzle. Following my interest and Wikipedia links I decided to settle on this as my topic for this blog.

The problem is as follows. Is there a way to place eight queens on a standard chessboard so that none of them could attack each other either diagonally, horizontally, or vertically? The history of this problem is part of my interest.

Max Bezzel first introduced the puzzle in 1848 in a German magazine. Over the next six years there were a variety of solutions published in various magazines. With a reposing of the question the problem shifted to a question of how many different solutions to the problem there were. Franz Nauck was the one responsible for this reposing and asserts that there were 60 solutions to the problem. This sparked a debate between two friends Gauss and Schumacher. They communicated with letters for around a month. During that time Nauck corrected himself and affirmed that there were actually 92 solutions, but offered no proof of this assertion. Gauss referred to Nauck’s correction and laid out how to go about proving it, though never actually proves it. It seems that this task wasn’t of extreme importance to Gauss since he didn’t spend the time to solve it. These letters were published about fifteen years later.

The solution that Gauss proposed was clever. I’ll try to communicate as best I can the solution that was proposed. Gauss reformulates this problem into an arithmetic one. Gauss begins by assigning a position to each permutation of the numbers one through eight. Thus 12345678 would have the first queen at a1, the second queen at b2, the third queen at c3, and so on. The permutation 45632178 would have the first queen at a4, the second queen at b5, the third queen at c6, and so on. Using this idea of position Gauss was able to work with the solutions more easily. He found that if each permutation was summed with 12345678 and separately with 87654321, and if these sums were all different in each number that this was a solution to the problem. This is difficult to understand without an example so I will share the example that Gauss sent Schumacher.

All these sums are different in this permutation of 15863724. This means that it is a solution to the eight queens problem. In the next permutation of 13425867 we can see that 10, 9, and 8 all appear twice. This means that it is not a solution to the problem.

  

By setting the problem up this way Gauss was able to more quickly sift through solutions. The setup itself prevents queens from lying on the same row or column. The addition of the permutation 12345678 prevents queens from lining up diagonally to the right; the addition of the permutation 87654321 prevents the queens lining up diagonally to the left.

The solution isn’t flushed out arithmetically that I could find, but was explained to have 12 fundamentally different solutions that, through rotations, are flushed into all 92 solutions. This geometric solution is nice, but I wanted the arithmetic solution.  

The problem continued to evolve into boards of differing sizes and got into complications that I won’t flush out. As stated earlier, Gauss’ passion didn’t seem to be evident in this problem since he didn’t bother to create every permutation, or offer an argument for why there were only 92 solutions. I wish that his argument had been clearer and that Gauss had had more passion in the subject. This was a fun problem to post on because it combines my passion for chess and math!

David Kung

In my capstone class one of our “dailys” (or homework that we could do for that day) was to watch a lecture by David Kung. I watched this video over a week ago but it is still on my mind. Deciding what to cover for this blog post I decided that it should be something that was relevant to me. Since this lecture was so fascinating, I decided to focus on that. I would label this as studying the history of mathematics since it is looking at the contributions of this mathematician to his field. I will give a summary of the lecture, my reactions to it, and end on how I think Kung could change his lecture.

To start the post I will simply give a brief summary of the lecture. The lecture was on diversifying the mathematical community. He began by presenting the audience with a series of statistics on the lack of diversity in all STEM education. He would start with other subjects represented in a line graph that would show something like, the number of women with a doctorate in that subject over time. He would then ask the audience to draw their best guess as to where mathematics lay. Every type of diversity had a dismal report in the field of mathematics. Women, Blacks, Hispanics, etc, were all underrepresented in every part of the mathematics field. From here, Kung laid a framework of how he would change the type of mathematics education to be more conducive to fostering those minorities who aren’t typically part of math education. He finally ended with a story that told the story of one student who was able to succeed in a mathematics degree despite the social and institutional hurdles in her way.  

The suggested method of closing this achievement gap was through pedagogy, though there is an acknowledgment that pedagogy alone will not solve the problem. Passive teaching in STEM subjects, especially at the colligate level, is keeping students away from these degrees. Passive lecturing is becoming available in more and more forms and has the added benefit of being “pauseable.” Kung calls passive lecturing “professional misconduct.” There is a shift to more interactive pedagogies in earlier grades, but the colligate level is not following suite. Kung sites studies that find that interactive pedagogies, which focus on a growth-mindset opposed to a fixed-mindset, are able to help close the achievement gap by being more accessible to a wide variety of students.

Another method for closing this achievement gap is through recognizing the inherent bias that exists in classrooms already. In order to recognize this Kung wants to look at questioning patterns in the classroom. It has been shown that more time is spent with males than females. A teacher is more likely to call on a male student and is more likely to ask deeper, more thought-provoking- questions to said students. This is also true for minority students as well. By recognizing our bias and understanding our prejudices, teachers will better be able to approach all students fairly.

The reason that this video has stuck with me so long is the inherent and unrecognized bias that David Kung brings to the table. Taking a step back, I need to be careful about how I phrase this since I will be publishing this online. I want to affirm that my opinion is not backed up by research and that it is only my opinion based on my experiences as a white male. That being said, I feel that Kung’s attempt at being diverse and accepting, purposely ignores the views of an entire group based on their skin and sex--i.e. white males. If his goal is to be tolerant and accepting of others in education he can’t ignore this entire group. The most telling part of this is during a story about how women will do poorer on a test if there is a question about their sex before the test begins. After his point is made he tells how this research is backed up because they tested male’s golf skills with a similar question about if it was testing athletic ability or just for fun. His first story gets a bunch of quiet and sad nods from the crowd but his second story draws a laugh. I’m not doing the story justice, but this story is marginalizing a group. And everyone laughs. If one wants equality, there cannot be jokes on others behalf either.

There is so much talk about prejudice and how we need to draw out our inherent bias, but bias is bias. One can’t have it both ways. There seems to be a culture of blame in this lecture and one that I have seen in many other situations. Blaming the majority is an easy thing to do, but it doesn’t establish the culture of respect and authority that one is so desperately seeking. The argument that the majority can take it since they are “ahead” is not accurate though; it is simply using a system that one wants to unravel. At one point Kung talks about a picture of a selection of excellent professors at his university. He jokes that he is the diversity factor in the picture of fifty or so white men. But isn’t this marginalizing their accomplishments? Why is it their fault that they succeeded? I struggle to understand his lack of equality while operating under its guise.

I am all for diversity in every facet of life. I understand the lack of diversity that is in higher academia. But whose “fault” it is that diversity isn’t as prevalent as it should be is up in the air.  According to Kung, Berkley professors in the 1970’s blamed students not working hard enough for the lack of diversity. Today that blame has shifted to society. But it seems that society has become synonymous with white males. It won’t be until this mentality of blame has shifted to encompass all that actual change can happen.

I don’t want to affirm the Berkley professors of the 1970’s but I want the burden to go back to the students. I want to create a society in academia that is supportive of all groups so that success is based on the student. If this culture is created, and all black females are the ones who succeed, I don’t want there to be a culture of blaming them for marginalizing others. Those who try the hardest and perform the best are the ones who should succeed in a true democratic society. There needs to be equal opportunity and no judgment of those who succeed. Kung needs to follow his own practices and recognize his bias. By removing that and his inability to not blame others for society's problems he could accurately address the problems.