Friday, December 11, 2015

Was math invented or discovered?

In our last class of the semester, we were asked two questions: is math a science and was math discovered or invented. We then went on to have a discussion on whether or not it was a science because the class was much more split. For the question of was math discovered or invented, basically everyone in the class said it was discovered except me and like one or two other people. I thought I'd make this blog post to share my thoughts on why I think it was invented.

I think the biggest influence for me is what I attempted to describe in class, and I'll admit didn't do a very good job vocalizing my thoughts. I personally feel that math is extremely similar to something like language. Early civilizations needed a way to communicate with each other and eventually different languages were born. In a similar fashion, I feel like mathematics came about in a way close to this. I will say this is purely my thoughts and for all I know they could be completely wrong. However, I think earlier people needed a way to describe how many of something they had or a way to describe how much they owed another person so they invented certain ways to say these things. As a result mathematics was invented as a tool to get fix certain problems they had.

On the other side, I will have to say that the biggest argument that would sway me into thinking math was discovered is how did people invent all these crazy theorems and find all these interesting patterns, like prime numbers. It seems ridiculous to say someone invented all of these crazy ideas in mathematics.

Even though I say math is invented instead of discovered, I would have to say that I lean more towards the middle of the spectrum. I understand there are valid reasons for both arguments and they both make a lot of sense. Do I think that it really matters if math was invented or discovered? No not really. At least to me, I can appreciate and use mathematics in my life without really caring how it actually came about. Others might be more concerned and have stronger opinions on the matter. If it were to come out tomorrow that math was discovered, in no way would that change my opinion of mathematics, and I'd still continue to enjoy it.

Wednesday, December 9, 2015

Emmy Noether

So this is a pretty late blog post but I still thought I'd share it with you guys. A while back we talked about Sophie Germain and her contributions in mathematics. The biggest thing that stuck with me about her life was how difficult it was for her as a female mathematician. She went so far as signing all her papers as a guy just in case people wouldn't respect her findings if they found out she was a girl. That got me thinking about how many female mathematicians I know about, where I came to the startling realization that I know absolutely no female mathematicians. Because of this, I thought it would be interesting to read up on some and pick one to share with all of you.

Emmy Noether was a German mathematician born in 1882. Growing up, her father was a mathematician at the University of Erlangen where she attended many classes. After finishing at the University, she went on to work at the Mathematical Institute of Erlangen. She worked there for seven years without pay, similar to many other women of the time, then in 1915 she was invited to join the mathematics department at the University of Gottingen. Most historians break down her contributions to mathematics into three different periods of times (Wikipedia).

In this first period (1908-1919), she worked in many different fields of mathematics. One of which is in algebraic invariant theory. Some of her work in this field included expanding off the ideas of Paul Gordon, "the king of invariant theory," by making it possible to study the relationships between different invariants. Also during this first period, she worked a lot in physics. With her work in physics, she proved her Noether's first theorem which solved a problem with general relativity, a geometrical theory of gravity (Wikipedia).

During her next two periods, she made a ton of contributions to abstract algebra. First, she published a paper in which gives one of the first ever formal definitions of a commutative ring. Also in this paper is the Lasker-Noether theorem which basically generalizes the fundamental theorem of arithmetic. She is also credited with the ideas that serve as the foundations for algebraic topology (Wikipedia).

Overall, I found it rather interesting to learn a little more about someone who had seemingly endless contributions to basically every field in mathematics and is yet never even mentioned in any math class I've ever had. I know its hard to recognize every single mathematician, male or female, but I do think classes should at least try to incorporate some of the leading females of the time as well as their male counterparts every once in a while. At least after writing this blog my count of how many female mathematicians I can name has gone from zero to two.

Sunday, November 1, 2015

History of Math: Seven Bridges of Konigsberg

The Seven Bridges of Konigsberg was an option we could investigate in either one of the dailies or one of the hand outs on Euler. I think the main reason why I found this problem interesting was that it shows how mathematical theory can apply to the "real world" and further shows the brilliance of Euler.

So, Konigsberg was a major city for commerce and trade in the 13th century, in what is now modern day Russia, that was divided into four distinct districts by the Pregel river. To connect all the land seven bridges were made. Apparently, it is said that the people living there at the time always used to play a game where they would try to see if they could travel around the city to each of the four districts while only going over each of the seven bridges once. For reference, here is a picture of what the city and bridges looked like:

In 1736, Euler proved that there was no solution to this problem. In other words, there is no way for someone to walk around the entire city and only go over each bridge once. The real beauty of this problem is the way in which Euler solved it, or at least to me. In class we discussed how he always had unique ways of thinking about and solving problems and this one is no different. He realized that only the sequence of bridges mattered and not the actual route inside each land mass. Because of this, he was then able to construct a graph of the problem, with the land masses as endpoints and the bridges as the connecting lines as seen below:
Now with this new representation of the problem, it is fairly easy to see that each land mass has an odd number of bridges connected to it, either 3 or 5. Euler then concluded that there is no way to only use each bridge once no matter where you start or end.

Overall, this isn't too complicated of a problem, especially in today's world of mathematics, but at the time the significance of it was huge. For one, it is now considered the first theorem in graph theory and the first true proof in the theory of networks. Also, the way he thought of the problem was so innovative at the time. Unlike most, he realized that the key information in the problem was not the exact position of the bridges and land but rather how many there were and their endpoints.

Personally, I just find it amazing how one man could have such an insight into so many different aspects of mathematics. I always knew about the impacts he had in other fields of mathematics like Euler's formula or Euler's identity, but I didn't know that he basically came up with graph theory as well. I know we talked about how great he was in class but I didn't really appreciate or believe it until looking in to this and a few other problems he solved.

Monday, October 19, 2015

The Joy of X

For class, the book I chose to read was The Joy of X by Steven Strogatz. Strogatz opens his book by stating that his goal of writing this book is to have "a guided tour of math, from one to infinity." He realizes that many people are off-put by mathematics early in their lives for some reason or another, and hopes that through his book, them (or anyone) can once again come to enjoy or at least not despise math.

Honestly, I struggled to get through the first few chapters of this book. As someone who enjoys mathematics, I found the beginning to be rather dull as it just went through basic ideas about numbers and counting which almost everyone already knows. Because of this, my hopes were not high for the remainder of the book, however I was pleasantly surprised with how much I enjoyed the rest of it. In chapter 4, he talks about how counter intuitive the commutative property of multiplication is and how it breaks down in nature, i.e. the Heisenberg Uncertainty Principle. I had never really given this much thought and it sparked my interest and helped make me want to actually read more. Throughout the chapters from this point forward, there were quite a few more moments like this. For example, in chapter 15, he talks about sine waves and how they show up almost everywhere in nature, and in chapter 24 he talks about linear algebra and how its the basis for google's web search. All these little examples of mathematics occurring in everyday life really made me appreciate this book much more than when I started out assuming it was going to be wall after wall of boring text.

One thing I noticed about the book that I thought was kind of interesting was that the order in which he presents his topics is pretty much identical to how we have been learning about mathematics in our class. The first few chapters are almost parallels of the few first days of class we had in regards to the subjects that we discussed. Another aspect of the book I thought was kind of cool was that Eugenia Cheng and Strogratz both used the example of mattress flipping when talking about group theory and bagels with mobius strips.

Overall, I ended up really enjoying this book and would definitely recommend it to anyone interested. The amount of real life examples he uses to show the importance of math really hooked me. The fact that its a book about mathematics written in a style that allows anyone to be able to understand what is being said is also really appealing and this may have been the biggest thing that drew me to it. The fact that you can just casually read the book and understand what is going on or have an active approach to look at investigate some of the notes he has on further proving some topics was really nice. There were days where I didn't really feel like thinking too much so I could just read it like any other book and then there were moments where something he said was super interesting so you could go to the notes part and there might be further discussion or proofs on it. In the end, I have nothing but praise for this book and would probably be tempted to read any other book similar to this.

Sunday, October 11, 2015

Is math a science?

The question of is math a science was posed in our last class. My initial thoughts on this question is that yes it I would say it is a science. Before really looking in to this question, I would define science as tools used to describe what occurs around us, which I believe math does. Also, math is present in many of the natural sciences like chemistry and physics, so based off this I would also say math is a science. However, after doing some more digging, I'm inclined to disagree with my initial thoughts and say math is not a science.

First, after reading the conveniently named article "Is mathematics a science," one of the reasons the author thought it shouldn't be considered one really stuck out to me. He explains that when determining if something is true, the natural sciences use the scientific method. More specifically they look at tons of observations and then make assumptions off that. For example, say a scientist wants to determine if our climate is slowly getting warmer. He may gather data on temperatures from many different months in the last 20 years. However, if we want to say something is true in math, we use proofs. It is not enough to just believe something is true based solely off observations in mathematics. For example, if we wanted to say that an odd number plus an odd number is even, it is not enough to just look at 1+1, 1+3, 1+5, etc. We use the observations to make a guess, but we need more than this to come to an actual conclusion. As mathematicians we would want a proof that shows this is true for any two numbers x and y, not just all of the ones we looked at in our observations. In this sense math definitely doesn't act like a natural science.

Furthermore, my opinion on this question changed when simply looking at the definition of what science is. Science, defined by, is systematic knowledge of the physical or material world gained through observation and experimentation. Another way of saying this is science is based on empirical reasoning (making sense of all the data you've gathered). Let me first say this is only my opinion, and it may not be correct, but I believe math is not based on empirical reasoning. This is not to say that it isn't commonly used in mathematics like when finding patterns of numbers, but I don't need to make observations and do experiments to be able to do algebra. If I was asked to solve some equation for x, I don't need any background on the numbers, observations, or experiments to be able to do this.

Overall, this is simply how I my take on the question after digging a little deeper into the question. I can understand why some people would say math is a science since it is pretty much used in every natural science in some way, shape, or form. Because of this fact, there is inherently a ton of grey area in the question and thus many ways to interpret it. I don't think one answer is right and one is wrong, which you can see by the fact that I changed my mind from yes to no, and I'd love to hear someone's argument for why it should be considered one.

Sunday, September 27, 2015

Nature of Mathematics: Infinity

The other day in class when we had the discussion of what zero divided by zero is: zero, one, infinity... etc. At some point during this discussion, I believe professor Golden briefly mentioned something about how there are some mathematicians who don't think infinity should be included in math. For some reason or another this really intrigued me, so I decided to research more on my own about infinity and the problems along with it. While investigating, I came across an article from Quanta magazine, which I wanted to share with you through this blog. Just a disclaimer, I'm definitely not an expert on this subject and some of still confuses me, however I'd like to share my thoughts and knowledge before and after reading this article and the subsequent research I did on the topics presented by it. I'll start by briefly summarizing the article and then move on toward my views on the debate.

Today in mathematics, the "ZFC" or Zermelo-Fraenkel set theory with the axiom of choice is the standard form of axiomatic set theory and the most common foundation in mathematics ( From my understanding its basically the most accepted concept of fundamental laws of mathematics and set theory. However, there are some holes in it, namely infinity. One of these problems that has brought up much debate is the "continuum hypothesis" (Wolchover).

The continuum hypothesis states: "There is no set whose cardinality is strictly between that of the integers and the real numbers." (Wikipedia). The problem is that with the "ZFC" there is no way to prove if this hypothesis is true or false. In 1940, Kurt Godel showed that the continuum hypothesis could not be disproved, whereas in 1960, Paul Cohen showed that it couldn't be proven either, both assuming the ZFC axioms to be true (Wikipedia). This is where to two differing theories presented come in to play. First there is the forcing axioms which say the continuum hypothesis is false and then there is the inner model axiom, "V=ultimate L," which says the continuum hypothesis is true (Wolchover).

The V=ultimate L model, from my understanding, says that you must construct some set L that is equal to V, the universe of sets. In this constructed L, you must start with the empty set and use it to build bigger and bigger sets until encompassing all of them. In this manner, the resulting universe of sets would prove the continuum hypothesis to be true (Wolchover). Hugh Woodin, a mathematician at the university of California, Berkeley, found that such an L could, in theory, be constructed. However, such an L has not actually been made yet (Wolchover).

On the other hand, the forcing axiom looks not to create a new model that encompasses everything, but rather expand and simply fill some of the holes of the existing model. Stevo Tordorcevic, one of the leading activists for this method, likens it to the idea of complex numbers (Wolchover), which I thought made it really easy to understand. Just like how complex numbers fill gaps in our understanding of the number system by adding on to it, the forcing axioms work in the same way.

Before reading this article, I honestly had no idea that this problem with infinity existed, or any problem with it for that matter. That's why I thought it was so interesting when it was mentioned in class. After doing some research, I found out that are actually quite a few problems with infinity. For instance, infinity has two distinguishable types, actual infinity and potential infinity. The difference between the two is also quite intriguing but for fear of digressing I'll leave it at that.

So on to my thoughts about the article. After learning about both sides of the argument, I think I'm more inclined to agree with the forcing axioms. The sole reason for this is because the forcing axioms have concrete results that deal with the problems we have now. Natalie Wolchover, the author of this article, puts it in to words much better than I could by saying, "while V=ultimate L is busy building a castle of unimaginable infinities, forcing axioms fill some problematic potholes in everyday mathematics." I've always enjoyed the more applied part of mathematics so perhaps that is why the forcing axioms appeal more to me. However, the idea that one day a set could be found that encompasses and fixes everything is also fairly appealing. As of right now, its only a matter of opinions on which one you like more until anything can be further proven, so I'd like to ask which side you would chose ?

Sunday, September 13, 2015

Doing Math: Archimedes' Stomachion

When I was younger, I had an obsession with puzzles. I would spend hours at a time putting puzzles together or playing with tangrams online. So when there was an option to try Archimedes' stomachion as part of work for this class I was stoked. I was fairly confident this wouldn't be too difficult, but soon found out how wrong this assumption was. If you don't know what the stomachion is, its basically a rectangle that has been cut into 14 distinct pieces and the goal is to put it back together.

I spent hours working on this but ultimately was never able to complete it. To start out, I just randomly started placing some of the bigger tiles into the corners and trying to put the smaller ones around them. Obviously, this strategy didn't really work too well. Here was my best attempt just randomly placing the tiles anywhere.
For a while, I thought the top left corner was right but I wasn't making any progress so decided to try a different approach. Instead of trying to fill in the rectangle, I started to focus on the individual pieces. The long light blue one in the bottom right corner was a real pain to find a decent spot for, so I literally spent about 20 minutes trying to find out the other piece that went along the top of it. As you can see the darker blue triangle worked pretty well.  I then spent the next hour or so doing the same thing for each of the other larger more annoying pieces like the big purple piece. Then I started to focus on filling out the sides of the entire rectangle. This was about the time my second big progress came when I filled out half of the rectangle.
At this point I was 100% positive I had gotten that half the rectangle correctly tiled and thought I would be done with the puzzle in no time. Not long after I came as close as finishing it as I possibly could without actually doing so.
Words cannot describe how disappointed I was when I found out that the last little piece didn't fit. At this point I finally caved and decided to look up the solution. As it turns out, Bill Cutler found out there are 17,152 different solution and 536 truly distinct ones. After finding out there are so many I'm a little upset I couldn't find one of them. However, it amazes me that Archimedes, or anyone, could have created something that seems so simple on the outside, but when actually analyzed is so much more complicated.