Sunday, October 18, 2015

"How Not to be Wrong: The Power of Mathematical Thinking" Review

Blog Post Review

 I recently read How Not to be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg. To say that I enjoyed this book would be a stretch.
The biggest problem with this book was that it had no idea who its audience is. There was a disconnect between much of the information being given. On one had the author would provide us with an overly simplistic explanation of math that a sixth grader could understand, and then go into overly complex math that is explained in an incredibly illogical and confusing way. His jumping from topic to topic without building on previous issues further muddies the waters on who he is trying to reach. His introduction needed to define his audience and explain his thesis for said audience. This clarification would have better organized his thoughts for how the book progressed.

This disconnect made for a very long and tough read. But the authors attempt to tie these concepts together is perhaps the one guiding light in the book. His attempt to connect concepts was by using an overarching example that he could keep referring to. This was done almost every chapter and gave relevance to the concept that her was trying to get across in said chapter. And these stories were fantastic and relevant. His use of the lottery story in Massachusetts allowed him to constantly give context to a series of mathematics that wasn’t incredibly interesting. If he had been able to tie the chapters together like he tied individual chapters together, it would have been a fantastic book. Instead the book continued to feel disconnected.

Unfortunately, this is where he fell apart. Though the stories were good and relevant, the author failed to come back to his overall thesis. The thesis of the power of mathematical thinking wasn’t addressed in a relevant enough way. The book read as a series of small vignettes and didn’t come to a conclusion. I wanted to know how and why the book was going in its direction. Even the movement of the mathematics wasn’t connected in a relevant enough way to even hint at a thesis. Individual chapters hold weight. I could see splitting this book into different articles if better concluded in each chapter. In this way, at least, the author could have better argued individual points about mathematical thinking.

I guess, in conclusion, that the real problem with this book is that the majority of the mathematics lies in an area I just don’t love. Statistics is imprecise and just isn’t up my alley. I understand that it is used in ways that get at the heart of mathematics, a way that describes the world around us, “it is the extension of common sense.” And statistics embodies this. I just like to live in the abstract world when it comes to mathematics. The author attempts to describe a mathematical world that we live in but falls short. A few of his insights give credence to his views, but he fails to share them in a cohesive way that argues for any type of point. Mathematics describes the world we live in and mathematical thinking allows us to navigate within it, but complex, unconnected stories does not, a thesis make. 

Monday, October 5, 2015

Chautisa Yantra

This blog had to be screenshotted from word since I couldn't figure out how to transfer it over once I wrote the equations. ya go!

Saturday, October 3, 2015

Blog Post One: Greek Influence

My first post since talking about the nature of mathematics is based on the influence of Greek mathematics. I enjoy history and especially the history of how ideas came into being. During my time as a philosophy major I was typically more interested in the history of how ideas developed; how different thinkers countered each other using their best logic. Mathematics interests me in a different way than philosophy does. I enjoy the computational logic of watching a long problem come together with perfect clarity. I enjoy the satisfaction of having struggled through a problem and achieved the singular correct answer.

Thus my tendencies lie in two different camps when comparing how I approach philosophy and math. I love being able to argue with no answer in sight as a philosopher, and I love the ability to come up with the “correct” answer in math. My hope throughout this course is to unify these two opposing forces and enjoy the history and development of how math arrived at my doorstep as a young mathematician.

The first part of this journey had me delve into the history of Greek mathematicians on modern mathematics that I have seen in my life. This requires me to look back on my own mathematical journey as a mathematician. To be brief…I grew up as a “gifted” child in mathematics. I never struggled in math and excelled in every class I ever took. I was frequently told how smart I was and never spent much time on math until I got to the math beyond calculus. Suddenly I was expected to struggle with mathematics, the only problem being that I had never struggled before. I didn’t know how to act and it took copious amounts of work to succeed. But the math content I saw was the typical American curriculum. It was everyday math, or Chicago math. It focuses on procedure and explaining ones reasoning. My teachers didn’t fully utilize this though and followed a more traditional method. I did the algebra, geometry, advanced algebra (trig), stats, pre-calc, calc route which most students who like math go through.

So when looking at the Greek mathematicians I have two things to look at when thinking how they influenced my mathematical journey. The first thing to look at is the culture of math that I was raised in and the second being the progression of the content I went through.

In terms of content I think that Euclid was a huge influence on how geometry is taught. I don’t think there is anyone who would disagree with me on that. His Elements has profound implications on how math was/is done for generations. Geometry in high school was a series of propositions that led to theorems and finally proofs. This logical structure for proof writing was instilled in us throughout the whole year. I loved the class. It relied on logic and used logical steps to move forward. This Euclidean influence could not be more obvious.

Strangely though, this is not how geometry was first introduced to me. From an early age we studied shapes and their properties. This type of spoon-fed information that I used to regurgitate volume, area, length, etcetera would have driven Euclid crazy. The logical structures for my understanding were not based in logical deductions; they were based in the authority of my teachers and textbooks. This is a culture that is against how Euclid would have looked at math; in fact this is against how many early mathematicians would have looked at math.

            I’ll speak more to this in the next paragraph, but math was taught in an abstract way to me as a child. Symbols and their meanings were taught to me with the express purpose of solving math problems. Using these new symbols and skills I could solve bigger and harder problems and discover new symbols. In essence, math was taught to me so that I could solve math problems. The Greeks would have scorned this and expected an application for the math being done.

In reading the blog by David Mumford I look at one of his key take-aways being that math was based in applied math. Math was accomplished in order to solve real-world problems. The Greeks would have had this in the back of their minds as they solved math problems. Thus, much of their math was geometry at its core. Math today separates much of geometry from other subjects, where the Greeks used it as a basis for math. Mathematics was a culture of discovery in the Greek day, mathematics in our schools today is a culture of solving problems. So the Greek influence is not strong on this front. Math is frequently not seen as a culture of discovery. Our schools need to teach what is at the core of mathematics again and follow our Greek brethren.

In summation, I can see the influence of Greek mathematicians in how mathematics was taught to me, but I can also see a blatant dismissal of much of the underlying principles that determined why mathematics was done.