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.