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.

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