The Mathematical Renaissance
A Personal Reflection on the History of Mathematics
History of Math
December 8, 2014
Help Received: None
In today’s society, we take for granted much of the basic concepts used in mathematics
today. From square roots, to long division, to the symbols we use to represent the ideas of
addition and subtraction, much of our basic understanding in the field of mathematics took
thousands of years to develop. I can easily remember my earliest memory in math; I was in the
2nd grade and we were covering long division which for some reason proved to be a tough
concept for me to grasp. And while I eventually got it with practice, it’s amusing to look back on
it now after taking this course and realize how much harder long division would have been to
learn if we didn’t have our current numerical notation system. Take for example a simple
problem like 8+2. We know almost immediately that the answer is 10, but written in a form that
existed before current notation you would see the problem written as ‘take 8 stones and add 2
more stone too it.” The answer is still very simple, but what if we were to solve a much more
complex problem like (8+2)/2. Written in older notation it would be written in a form similar to
“take 8 stones and add 2 more stones too it, then take half it.” The reason why these problems are
so easy for us to solve is because they have been reduced down to its basic form using notation
that includes addition (+), subtraction, (-), multiplication (x), and division (/) symbols. But older
civilizations did not have this type of notation set up and had to use things like shapes and real
life objects to describe mathematical concepts. By taking this course, I feel like I have a greater
appreciation for our current system of notation used in math as well as have a wider
understanding of just how earlier civilizations were hindered by the limitations of their own
notation system.
The Greeks were a fairly advanced society having notable achievements in art, poetry,
architecture, as well as in the field of mathematics. In fact, the Greeks made great strides in the
branches of arithmetic, algebra and geometry which they inherited from the earlier
Mesopotamians. But after the first few months in this course I found that I didn’t really have a
good appreciation for the contributions made by the early Greek mathematicians. I first began to
realize this when I began work on artifact 1 which was a short essay about the history of the
Pythagorean Theorem. I chose to write about this simple theorem because I consider it to be one
of the greatest mathematical contributions made by the Greeks. I remember learning about the
Pythagorean Theorem for the first time in the seven grade which became the focal point where I
started to develop a passion for math. The concept of the theorem is actually quite simple, if you
square the sides of a right triangle and add them together they equal the length of the hypotenuse
squared (a2 + b2 = c2). However, up until this point I kind of just accepted this relationship as a
fact, it just exists, but I never really thought why the Pythagorean Theorem worked. It was doing
artifact 1 that I found out the proof behind my childhood theorem and was amazed at the logic
the Greeks used to prove it. When you square a number, that is to say, multiply one number by
itself, we often write it in simplified modern notation as x2. But this notation will not develop for
a few thousand years and the Greeks will rely on shapes, more specifically the square, to explain
the idea of squaring a number. How do the two relate? The area of a square is just the length of
the height and width multiplied together, and since the sides are the same for a square, the
answer will be the sides squared. Because of this, when completing artifact 1 I saw the
Pythagorean Theorem in a completely new way than I had done when I first learned it back in
the seventh grade. The Pythagorean Theorem can also then be rewritten as “the addition of the
areas of two squares will equal the area of a square which matches the hypotenuse.”
While this concept seemed so novel to me, it’s also important to keep in mind that during
the early Greek period, this was how the Greeks approached problems which called for squaring
a number. Now the Greeks were marvelous mathematicians, who created marvelous architectural
masterpieces in which rigorous knowledge of mathematics was required. But the Greeks were
also limited in their knowledge because of their inability to condense these concepts into simpler
ones. They were not able write out the theorem in simple notation such as a2 + b2 = c2 and
instead had to use shapes in order to describe it. If they had to ability to do so it would be
amazing to imagine what the early Greeks would be capable of. More advanced math would
have meant greater architectural feats, stronger defensive positions, greater weapons, and overall
a much stronger military position to fight off invading countries. In a sense their knowledge of
math made the Greeks powerful, but it also limited how powerful they would become.
Unlike ancient Greece, where knowledge of their new discoveries in math traveled
throughout Europe upon trade route, the Chinese civilization spent most of their time in isolation.
Much of the work completed by the Greeks will not penetrate into the Chinese populace for
hundreds of years and as such the Chinese will not build off their work when Greek influence
declines during the dark ages. But even while they remained secluded to eastern Asian, Chinese
mathematicians published advancements in math that are still used today in the fields of
geometry, statistics, and arithmetic. It was here that I worked on article 2 which was a homework
problem in which I solved square roots using the Chinese square root algorithm. For those who
have never done this, I would like to point out that the process is quite extensive, in fact I turned
in artifact 2 with five pages of work for finding the square root for a three digit number. But it’s
amusing to reflect on the fact that for modern mathematics, computing square roots is simple
even for numbers that are infinitely large. But back in early China, solving square roots was a
long and arduous process. I noticed that like the Greeks, the Chinese also chose to use the area of
a square to represent the concept of a squared number. But the Chinese took a step ahead of the
Greeks and thought of each digit as a fraction of a much larger area. Now I admit that I
thoroughly enjoying working on artifact 2 because it was like solving a puzzle, but I can’t
imagine Chinese mathematicians sharing similar views. Thinking of how often we use square
roots in engineering, architecture, and mathematics, if you solved every square root using this
process it would take you days or weeks to solve the same problems we solve now. And even
with modern notation, it still took me 5 pages to complete the artifact along with an entire
evening of time. After completing artifact 2 I realized not only should I be grateful of our
notation system in math but our advances in technology that made the calculator possible.
Another thing to consider is that if these societies were hindered by their lack of
simplified notation, what makes us any different? Typically when we think about mathematics,
we think about all of the concepts and theorems that we currently use, derivatives, integrals,
remain sums, the invertible matric theorem, are just some examples of this. But has anyone ever
thought of all the concepts in math that we have yet to discover, what do we not know yet that is
preventing us from further advancements? The same is true for earlier civilizations. Would
Archimedes have been able to understand more about the area between curves? Would
Hipparchus been able to decipher more about the relationship of cords and their relationship to
the radius of a circle? Both of these mathematicians were hindered by their system of writing,
and the concepts within math that they took to be true. If that is the case, how are we hindered in
our current era in the field of math? Does having a base ten number system prevent us from
understanding more complex math that cannot be understood in a base ten system? We can do
double and triple derivatives, what about 1.5 and 2.5 derivatives, what prevents us from doing
this? It is often said that knowledge is power, and if this is true than what about the great Roman,
Chinese, and Mesopotamian empires. Would their empires have fallen as quickly if they were
more advanced in the field of math? More advanced math would have allowed for greater
advancements in weapons and better engineering resulting in better hospitals, schools and
medicine. Taking this course, I found that I developed a much greater appreciation for our
simplified method for solving mathematical problems. In society we often take for granted what
we currently know and it was only through studying the past that I found there is still much to be
discovered.