Exploring Limits

In calculus, a limit is defined as the value of a function as it approaches some point. Sometimes, a function has no finite limit at a point because it just keeps growing, and we say the limit is infinite. In this case, the function never reaches the limit but the value grows arbitrarily large as it gets nearer and nearer to the limit. In our reading, I have been considering limits in a different light. I have been thinking about the limits of civilizations as they progress in their development of mathematics. Some civilizations seem to reach a limit of understanding and because of cultural restraints, their limited number systems, or even because they outwardly reject an idea, they stop progressing. Fortunately, sometimes their discoveries shape and influence other cultures and, as a whole, progression continues. I would like to explore different limits in the progress and development of mathematics and consider what limits us today.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

In ancient Mesopotamia, more than 4000 years ago, the Babylonians used the base of 60 to develop a high level mathematical system. They developed positional notation and could use fractions as well as whole numbers. They developed systems to figure square roots. Clay tablets from that time show tables with logarithms, multiplication facts and reciprocal pairs. There is information about calculating compound interest and solving quadratic equations. Writings on the tablets suggest that math was a subject that was taught and studied. In many ways, they seem to have exceeded the capabilities of other civilizations that came much later in history. No one can question that their accomplishments were amazing, to say the least, and perhaps influenced other cultures. However, because most of their mathematics were only for very practical purposes like conducting business, surveying land and constructing buildings, they stopped short of exploring some of the deeper meanings of things. For example, our text points out, “In the Babylonian square-root algorithm, one finds an iterative procedure that could have put the mathematicians of the time in touch with infinite processes, but scholars of that era did not pursue the implications of such problems.” (Merzbach and Boyer, pg. 26) What might have been the implications if they had? As they approached the limit, they stopped rather than exploring the infinite possibilities. They stood on the brink of even greater discovery, but did not pursue it.

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

One of the most dramatic examples of cultural influences limiting the progress of mathematics is the example of the progression of Indian positional decimal arithmetic to Europe. Mathematicians in India had developed a number system with ten digits, including zero, and used it to develop methods of computing fractions, square roots and π. In the tenth century, Gerbert of Aurillac attempted to introduce the system to Europe. He had learned the system first hand from Arab scholars in Spain.   However, he was rejected and during this time of the Crusades in Europe, he was rumored to be sorcerer. He died after a short reign as Pope Sylvester II. “It is worth speculating how history would have been different had this remarkable scientist-Pope lived longer” (Bailey and Borwein, 6).” The Indian system was reintroduced 200 years later by Leonard of Pisa, but was rejected again and considered “diabolical”. It wasn’t until the beginning of the 1400’s that scientists began using the system. “It was not universally used in European commerce until 1800, at least 1300 years after its discovery” (Bailey and Borwein, pg. 6). While many other areas of the world were able to do complicated computations using the Indian system, Europe, because of its cultural restraints, was still laboring with Roman numerals. Imagine what the brilliant minds of the Europeans might have discovered or developed if they had the ease of the Indian number system? In this case their culture may have created a limit that kept them from infinite discoveries.

Today in our world we have amazing tools to help us progress. Not only do we have the combination of a well-developed number system, thousands of theorems and laws and the knowledge of centuries of learning, we also have technology that assists in remarkable ways. Indeed we have all the tools of the past plus the technology of our day. However, are there things yet to be discovered, or have we reached a limit? Are there obstacles in our society or ways of thinking that limit us? As recently as the early 1900, women had a difficult time pursuing their mathematical interests. Even today, women and minorities continue to be underrepresented in the math and science fields. What might have been the result if woman had been afforded the same educational opportunities as men over the years? Do we limit ourselves by the way we approach math? Are there different number systems or “languages of math”? In recent years, computer scientists have given us other “languages” for coding. Are there similar languages for math? The challenge for our day is to not be content and accept that what has been learned is all there is.

In our reading for class I have been amazed at how often a group or civilization is on the brink of great mathematical discovery, but because of varying reasons they stop short of the mark. Sometimes cultural influences limit the progress and other times it seems individuals do not look far enough to find deeper meaning or answers. It is true that hindsight may be twenty/twenty, but I can’t help wondering what future civilizations may look back on and see that we barely missed. What are we on the brink of discovering if only we would look forward and push closer and closer to the undefined limits?


Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Bailey, David H., and Johnathan M. Borwein, “The Greatest Mathematical Discovery?,” 2011.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s