September 28, 2018
Thanks to Alissa Simon, HMU Tutor, for today’s post.
I hear a lot of teachers complain that students are not willing to spend much time trying to struggle with a difficult problem. Some of these teachers lament the fact that gadgets have become a main source of information, rather than logic. In other words, we can answer our tough math problems with Google or some other device. I think there is more going on than the implementation of technology, though. It seems that in addition to new technologies, our students are also handed a lot of information at one time and asked to decipher it. Technology can be fun, enticing and extremely helpful, but I also agree that we (all of us!) would benefit from sitting with a particular problem or question for a long time, and puzzling it out on our own. This type of solitary work reminds me of the concentration required from composers, authors, and mathematicians. Our current education model often emphasizes group projects because, it is true, that we learn a lot from groupwork. But I hate to see it come at the expense of solitary contemplation. No one else can tell me what I myself think. Instead, I have to understand it for myself, and that is often the result of hard work, struggle and problem-solving.
Choosing Archimedes as the subject of today’s blog is a bit surprising since I have never truly enjoyed math. I always had to work so hard to understand the concepts. However, I do enjoy Archimedes, and so, as ironic as it seems, I wanted to explain a few reasons why. On the one hand, I should never speak authoritatively about math. On the other hand, however, I am a great representative of the “struggle-it-out” style of learning. And I’ll be honest, while I have not resolved my fear of math, I have come to see elements of beauty in it. A few years ago, I worked my way through pieces of Euclid and came up with some very rewarding ideas. (I wrote about some of these ideas in three separate blog posts. Scroll to the bottom of this post for links to those.) That these ideas reward myself alone is inconsequential because they often connected with yet other interests. In other words, they enlightened studies in other areas. I find these connections particularly fascinating because this capability mimics one of the ways in which knowledge grows. It is also how houses are built. I do see vital connections between theoretical knowledge and practical applications.
Recently, I discussed some of Archimedes’ writings. “The Sand-Reckoner” caught my attention because he quickly develops a mathematical account of the universe. Furthermore, Archimedes proposes that human knowledge would benefit by increasing the current understanding of large numbers. Previous to his work, Greeks used the word “murious” which roughly translates to “uncountable.” (The Romans later changed murious to myriad.) In “The Sand-Reckoner,” Archimedes argues that by using a myriad as the number base, he can learn information about pieces of our world and our universe. In his proof he uses larger numbers than anyone had previously used. In fact, he repeats the desire to push the envelope.
This text surprised me, not because of the difficulty of the math (which is astounding when coupled with the difficulties of doing precise equations in such an old system.) Rather, I understood that whether or not the calculations are factually accurate for us today is not the reason why we continue to read Archimedes. I believe that we still read Archimedes because he asked humans to combine calculation with imagination. To think of a problem, such as the size of a grain of sand, and then try to measure it. He did the same with the universe. So, while claiming that the diameter of the universe can be measured by the diameter earth may not be precise, it does capture the imagination.
In addition to mathematics, this proof places importance on theoretical knowledge. He elegantly demonstrates that it is good to think about things. To sit with a problem, even if it is potentially unanswerable. He writes, “It is true that some have tried, as you are of course aware, to prove that the said perimeter [of the earth] is about 300,000 stadia. But I go further and, putting the magnitude of the earth at ten times the size that my predecessors thought it, I suppose its perimeter to be about 3,000,000 stadia and not greater” (521). He repeats this phrase later as he claims to go further than anyone has regarding the dimensions of the universe as well. It is this unlimited imagination that nearly reaches the heights of today’s astronomy. In “The Sand-Reckoner,” Archimedes demonstrated the importance of philosophical thought through a mathematical proof.
I want to emphasize that conversation enhanced many of my own ideas about Archimedes. In order to access this reading as best as I could (and I am still far from an expert on this reading), I completed the following steps. First, I read the text twice, taking notes and writing questions. Next, I tried to answer a few of my own questions based upon his text. And finally, I discussed the whole reading with a group. The discussion leader asked tough questions that gave further insight into Archimedes’ text. These difficult concepts came alive during our discussions, for which I am grateful. So, while I first struggled on my own with his proofs, I also found conversation as a necessary accompaniment towards better understanding. Thanks to those who discussed these works with me!
Access the blogs on Euclid at our website, hmu.edu, or by clicking on the links below:
(First post: http://www.hmu.edu/hmu-blog/2016/1/28/january-quarterly-discussion-review)
(Then this: http://www.hmu.edu/hmu-blog/2016/2/3/euclid-and-whitehead-found-poem)
(Finish with this: http://www.hmu.edu/hmu-blog/2015/12/22/euclidean-utopia)
To leave a comment, click on the title of this post and scroll down.
