Trivium & Quadrivium Homework

Image source: https://humanities.byu.edu/wp-content/uploads/7-liberal-arts.jpg

The first thing that struck me was that the study of liberal arts was required before the study of theology. I have taken a course on Religious Studies, and a few relating to Indian religions and I always found it interesting that in both Christianity and Hinduism, religion was something that was mostly studied by people who held religious titles, but that this was tied to their knowledge of a particular language. When religious texts were translated, they became accessible to the general public. I hadn't realized that, in the Christianity case at least, such high level of knowledge was required on top of knowledge of the language before a person was allowed to study theology. I wonder if studying theology used to be an intensive process and if that is why all these prerequisites were required or if it was something to do with creating separation between those who had access and those who did not (similar to a "free man" being allowed to study while a slave was not). 

The second thing that surprised me was that there was no controversy between algorists and abacists when Hindu-Arabic numbers were first starting to appear in Europe. I'd imagine this would have increased the difficulty in communicating math to each other, and I'm also surprised they didn't become defensive of the method they preferred. I think of Pythagoras and Hippasus and what types of change or suggested change are more offensive than others. For instance, I see a metaphor of language in the rise of algorists in Europe - it's essentially the introduction of a new "language" which expresses the same ideas as the native language but in a different format that the native language speaker may not understand. Languages don't always coexist either, depending on the politics of the area they are being introduced, but it has still been seen to happen. However, when Hippasus discovered irrational numbers, this went against what Pythagoras thought and he was sentenced to death for it. If the metaphor could be applied here, Hippasus was essentially put to death for discovering a new set of vocabulary in an existing language, which seems less extreme than speaking a different language. But maybe the metaphor doesn't quite work here. Or maybe there were some politics in play here. A professor once described math at the time of Pythagoras as a cult, using Hippasus as an example of what happened when someone strayed from what was accepted. I would be interested to look into when it stopped being cult-like and if there were other instances similar to Hippasus, but also other stories of coexistence because those would be helpful in validating students when they use different methods instead of pinning those who favour algebra against those who favour pictorial representations, for example.

Lastly, I thought it was funny that "extraordinary" math lectures were "diversions for holiday afternoons" at the University of Vienna, and interesting that this was one of the few places where math debates were held at the time. Going back to the "math as a cult" idea, it seems that, at least in universities like this, math became less absolute fact and more open for discussion around the 14th-15th century. I assume introduction of debates is what led to the decline in universal usage of Euclid's Elements in Europe (but it is still impressive that Euclid's books, although sometimes varying in emphasis, were a requirement for so long).

Numbers With Personality

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It's interesting to think that Mayan head-glyphs may have held more information than simply being representations for a numerical system. I thought Major's theory about the names of numbers contributing to their individuality and reasons for why certain cultures assign significance to them was an interesting one; although I knew the word for four in Chinese languages sounded like their word for death, I hadn't really thought too much about what that did to the personification of the number. 

In terms of the quote, "each of the positive integers was one of his personal friends," this initially struck me as overwhelming. If numbers are personified in the way Major noticed of Mayan culture, then in a set as massive as "the positive integers," how could each be his personal friend? Then I thought, would he view digits 1-9 as unique friends (like the Mayans had head-glyphs for 1-20) and anything above as forms of these nine friends? Could different forms be considered different life stages of those friends' lives? Different moods? 

The personification of numbers seems to be a very open field that could be a very good way to connect students to math. I remember being in a French class that I didn't particularly like, but I made it my own by using the names of characters from whatever show I was obsessed with interested in at the time to make it more fun for myself. Similarly, a student who has had bad experiences with math or doesn't particularly resonate with the subject might connect with it better if they had this method of storytelling/personification introduced at some point in their math career. I think it might be useful to introduce them as a stand alone lesson, perhaps with an introduction to Mayan math first before branching into making connections with numbers like 13 in Western culture or 4 in certain Asian cultures. Alternatively, it could be a class-long endeavor at a lower grade level; that is, it could be introduced alongside integers as a fun fact, then referenced continuously throughout the course as a callback.

For me personally, I don't think of numbers as having personalities. I am drawn to certain numbers more than others; for example, I prefer even numbers and those divisible by 5. Four is my favourite number but that's because April is my birth month. My birthday, like other calendar dates, is only special to me because there is some sort of celebration with other people. However, I have never really thought of a number with fondness. For example, if a calculation results in the number 4, I would be unaffected. If I step away from numbers, I definitely have symbols that are dear to me as a friend might be, and considering numerals are representations of an amount, it's a similar idea. For example, the Om symbol reminds me of many different things and when I look at it, I feel calm and collected. If I personified that symbol, it would definitely be a wise, reassuring being that celebrates creativity, yet moves at a slow pace.

Dancing Euclidean Proofs

Image source: https://vimeo.com/330107264

One of the ideas that stood out to me was using "imagination" to assume both dancers arms were the same length. In proving one of Euclid's Propositions when I took a geometry course in my undergrad, I remember the professor putting emphasis on making sure our proofs followed logic over diagrams. At first, it seemed possible to draw a diagram that disproved the proposition, but once we stepped through the proof, we saw that it was impossible. In this way, I think these dancers could use their dance to show students that, although the "lengths" (arms) don't look equal, because they are defined as such, the proof still holds true. 

It was interesting to me that they pointed out that the math and choreography would fall into place after some rethinking. I hadn't really thought of the similarities between choreography and developing a proof before but after reading that, it seems quite clear. When initially attempting a proof, I often had a long-winded process of achieving a result but the more I worked with it, the cleaner and more pleasing it became. The same process happens with choreography - sometimes I would think, "I want to go from two horizontal lines into an "X" formation," and would probably change the way the dancers got to that position three to four times because I found a way that clicked together better and just made for a method that felt more natural overall. It was a realization that consciously or unconsciously, we're all developing intuition in a field by spending more time in it. To extend the similarities, I remember being a student in dance and being frustrated when the choreography would constantly change, but now I understand that it's less the teacher's inability to commit to a choreography but more them gaining familiarity with the song/students/situation and their increased understanding of the best execution of the dance that leads to those changes. Similarly, it seems like an exhausting and tedious task to rework a proof, but it often leads to a better understanding of the content and also a better "product" that others can more clearly understand and appreciate as well.

The idea of collectively being responsible and embodying the proof with someone else was a powerful one for me. I remember studying Euclid with my friends prior to tests and it was funny because even though we were all in the same course and all working on the same problem, each of us had a separate method for obtaining the result depending on how we began the process (for example, I tried to use circles wherever I could, while my other friend was partial towards angle relations). We often found it useful to explain our own processes to each other to make sure they made sense outside ourselves, and in case we noticed mistakes in each other's' reasoning. To be physically stepping through a proof with another person would not only be technically challenging, but also mentally challenging because you are having to follow a path that may not necessarily be your vision. However, this is quite valuable as well because there is a lesson in reliance and dedication to collaboration in deciding to pursue this method.

Euclid Poems

Image source: https://history-biography.com/euclid/

 Euclid's collection of different works in "Elements" has had a major influence on mathematics and different scientific fields since it was written. In particular, the geometrical principles influenced physics, astronomy, chemistry, some engineering (https://history-biography.com/euclid/). I believe the poem by Edna St. Vincent Millay is praising Euclid's understanding and appreciation for geometry. My interpretation is that the "Beauty" referenced in this poem is referring to geometry in particular because of the lines, "intricately drawn nowhere/ In shapes of shifting lineage." I think "intricately drawn nowhere" refers to the abstraction of common shapes and patterns found in nature but documented mathematically in "Elements," while I think "In shapes of shifting lineage," is a reference to the work of others ("lineage") that Euclid collected. Thus if "Beauty" is geometry and "Euclid alone has looked on Beauty bare," then the poet claims that Euclid is the only one who was experienced enough to see geometry in all of its glory. The lines "Let all who prate of Beauty hold their peace,\And lay them prone upon the earth and cease\To ponder on themselves," further perpetuates the idea that no one else is close to Euclid's understanding of geometry and those who speak of it aren't qualified to. The line, "Fortunate they\Who, though once only and then but far away,\Have heard her massive sandal set on stone," tries to convey how far away everyone else is from Euclid's connection to geometry by saying it is rare for people even to have "heard [Beauty's] massive sandal set on stone." 

In contrast, I believe the parody poem by David Kramer criticizes the idea that Euclid alone held a superior understanding and appreciation of geometry. I believe he is saying that many have achieved greatness in geometry, even if it was only a very specific section, when he says, "Has no one else of her seen hide or hair?/Nor heard her massive sandal set on stone?/Nor spoken with her on the telephone?" Then by addressing the poets and saying "For Beauty bare you never yet had seen," I believe the Kramer is saying both that the poets were unqualified to speak on the matter and that Euclid was a starting point but definitely not the be-all end-all of the field of geometry. It's almost as though Kramer expresses that Euclid couldn't have been so special to have been the only one allowed access to geometry when he says, "Would you, for this mathematician, \Remove...Once only, and then but far away, your sandal?" Thus, I think the first poem is expressing that very few brilliant minds besides the great Euclid have been fortunate enough to really grasp geometry in its entirety, while the second one rejects this idea and instead implies that geometry is accessible to everyone.