Math According to Archimedes and Hardy

February 1, 2019

Thanks to Alissa Simon, HMU Tutor, for today’s post.

I have a number of questions still rumbling around after Harrison Middleton University’s January Quarterly Discussion. We read Archimedes’ Sand Reckoner and G. H. Hardy’s Mathematician’s Apology. I put these two pieces together because I am interested in mathematical discourse separated by thousands of years. More than time, however, they also came from different parts of the world, encountered very different technological advances, and lived immensely different lifestyles. Archimedes of Syracuse was a Greek mathematician and inventor who lived around 287-212 BC. Hardy, on the other hand, was born in 1877 in England and showed an early aptitude for numbers. He continued with math through college when he became largely interested in “pure mathematics” which, he claimed, is more noble than practical math. So, my first question is whether or not Archimedes’ Sand Reckoner corresponds to pure math, or practical math?

In The Sand Reckoner (which I have written about before), Archimedes sets out to demonstrate that math has strategies to break down something as large and abstract as the measure of the universe, or the grains of sand on earth. His proof begins with rather large assumptions, such as “I suppose the diameter of the sun to be about 30 times that of the moon and not greater.” Initially, I did not understand why Archimedes would base a proof upon such unknowns. However, I have always thought that the exercise was more to inspire imagination than prove an actuality. And now, based upon conversation during the Quarterly Discussion, I see that Archimedes wanted not just to inspire imagination, but to demonstrate the potential of math. He was explaining that math functions on strategies which engenders new information. This would be important, of course, living in a time when math was largely unknown and therefore, seen as untrustworthy. So, to me, The Sand Reckoner is not a proof of any one thing, but a proof of math itself. He asks his king, other educators, and perhaps his community to believe in the potential of math and to contemplate questions of great size.

Jumping forward to Hardy’s piece, then, he draws a very decisive line between practical mathematicians and pure mathematicians. Practical math builds things like bridges and steam engines. Pure math contemplates greatness. For some reason, Hardy’s differentiation always brings me back to Archimedes, who built levers and invented all sorts of practical things, but yet also contemplated the universe. Does the mathematician who builds the bridge not also dwell upon other possibilities? Surely not all of them do, but I find Hardy’s approach very severe and limiting. I am not sure if his words are meant to inspire others to attempt a career in math, or to explain to the masses how little they actually know. Either way, I feel that the work fails when placed next to something like Archimedes’ proof which shows math’s potential rather than belabors the value of ambitious men. Perhaps, though, my perspective is naive, since I do not grasp much of the math that would place me in this elite group.

Clearly Hardy values creative thought over any other pursuit. I can identify with this, but I wonder if his criticisms speak to moral dilemmas of his day. Hardy wrote A Mathematician’s Apology in 1940. I have to think that war-time inventions must have been on his mind when he differentiated between practical and pure mathematics. And yet again, I return to thinking about Archimedes who built many machines of war such as the Archimedes Claw and catapults. Does this remove him from the rank of pure mathematician (if he was ever considered such)? In theory, I believe that I understand Hardy’s point. In fact, I relish the idea that a life of creative thought or philosophical discourse is as worthy as shipbuilding. This would justify my own life as well. However, it seems rarer that society allows such thinking to exist. Rather, society is structured in a way in which we must all pay for food and shelter, and creative thought does not pay. I think that perhaps Hardy might have been trying to tell us, the public, that we should value creativity more than we currently do.

Additionally, his message does not address morality at all, which the group found interesting. I wonder how Hardy would tie ambition to morality. He glories in the uselessness of math because it cannot be tied to evil. He writes,

“If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither [Carl Friedrich] Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in times of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science, at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.”

According to Hardy, pure math never filters into practical applications. I find this reasoning illogical, though since again, levers as created by Archimedes were once thought impossible and are now the foundation of much greater machines. In my mind, the lever was purely theoretical at one point and is now elementary science. Also, once public, how can anyone protect the ways in which their work will be used (or not used)? How can Hardy surmise that the pure math of today will not be the applied math of tomorrow? And does its application make it any less pure?

As always, I am indebted to a wonderful group who wanders through these questions with me. The next Quarterly Discussion will be held in April 2019. For more information email I look forward to hearing from you!

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What's Your Sign?

January 11, 2019

Thanks to Alissa Simon, HMU Tutor, for today’s post.

“Let’s listen with our eyes and not just our ears. That would be the ideal.” -Christine Sun Kim

Early exposure to language seems to be the key in all languages. The key to what? Success in that language, or with language in general? Or in society? What is the bar for success and how is it measured? With today’s blog, I want to better understand how language expresses thought, particularly through the lens of sign language.

Most of us have some experience with a young, non-verbal child (your own, someone else’s, or even through movies, etc) in which the child wishes to communicate. Think of the crabby one or two year old demanding something by screaming or crying, body language and/or facial expressions. Also, think of the parents’ joy at their child’s first word. It really is amazing to have these tiny humans mimic words at such a young age. This seems to be proof of an innate desire to communicate with others. Even if it is mimicry at this stage, they come to grasp the idea of language and communication, which still leaves me wondering how far language is learned versus innate. The child simply sees adults talking, or perhaps older siblings, and they want to join. Spoken language has existed for over 60,000 years, making it difficult to differentiate natural from learned behaviors. However, a lot of superb scholarship is currently coming out of sign language due to two facts: a) it is a relatively new language and b) sign language is not linked with verbal language. These differences offer some key insights into how languages act.

According to the National Institute on Deafness and Other Communication Disorders (NIDCD), sign language relies upon visual cues alone. It creates language with bodies and space. Though there is an alphabet, it is used mainly for spelling new words or names. While there are a number of popular sign languages, today’s blog focuses mostly on American Sign Language (ASL).

First, let’s start with fingerspelling. This is the process of spelling a new word, such as the name of a person or organization. But it could also be some new piece of information. According to this lesson plan on ASLU, many flowers lack specific signs in ASL. Therefore if you want to say daisy, you would fingerspell D-A-I-S-Y. In further explanation of the utility of fingerspelling, the instructor continues: “How about food? While there are quite a few signs for various food items, there are thousands of types of foods that have no established sign. Hold on to your chair when I tell you this - there isn't even a widely accepted sign for burrito. (As opposed to a burro, which is a small donkey. We do have a sign for "donkey," but try showing a picture of a both a donkey AND a mule to 10 different Deaf people and watch 'em tell you ‘mule is spelled.’) And a mule is a relatively common animal -- don't even get me started on ‘ring-tailed lemurs!’”

Another use of fingerspelling is when you have a common name like John or Bob, you can fingerspell the name, but refer to a particular space in front of you which will equal “Bob” for the rest of the conversation. So, as a shortcut, you can point to that space to indicate Bob, rather than fingerspelling the name each time. There are some instances, however, in which a name does have a particular sign. Think of names like Dawn or Penny. These are somewhat common names which also have a corresponding sign. In this case, you would fingerspell the name to explain that you mean to designate a proper noun and not the concept of dawn, for example.

In fact, names offer such a rich area of study in terms of sign language. There is an entire culture dedicated to naming things in sign language. Names often depend upon a person’s characteristics as decided by the Deaf community. This short video from My Smart Hands gives a little more understanding into how names are assigned. (among others) explains:

“Fingerspelling your name can seem a bit impersonal, especially among friends. So, members of the Deaf community often give each other sign names. Your sign name is often related to something about you (a characteristic). For example, if you have curly hair, your sign name may be a combination of the first letter of your name and the sign for curly hair. Culturally, it is not appropriate to pick your own sign name and only Deaf people assign sign names. When you first use a sign name in a conversation, you would fingerspell the name and then show the sign name. Once the people know who you are talking about, the sign name makes it easier and more personal to refer to the person during the conversation.”

Many cultures have not been able to control their own language due to outside oppression and/or language mixing. While French sign language and ASL do share a common history, today they are distinct languages. Due to the unique nature of the Deaf community, it is not clear (to me, at least) to what extent an outside force could actually affect or alter a particular sign language. (For more on this, check out Nicaraguan Sign Language or the fascinating Al-Sayyid Bedouin Sign Language.) It makes sense that the Deaf community prefers to assign their own signs for names, since much of sign language relies upon a cultural understanding of how to use space, body language and gesture itself. In the same way that typing in all capital letters looks like shouting, some subtle body change (such as pointing, shaking your hand, or raising an eyebrow) may greatly affect the nature of the word that you mean. The artist Christine Sun Kim explains that “In deaf culture, movement contains sound.” (She explains the concept of full-body language in this TedTalk.) Signs can express irony, sarcasm, anger, humor, etc. just as capably as voiced speech. Understanding humor or sarcasm in a second language, however, is one of the greatest challenges of learning a new language.

Due to the way that sign language involves full body language, I find that it is an extremely emotive language. While the Syntopicon does an adequate job of outlining topics related to Language, I find fault that there is not a cross-reference for Language in terms of Emotion. I believe there is much development to be made between the cognitive associations of language making, language preferences, language learning, memory recall and emotion. Since memory and emotion are both contained in the limbic system of our brains , it would make sense to pair language recall with memory and/or emotion. (To Adler’s credit, he does suggest a link between Memory and Imagination). There is room though, I believe, for exploration regarding emotion as it relates to language cognition, use, and development. One other area of great interest to me is the bridge between Language and Art (which Adler does address in the Syntopicon). (For more about the deaf experience in the world of art (and sound), check out artist Christine Sun Kim’s exploration of noise.)

I also wish that there was a cross-reference between Language and Nature. Is language natural? In what way(s) does the early language-learner express a natural tendency to communicate? If the young child desires to communicate, is this concrete evidence that natural language exists, that language is natural to humans? What does it mean to say that language is natural? And finally, what can a developing language such as sign language tell us about language itself?

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Imagination in Flight

November 16, 2018

Thanks to Alissa Simon, HMU Tutor, for today’s post.

In Left Hand of Darkness, Ursula Le Guin has her protagonist, Genly Ai, travel to the distant planet Gethen which has no birds or flying insects. As a result, the communities there never even thought to attempt flight and their language has no word for flying. It is no wonder, then, that the people mistrust Genly who arrives by airship. It is also easy to see why Le Guin chose this scenario. She masterfully removes something which we often take for granted (that there are flying animals and insects) and then demonstrates how it impacts imagination. (For the record, there are many other major differences between our earthly world and Gethen, but I’m only talking flight today. I definitely recommend the book for all of those who are curious about science fiction experiments.) In chapter thirteen, Genly Ai and another man are sharing folktales about the places where they are from. Genly shares the story of flight. He remarks that he is not talking about a spirit world, but the real world. He says, “’Not by flapping their arms, you know. They flew in machines like cars.’ But it was hard to say in Orgota, which lacks a word meaning precisely ‘to fly’; the closest one can come has more the meaning of ‘glide.’ ‘Well, they learned how to make machines that went right over the air as a sledge goes over the snow.’” Of course, in order to communicate, language restricts Genly Ai to analogies of the place where he is, so he focuses on a common machine from this icy climate, the sled.

Albuquerque International Hot Air Balloon Fiesta. Photo credit: Alissa Simon

Albuquerque International Hot Air Balloon Fiesta. Photo credit: Alissa Simon

The history of flight is extremely curious and inspiring. The history of aviation includes such fascinating, bold, strong personalities as Emilia Earhart, Charles Lindbergh, the Wright brothers and many, many others. However, I was caught by surprise recently when I discovered how little I know about lighter-than-air ships. In reading Ships of the Air by Lyn Curlee, I saw again that same spark of curiosity that often drives human invention. Curlee writes, “One day, after watching ashes from a fire float upward, Joseph Montgolfier folded a piece of paper, held it above a fire, then watched it fly up the chimney. Joseph believed that the smoky fire created some kind of gas that was lighter than air. Only later did he and Étienne understand that hot air rises. But Joseph did understand that if a big enough bag could be filled with hot ‘gas,’ the bag would rise off the ground – and could carry a person with it.” From there begins a wonderful, rich, global history layered with politics and science. After Montgolfier demonstrated a hot air balloon flight to Marie Antoinette, the world took note. Furthermore, his balloon contained a flight crew of a sheep, rooster, and duck, whose survival proved that the atmosphere was higher than previously imagined. Many people became interested in designing and flying airships. In the late 1800s, they became popular sights in France, London and Germany. And as war broke out, the zeppelin famously became a machine of war, rather than leisure.

Back when the Montgolfier brothers were experimenting with cloth and paper balloons, however, there were many misconceptions regarding flight. Curlee writes that in 1766, “Professor Charles’s balloon floated 15 miles into the countryside, landing near a small village. The villagers, who thought the balloon was a monster, destroyed it with pitchforks.” This mentality echoes what Le Guin describes on her science fiction world, Gethen. It took an incredible amount of imagination to believe in flight. Furthermore, imagination is, in part, problem-solving. For the story of airships to become any kind of success indicates that man must often think outside the box. I return to Joseph Montgolfier watching ashes rise. With possibility comes the calculated risk of burning the paper. Understandably, then, the airship has faced many problems, such as weather, flammability, size versus weight ratios, etc. Curlee continues, “The story of lighter-than-air travel is mainly the story of failures. People who designed airships made many mistakes – often because they were experimenting with new technology, sometimes because they were careless.” Even so, hot-air balloons still inspire our imaginations. They predate airplanes, have been created by humans all over the globe, and have been put to many uses (including a German mail service). One thing is clear, flight of any kind captivates humans. The ability to defy gravity, even for an instant, sparks the imagination.

Darth Vader at the Albuquerque International Hot Air Balloon Fiesta. Photo credit: Alissa Simon

Darth Vader at the Albuquerque International Hot Air Balloon Fiesta. Photo credit: Alissa Simon

These photographs were taken at the Albuquerque International Hot Air Balloon Fiesta. Every October, over five hundred balloonists visit Albuquerque for its unique landscape and wind patterns. Balloons feature colorful designs, brand names, and cultural icons (Darth Vader is often a big hit). To see five hundred balloons floating up in the sunrise certainly inspires the imagination!

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November 9, 2018

Thanks to Alissa Simon, HMU Tutor, for today’s post.

Manifesto, directed by Julian Rosefeldt, disorients the viewer. Locations change without warning, as do characters. The speaker is a sometimes character and sometimes narrator. So many things happen simultaneously that it would be difficult to express them all. I will wander through a few of the things that caught my attention, though I am, unhappily, missing the majority.

The precision of this movie astounds me. As an introduction to the puppeteer, for example, the camera begins with an aerial view of the puppeteer working on a puppet that we soon come to meet quite closely. The viewer is yet unaware of this puppet’s meaning. It is only as the puppeteer rises from the desk and leaves the scene that the viewer is introduced to the the rest of the room. As the camera slowly pans across a small, brick room filled with puppets, we begin to recognize famous or historical figures. Some wear shocked expressions, some bored, some guilty, some blank. Sirens and street noise sound in the background. The collection includes figures such as Mother Teresa, Gandhi, Hitler, and the Planet of the Apes. My favorite is Marilyn Monroe, who lies face-up on a dingy, toy sofa with a man’s hand suggestively patting her exposed knee. The expressions on these two linked figures do not correspond to each other. Marilyn Monroe looks pained, while the man smiles joyfully. This quick snippet implies significant action. Small details such as this connect the viewer to the image in a way that creates life. Furthermore, the room contains such a dense amount of history, that the viewer is bound to connect with the overwhelming feeling of the passage of time. As with all the other scenes in Manifesto, Rosefeldt overloads the scene with sensory information. Therefore, by the time the camera returns to the puppeteer, the viewer has formed an emotional attachment to the puppets. (Emotional regard in any sense forms an attachment.)

Rosefeldt excels at a minimalism that forms some emotional bond between viewer and screen. This perplexes me, particularly in this scene with the puppeteer, whose hands we see and whose voice we hear, and yet we have no real knowledge of this figure outside of the hundreds of dolls placed about the room. The things I know about the puppeteer: the person is intense, passionate, exacting, precise, and interested in history or characters. The viewer is reasonably startled, then, to find the puppeteer pinning a wig onto their very own puppet. This most startling event astonishes us because, in my mind, the puppets have become hyper-real. They express emotion, and therefore, they must themselves convey some sort of “real” emotion. They are the placeholders of our emotion, and to see (and hear!) their creation is grotesque and surreal. Moreover, this particular puppet enters creation only once hair has been added. This grants an identity. Now the viewer connects with the puppet by labeling it. Therefore, the viewers themselves restrict the doll’s freedom. This is, of course, the intended effect. These hyper-real puppets exist in this box-style room, piled on top of each other in some sort of eternal limbo for the voyeurs’ eyes. They represent an act of creation, but the viewer agonizes over their apparent lifelessness coupled with their inability to die.

This reinforces the larger narrative in Rosefeldt’s film: that a manifesto is born out of a particular moment. It explosively responds to specific cultural content. What, then, is the next generation to do with it? What is the viewer, who likely feels no connection to the particular character of a movement, to do with the incongruities and absurdities presented by a manifesto’s anger? Is the manifesto to be shelved among others, dismissed from its specific argument, but unable to die? Should it be removed from view altogether? However, if we dismiss the manifesto as completely irrelevant, then perhaps we overlook some important cultural revelations.

Rosefeldt masterfully incorporates incongruous images. Cate Blanchett brilliantly plays all of the characters and delivers all of the manifestos. In each scene, she also slips into a monologue, delivered in a different pitch from the character’s voice. She plays both men and women, young and old. She plays rebellious, ridiculous, hallucinating, strict, poor, wealthy and conservative characters. She incorporates accents and body language. Meanwhile, Rosefeldt pays particular attention to sound and image. Each scene offers some startling revelation, such as we saw and heard with the pinning of the puppet’s hair.

Irony rules this movie. I offer a few examples of scenes which struck me, though there are many to speak of. First, I found the CEO at a fancy party particularly ridiculous. We meet them at a beautiful house, located along a gorgeous body of water, during a cocktail party. The horizon dominates the viewer’s gaze. While the viewer contemplates the horizon and natural beauty, however, the party-goers never even glance in the direction of the water. Furthermore, they discuss beauty in such abstracted, idealistic and theoretical terms as to make it unobservable. They literally miss the point. In another scene, a woman rises to a podium above a gravesite to give a speech of mourning. Instead, she desecrates the body and offends emotion. Standing over the corpse, she claims that “to sit still is to risk one’s life” and that “one dies as an idiot or a hero, it’s all the same.” The movie ends with a teacher instructing her students to steal and plagiarize. She claims that “nothing is original.” She does warn to steal only the things that speak directly to the student and that connection validly makes it their own. She then walks the room indoctrinating the children while erasing their ideas.

Part of Rosefeldt’s point is that manifestos become absurd when taken out of context. They burn brightly for a moment, but cannot last in an ever-changing world. Rather, if they do last, they become ridiculous. He achieves this through a balance of image, sound and speech. Rosefeldt compiled the script from actual manifestos written from the early 1900s through the early 2000s. During the span of 100 years, we see a variety of ideas demonstrated through art, architecture, drama and impersonation. I have only mentioned a few of the scenes in Manifesto. I encourage you to watch the scenes (or the whole movie) and post your thoughts. What does it say about art? About history? About characters or artists? Prepare for a disorienting experience, which is well worth your while. Visit Julian Rosefeldt’s website to view the film scene by scene.

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