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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How Scientific Language Is Created

December 21, 2018

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

Last week, I posted a blog about Bohr’s use of language. Specifically, I wanted to investigate how the field of science will find ways to accurately describe indescribable events. I discussed the way that modal verbs (helping verbs which express doubt or uncertainty like “might” or “could”) can negatively affect the reception of a scientific article. I think Bohr embraced this idea of uncertainty. In fact, he claims that areas of uncertainty become the best areas for advancement because they point out specific questions. Rather than formulating science as if it were static he asked that we (both the scientist and the reader) investigate our use of language, our preconceived notions, and our unknowns. Bohr accepts, in fact, desires to imbue scientific language with doubt. I think he goes to great lengths when discussing language in order to enlighten future generations of scientists and readers as to the complexities involved in atomic sciences. That science can be grounded upon facts but still involve many, many questions is part of the reality of science. Therefore, language must reflect this reality. Really, we do not have all the answers and should not proceed as if we do. The problem is, however, that journal articles which include doubtful language are often regarded as less rigorous, less accurate, and less scientific. Bohr, however, would applaud these articles as attempts to base the unknowns upon the knowns. Moving forward, moving into an era of atomic theory, then, will demand a higher sense of intelligence from both readers and scientists.

In today’s blog, I want to better understand two parts of the question of scientific language. First, I am interested in the perception and reception of modal verbs in languages other than English. If modal verbs in English are perceived as unscientific, are they also perceived this way in other languages? Much of science is presented in English. In limiting our scientific language to a handful of languages, do we limit our ability to describe the indescribable? Scientists often think outside the box in order to find terms that reflect what they find. For example, names of celestial bodies refer to mythological beings. Latin terms classify plants. Clouds, too, were named in Latin according to observable features. What then, do we use to describe atomic energy: metaphor, mythology, ancient languages, compounds? If scientific articles are published in only a handful of languages, does this exclude some metaphoric understanding or phrasing from an outside culture? Does the way that we currently publish scientific findings prohibit (or at least discourage) any culture from entering the dialogue? Also, how do we adequately translate any scientific finding into another language? It is common in the scientific realm to stick to the original language when using a specific term. So, the Latin name “cirrus” is often used in the translation, rather than a word from the target language. However, using a term for an identifiable object, such as a cloud (or plant), is very common and accessible which is not true of atomic theories. In other words, it is incredibly difficult to adequately express the experience of atomic behavior in any accurate, identifiable, universal language. I just wonder if this dependence upon one particular language limits us in some unforeseeable way.

My second question today deals with Bohr’s insistence that we continue to use classical terminology even for unobservable data. I understand the importance of adherence to non-abstract language as a way to describe abstract ideas. However, language is never static, which may present problems for the idea of classical terminology. For example, atomic theory is so named only because at one time we assumed that atoms were the smallest pieces of material in existence. We now know that this is not true, so we have adjusted the definition of atomic as well as the public perception of the science. Furthermore, from Bohr’s Atomic Theory I chose to look up the term “ion” and am still uncertain about the definition’s accuracy. According to Merriam-Webster, “ion” is defined as either “1: an atom or group of atoms that carries a positive or negative electric charge as a result of having lost or gained one or more electrons; or 2: a charged subatomic particle (such as a free electron).” The terms “lost” and “gained” included in this definition make it sound as if an atom has a natural state, and that the ion is not the natural state. I struggle with this because having an electric charge may be considered just as natural as any other state. It may be important to note that the ion is less stable than another state, but that is not what the definition explicitly says. So, even if we stick with classical terminology, definitions will change over time. In fact, just in scanning the Wikipedia page for “ion,” our understanding has rapidly progressed in just under one hundred years. Furthermore, scientists such as Faraday (who first discovered ions) may have used the term differently than contemporary scientists. This is, of course, something that Bohr was intensely aware of, but perhaps the layperson will not understand the subtleties of these changes. I do understand his explanations regarding classical terminology, yet still, I am left wondering how one might be conversant in the language of science without knowing the history of an innumerable amount terms.

Clearly I am not a scientist, and I do not have the necessary skills to examine a lot of the terminology in Bohr’s Atomic Theory. However, I do spend a lot of time thinking about the effect of language on communication, society, and human life in general. I feel that it is of great importance (and benefit) to consider these larger questions as they relate to specific fields. I am grateful to Niels Bohr who used language as carefully and precisely as possible, so that even someone such as myself could attempt to understand the complexities of Atomic Theory.

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Bohr's Use of Language

December 14, 2018

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

At the end of the fourth chapter of Atomic Theory and the Description of Nature, Niels Bohr writes, “Besides, the fact that consciousness, as we know it, is inseparably connected with life ought to prepare us for finding that the very problem of the distinction between the living and the dead escapes comprehension in the ordinary sense of the word. That a physicist touches upon such questions may perhaps be excused on the ground that the new situation in physics has so forcibly reminded us of the old truth that we are both onlookers and actors in the great drama of existence.” I love the stage analogy that Bohr uses. I picture a camera forever panning backwards. When the scene begins, we are looking at a stage, but as the camera moves backward the audience is on the stage. Included in my visualization is that both the stage and ourselves become increasingly smaller. This is important to the way that I see Bohr’s argument. Bohr suggests that even if we can claim to know pieces of the whole, we will never see the complete picture at one time. This is not to say that we cannot connect pieces in the way that we do a puzzle, but that no single piece can stand as significant of the whole. Atomic Theory and the Description of Nature explains that the future of science will be (and already is) beyond our senses. Instead of seeing reactions and experiments, we must rely upon a variety of tests, the accumulation of which will grant a picture of the whole. At no one time, Bohr reminds us, will we be able to actually see the whole, however. In both this piece and in “Discussion with Einstein on Epistemological Problems in Atomic Physics,” Bohr explains how his view differs from Einstein. Unlike Bohr, Einstein believed that at some point we will have a complete picture of atomic physics.

A recent discussion of these readings sparked my curiosity about the things which validate science, such as observable data. I am also interested in the way that Bohr compares atomic theory to classical philosophy. By this, I mean that he understands that there are unknowns in atomic theory. Finally, I also want to know more about the way he emphasizes that the scientist is a part of the experiment. In Atomic Theory he writes, “The resignation as regards visualization and causality, to which we are thus forced in our description of atomic phenomena, might well be regarded as a frustration of the hopes which formed the starting-point of the atomic conceptions. Nevertheless, from the present standpoint of the atomic theory, we must consider this very renunciation as an essential advance in our understanding. Indeed, there is no question of a failure of the general fundamental principles of science within the domain where we could justly expect them to apply. The discovery of the quantum of action shows us, in fact, not only the natural limitation of classical physics, but, by throwing a new light upon the old philosophical problem of the objective existences of phenomena independently of our own observations, confronts us with a situation hitherto unknown in natural science. As we have seen, any observation necessitates an interference with the course of the phenomena, which is of such a nature that it deprives us of the foundation underlying the causal mode of description.” As with classical philosophy, we are at a crossroads. This new path is filled with unknowns, and not only that, but unobservable unknowns. Despite this complication, Bohr asks scientists to depend upon established terms which maintain a sense of cohesiveness, but also give us some concrete foundations for theoretical science. This technique hearkens back to the beginnings of philosophy as humans grappled to find language suitable for metaphysics.

The “old philosophical problem of the objective existences” outside of our own hearkens back to the roots of philosophy. In fact, as science moves forward, it must address many of the same questions that began as early as 2000 years ago. To address some of these unknowns, Bohr demands precise language without straying from classical vocabulary. Both Atomic Theory and “Discussion with Einstein” address the difficulty of language for the scientist and for the public. He explains that unknowns do not equal a lack of knowledge or a scientist’s uncertainty about the validity of their research. Rather, an unknown is in itself useful. He labels this dilemma an “intricacy of language.” Bohr writes, “[Q]uantum theory presents us with a novel situation in physical science, but attention was called to the very close analogy with the situation as regards analysis and synthesis of experience, which we meet in many other fields of human knowledge and interest. As is well known, many of the difficulties in psychology originate in the different placing of the separation lines between object and subject in the analysis of various aspects of physical experience. Actually words like ‘thoughts’ and ‘sentiments,’ equally indispensable to illustrate the variety and scope of conscious life, are used in a similar complementary way as are space-time co-ordination and dynamical conservation laws in atomic physics. A precise formulation of such analogies involves, of course, intricacies of terminology, and the writer’s position is perhaps best indicated in a passage in the article, hinting at the mutually exclusive relationship which will always exist between the practical use of any word and attempts at its strict definition.” The imprecision in language exists in all fields, and grows as the field grows. Bohr’s insistence upon utilizing classical terminology is twofold. First, He asks that we use exact, well-defined terms so as to limit misunderstandings. Second, he wishes to avoid further abstraction of an already abstract subject.

Bohr’s focus on the language debate reminded me of a recent article on modal verbs, or verbs which predict rather than describe simple facts. The article claimed that scientific papers often get buried or dismissed because they include words such as “might,” “could,” “may,” “ought,” or “will.” Of course, these verbs reflect the fact that scientists do not have all the answers, and each experiment leads to further unknowns. This dismissal is something that Bohr feared and a reason for his insistence upon classical terminology. Incorporating existing terminology with atomic physics, science remains valid and as independent of the scientist as possible. Again, I am reminded of the fact that, according to Bohr, the scientist is a part of the experiment as much as they are observers. Therefore, if the scientist were to also alter terminology in a way that best suits their vision, they would further insert themselves and their view into the experiment. Furthermore, modal verbs signify opportunity for further experiment. They also reflect Bohr’s insistence upon the fact that we cannot know the whole picture anymore. As we interact with and learn from the world, the complexities in science grow larger. However, while uncertainty can be off-putting, uncertainty in science should be celebrated.

Bohr’s focus on language makes me think that there are opportunities for educators here too. In teaching science (to both scientists and non-scientists), we should include a better understanding of the specificity of language. We can also explain the benefit of things like modal verbs. Perhaps this will better enable us navigate complicated theories and unobservable data. We could also better educate young scientists with writing skills. Integration of these fields seems inextricably tied together. Bohr speaks of the writer’s dilemma which he calls, “the mutually exclusive relationship which will always exist between the practical use of any word and attempts at its strict definition.” In some senses, the scientist is now also a writer. In other words, language is of extreme importance for the future of science and we would do well to also teach according to these principles.

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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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