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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 asimon@hmu.edu. I look forward to hearing from you!

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Discussing de Tocqueville

November 2, 2018

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

For the October Quarterly Discussion, we read four chapters from Alexis de Tocqueville’s Democracy in America. As usual, I distributed some questions beforehand intended to help start the conversation. Each discussion lasts 1.5 hours in which I (mostly) lead. I enjoy the responsibility of organizing these discussions because I get to begin with the questions that I have about a specific text. Due to the fact that so much of de Tocqueville’s writings resonate with me, I really struggled to refrain from participating too. His writings also speak to current politics, and therefore, it was doubly hard to avoid participation. I have to thank the participants in Harrison Middleton University’s October Quarterly Discussion who did an admirable job of sticking to the subject.

We began with the formation of political parties in general. He writes, “But when the citizens entertain different opinions upon subjects which affect the whole country alike, such, for instance, as the principles upon which the government is to be conducted, then distinctions arise that may correctly be styled parties. Parties are a necessary evil in free governments; but they have not at all times the same character and the same propensities” (88-9). So, while he finds parties to be a necessary evil, he also does not find them equal in character. From there, we tried to understand de Tocqueville’s delineation between “great” and “small” parties. Despite the way that it sounds, these two types of parties have nothing to do with size. Rather, in de Tocqueville’s mind, the great parties are those that discuss issues and have, what he calls, a “more noble” pursuit. On the other hand, small parties form around an issue or two. The small parties, according to de Tocqueville, care more about a single issue or a private interest than about ideas or the good of society, whereas great parties are concerned with principles and their general application. In 1830, he writes, “America has had great parties, but has them no longer; and if her happiness is thereby considerably increased, her morality has suffered” (89B). According to de Tocqueville, the great parties arose out of necessity and strife, a time when America was suffering. These parties looked at broad issues that would impact all of America. The focus, therefore, was more holistic. However, once these changes were implemented and the need for social cohesion lessoned, special interests overtook the general cohesion of the great parties and replaced them. De Tocqueville describes the effects of the small parties as those which “agitate” society rather than revolutionize it.

Furthermore, de Tocqueville’s use of happiness and morality is of great interest. In this section, he seems to define happiness as a level of individual comfort and perhaps peace. It appears that his version of happiness in America is one which leads to a sort of immorality. He suggests that the more comfortable we are, the more self-involved we are and therefore, less moral. In other words, morality may demand an ethic that lessens our ease of living. In the future, I would like to further investigate de Tocqueville’s idea of happiness by moving outside of this single chapter. I am curious how happiness (in his terms) aligns with morality throughout the text. Furthermore, I wonder how different translators have dealt with this idea. Is happiness the most appropriate word choice for the original French? How have others translated this section? (The Great Books version was translated by George Lawrence.)

From there, we moved into the chapter on Freedom of the Press. De Tocqueville begins this chapter by stating that he has reservations about a free press. He writes, “I confess that I do not entertain that firm and complete attachment to the liberty of the press which is wont to be excited by things that are supremely good in their very nature. I approve of it from a consideration more of the evils it prevents than of the advantages it ensures” (92A). First, he finds that a free press is invaluable to a democracy because information distribution would be limited by a single entity. On the other hand, freedom implies that nearly anyone can create news if they choose to do so. In the first case, news is singular and perhaps biased or incomplete. In the latter, news may lack data, information, facts and anything pertaining to reality. Furthermore, he writes, “[T]he hallmark of the American journalist is a direct and coarse attack, without any subtleties, on the passions of his readers; he disregards principles to seize on people, following them into their private lives and laying bare their weaknesses and their vices. That is a deplorable abuse of the powers of thought” (95A). He continues that, despite the abuse of thought, each individual newspaper carries little weight, which makes many small voices. This cacophony creates the “spirit” of the press. The multitude of voices also ironically removes the danger of a single voice reaching the level of despotism.

These chapters address very complex issues inherent in America’s being. They are worth more than 1.5 hours of discussion. Rather, de Tocqueville addresses so many contemporary issues that the entire volume is worth (re)reading. Additionally, discussing a work like this one is vital to understanding the depth of democracy’s issues. Democracy in America explains some of the foundations of our country in a way that is both poetic and holistic. My gratitude goes to those who spent time in discussion with me. I look forward to our next conversation!

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Questions on Augustine

August 3, 2018

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

Each quarter, Harrison Middleton University hosts a Quarterly Discussion. This discussion is open to students and non-students alike. They focus on a short text which everyone reads prior to the discussion. I thoroughly enjoy these because they give me a chance to break away from my own studies, to focus on something in a small group which is a great listening opportunity. This month I was blessed to have Jim Keller, a current HMU master’s student, assist with the discussion topic, reading, and questions. He even led the discussion so that I could participate. What a treat! I think that anyone interested in Shared Inquiry style discussions should try their hand at leading. While it may seem intuitive, there really is a lot to learn about managing the flow of a conversation. Whatever your style, trying to put together a successful discussion requires a great knowledge of the text, but also an ability to listen to disparate voices in a conversation. I find this to be the greatest struggle, but also the greatest benefit, of Shared Inquiry. Many thanks to Jim for the assistance in setting up the conversation, and to the participants for some inspiring conversation.

This month, we read Book XIX from St. Augustine’s City of God. We began with a passage from Chapter 4 which reads, “And justice, whose office it is to render to every man his due, whereby there is in man himself a certain just order of nature, so that the soul is subjected to God, and the flesh to the soul, and consequently both soul and flesh to God – does not this virtue demonstrate that it is as yet rather labouring towards its end than resting in its finished work?” (580B). From this statement, I believe that Augustine’s version of justice can be defined as: “to render every man his due.” Upon first reading, I assumed the implication being that each man received an equal portion. However, Chapter 13 squarely denies that assumption. In Chapter 13, Augustine writes, “Order is the distribution which allots things equal and unequal, each to its own place” (588A). In other words, we all receive a lot in life, and it may partake of greater or lesser as fits our being. I am still contemplating how this reflects a sense of justice. So, taking both of these statements together, I see that Augustine’s world relies upon order. In the city of man, order is granted as best as can be expected, but imperfectly to say the least. Order is a form of justice in that it is at least an organizing principle. Justice, also, stems from God (or from the City of God) which exists in perfect peace. This ultimate ideal of peace is the justice that Augustine seeks. So, man’s flawed implementation of justice is at least an attempt to model the city of God. I do see how the city of man is flawed and he consistently revisits that throughout the chapter. I still cannot quite come to terms with the idea of inequality as foundational to this sense of justice. I always assumed that God granted portions to each man, so why would he perpetuate inequalities?

I also struggle with the way in which Augustine proves his point. Throughout the book, he claims that human life is flawed and poor in comparison with the life of the soul. And yet, Augustine’s proof always stems from examples of human life. I see the obvious reason for that, being difficult to capture universally-accepted empirical data which proves of the soul’s existence, yet to claim that human life is worthless and then turn around to exclaim its worth seems complicated at best. Chapter 6, for example, describes the ways in which it is acceptable for judges to implement torture. While admitting the system is flawed, Augustine also allows that the wise judge may need to torture innocent persons in order to understand the truth. Though he acknowledges that often tortured persons are innocent and at times the innocent are killed, he finds it to be a necessary part of the process towards the greater good. Augustine writes, “These numerous and important evils he does not consider sins; for the wise judge does these things, not with any intention of doing harm, but because his ignorance compels him, and because human society claims him as a judge. But though we therefore acquit the judge of malice, we must none the less condemn human life as miserable. And if he is compelled to torture and punish the innocent because his office and his ignorance constrain him, is he a happy as well as guiltless man? Surely it were proof of more profound considerateness and finer feeling were he to recognize the misery of these necessities, and shrink from his own implication in that misery; and had he any piety about him, he would cry to God ‘From my necessities deliver Thou me’” (583). In other words, while the judge may feel some level of guilt, he is to be absolved of any sin because he is fulfilling the duty required of him. Rather than a reflection on the individual, this scenario is meant to demonstrate man’s absolute depravity. The city of man grants a judge power and it is better for him to pursue this grave responsibility in the manner of the times than to avoid unpleasantness by shirking the judge’s sole responsibility. Duty compels the judge to act.

Contrary to all the questions I have raised above, I did learn quite a bit from these conversations. Reading Augustine begs conversation simply because of the complexity of terms and the text’s density. In this chapter alone, we discussed virtue and vice, good and evil, peace, eternity, eternal life, and justice, just to name a few. I would encourage anyone to pick up a chapter of Augustine and struggle with it as we have. Better yet, pick up the chapter with a few friends and struggle to define these terms in both his context and our contemporary world. My appreciation to the folks who struggled alongside me and listened patiently as we explored the text together.

As usual, I am already looking forward to October’s Quarterly Discussion on de Tocqueville. You can join! Simply email asimon@hmu.edu. I look forward to hearing from you!

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Translations of Chaucer

May 4, 2018

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

Last week’s Quarterly Discussion focused on Chaucer in translation. I opened the discussion with the question as to how one would determine what elements make a “best” translation. This seemingly generic question is actually really difficult to answer. Some authors excelled at rhyme schemes, while others performed better with word choice, for example. Truly, there is no single response and my intent in opening (and closing) the discussion with this question was to explore the benefits of having access to the primary source (and primary language).

Necessarily, translation contains many layers. First, there is the original language barrier. Most people who read a work in translation do so because they cannot access the language of origin. Secondly, translations must navigate not only word choice, but also context and cultural information carried within a language. Stylistic issues present a third difficulty. These are things such as alliteration, rhyme scheme, line length, etc. And finally (though this list is hardly exhaustive), the translator faces the issue of the target audience. By that I mean that the translator must weigh all of these decisions by postulating how successful the translation will be in the target language. In other words, there is a very delicate balance between risk-taking and minimizing risk, all driven by the target audience.

I selected Chaucer in translation for a couple of reasons. First, I was surprised that the Great Books version was Neville Coghill’s translation. My initial introduction to the Canterbury Tales was twenty years ago in an undergraduate class where we worked slowly and diligently (if impatiently) through the Middle English. I think this text is still fairly accessible with footnotes. Having said that, I do understand the amount of labor involved in combing through a footnoted text. (Personally, I believe that Gibbon’s texts include more footnotes - and more foreign languages - than the Middle English Chaucer, but that is perhaps just me.) Since I study the idea of poetry and occasionally attempt translations, I have become very interested in the differences (and similarities) between the original and translated versions of Chaucer.

For this discussion, we compared four different texts of “The Nun’s Priest’s Tale”: Nicholson (Librarius version, published 1934); Neville Coghill’s translation (published 1951; this is also the Penguin Classics text as well as the Great Books version); A. S. Kline’s version (dated 2007); and the original. Below is a short passage from all four texts which will hopefully illuminate many struggles both from a translator’s standpoint, as well as the audience’s.

First of all, Chaucer writes in rhymed iambic pentameter. Since this sounds archaic to the modern-day ear, the translator first has to decide if they will stick with that same rhyme scheme. Secondly, Chaucer includes jokes throughout, using words with double meanings that may have faded away over time. In some cases, the translators chose to create a new joke, one that would work with the contemporary audience. In other cases, the translator ignores the joke. What license does a translator have in making these decisions? How much introduced content is actually new content? For example, in order to replace the Old English forslewthen, which means slow to act, or to delay, Coghill adjusts the metaphor and loses the layered meaning of a man about to set sail who is incapable of stemming his own “tyde”. Coghill writes, “But as I see you mean to stay behind/ And miss the tide for willful sloth of mind” instead of the original line: “But sith I see that thou wolt heere abyde/ And thus forslewthen wilfully thy tyde”.

Analyzing translations can be a tedious process, but it is the most enriching language experience I can imagine. In studying these texts, we gained insight into: root words, etymology, cultural adaptations, similes and metaphors (both new and old), effects of acculturation, and more. Furthermore, when translations of Chaucer differ, then the moral, main idea, character or understanding of Chaucer and his times, might also change. The stakes may be even higher when thinking about something like Tocqueville’s Democracy in America or Plutarch’s Parallel Lives, in which the moral is educational. The following passage demonstrates just one (of many) deviations that we analyzed during this discussion.

 

“Ful sooty was hir bour, and eek hir halle

In which she eet ful many a sclendre meel.

Of poynaunt sauce her neded never a deel.

No deyntee morsel passed thurgh hir throte;

Hir dyete was accordant to hir cote.” – Original

 

“Right neatly was her bedroom and her hall,

Wherein she'd eaten many a slender meal.

Of sharp sauce, why she needed no great deal,

For dainty morsel never passed her throat;

Her diet well accorded with her coat.” – Librarius (Nicholson)

 

“Sooty her hall, her kitchen melancholy,

And there she ate full many a slender meal;

there was no sauce piquante to spice her veal,

No dainty morsel ever passed her throat.

According to her cloth she cut her coat.” – Coghill

 

“Full sooty was her bower, all melancholy,

In which she ate full many a scanty meal.

No pungent sauce was needed for her veal;

No dainty morsel ever passed her throat.

Her diet, her cottage struck a single note.” - Kline

After reading through these passages, we wondered: Do we have the same impression of the widow after reading these four lines? Why do Coghill and Kline introduce the term melancholy? Did Chaucer intend for the widow to be surrounded in dirt, or simply soot from the ever-burning fireplace? Did the widow feel impoverished? How does “veal” change the passage, if at all? If we assume that Chaucer is intentionally including words from Latin, French and Old English, does the word “piquante” fit here, or does it distract (since we no longer navigate language tri-lingually or are, at least, not aware of it)? “Cote” is the Middle English term for cottage, so is the introduction of “coat” appropriate?

Basically, if this is the character that Chaucer uses as the introductory frame for his morality play, how much change still maintains the same message? And, is it important to aim for the same? At the end of our discussion, we still struggled to state precisely what is “best”. The translator must make value judgments and so must the audience, thus complicating the very business of translation!


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