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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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Why Read Archimedes

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)

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Euclid and Whitehead: Found Poem

February 5, 2016

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

A colleague suggested that I re-read Euclid as if his definitions were a poem. Considering I study poetry often, this sounded like a fun exercise. I was unprepared, however, for the depth of insight that followed. I already discussed how important it was for Euclid to be both precise and abstract, but I did not realize how truly applicable this revelation was. Euclid's Elements contain nearly every characteristic of poetry: precise, concise, abstract, yet specific, and most of all, endlessly interconnected. From that exercise, I decided to arrange a few of the passages into a found poem. (A found poem basically takes words and phrases of others and connects them in a poetic format.) The following poem combines words from both Euclid (in italics) and Alfred North Whitehead. Enjoy!

Our Inaccurate Laws

I

[T]he first noticeable fact

about arithmetic

is that it applies

to everything

to tastes and to sounds

to apples and to angels

to the ideas of the mind and

to the bones of the body

 

A point is that which has

position

but not dimension

 

The nature of things is perfectly

indifferent

 

A line is length

without breadth

 

In a mountainous country distances are often reckoned

in hours

 

A line which lies evenly between its

extreme points

is called

a straight line

 

To see what is general in

what is particular

and what is permanent in

what is transitory is

the aim of scientific thought

 

Any combination of points, of lines, or of points and lines

in a plane

is called a plane figure.

 

any number x which is greater than 1

gives x + 2 > 3

there are an infinite number of numbers

which answer to the some number

in this case

 

II

The vital point

in the application of mathematical formulae is

to have clear ideas and

a correct estimate of their relevance to

the phenomena under observation.

 

our inaccurate laws may be                   good enough

 

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January Quarterly Discussion Review

January 29, 2016

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

Let me begin by stating that organizing and leading Harrison Middleton University's Quarterly Discussions is one of the best parts of my job. The participants always challenge me as do the texts and authors. This quarter, I chose something completely foreign to me and I am absolutely ecstatic with the result. For January's discussion, we read from Euclid's definitions in The Elements as well as a selection from Alfred North Whitehead's Introduction to Mathematics. The discussion truly brought the two works to life for me, especially in comparison with each other. I will briefly describe some of our conversation below. Please enjoy this segment of our conversation, and consider joining us for our next discussion (Kafka in April). Email me at any time at: asimon@hmu.edu .

Euclid begins The Elements with the definition of a point. It is ironic to think that it was necessary to unpack the definition of a single point, but it was. We discussed this idea as a basic, abstract unit from which structure is created. In other words, the point can be viewed as the first building block in Euclidean geometry. This sounds obvious, but consider the concept of creating a number of definitions from scratch. Euclid had the benefit of some previous mathematicians and philosophers around him, but he essentially created this mathematical process. He explained a logical structure where, previously, none existed. Euclid explained abstract definitions with simplicity and precision. Mathematics is often a field of precision, and his simple clarity of reasoning may be one of the best examples of logic to this day. That this ancient Greek text continues to survive at all may be a testament to its worth. At first, the definition of a point did not appear to be innovative, however, as I began to understand the scope of Euclid's theories and applications, I wondered at his ability to locate and define the point at all.

Euclid's Elements stem from his studies of the earth. (Geo is Greek for earth, and metron is Greek for measure.) Therefore, Euclid had the foresight to realize that his observations and measurements of the earth had larger implications, but he also realized that his observations were limited by a definiteness. In order to open them to wider applications, he needed to find a way to abstract these measurements into definitions. Therefore, Euclid began to synthesize, simplify and apply his data. While Euclid discovered geometry in a very concrete way, he catalogued it in abstract language in order to increase its applicability.

Alfred North Whitehead calls mathematics an 'abstract science'. Whitehead reinforces the need for simple, direct mathematical language when he says, “The reason for this failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception.” His statement reinforces the same instinct that drove Euclid to begin The Elements with basic definitions. Geometry exists as a bridge between the natural world and human understanding. Yet it is written in abstract language which allows for a wider application. In other words, grappling with mathematical definitions allows all of us an opportunity to solve problems for ourselves, even in unrelated fields. Geometry, discovered through concrete means, has abstract applicability, while understanding the abstract may lead us back to a specific resolution.

Therefore, when Whitehead states that our abstract and inaccurate laws may be 'good enough', he actually means to say that our approach is vital, and our precision may be good enough. The definition and application of infinity supplies one example of this idea of 'good enough'. After we attempted to unpack the idea of 'infinity', the group concluded that: if infinity is the negation of the finite, then neither of those terms are representable. Instead, they are abstract definitions meant to guide us in some way. However, the adjective 'infinite' grants us an ability to talk about infinity without addressing specific quantities. Things can be infinitely large or infinitely small...both of which signify something endless or boundless. Furthermore, while infinity cannot be grasped with certitude, indefiniteness is merely an unknown.

These beginning conversations could spark endless conversation in my mind. I find applications in science, poetry, nature, social sciences, etc. Euclid's graceful simplicity combined with Whitehead's invitation to study our world led to an exemplary conversation. Whitehead states, “The vital point in the application of mathematical formula is to have clear ideas and a correct estimate of their relevance to the phenomena under observation”. Understand the idea, clarify your parameters and explore a new phenomena. The world is as big (or small) as we make it.

Thanks to all of our participants!

 

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