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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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