Saturday, January 26, 2013

History of Mathematics: Charles Dodgson

Mainly, I talk about proofs, patterns, and ways to apply mathematics to the real world. However, there are other ways that mathematics is cool too. Not only is the actual content interesting, but the stories are as well.

Today, I will tell the story of a mathematician named Charles Lutwidge Dodgson. Dodgson was born in 1832 in Cheshire, England. He graduated from Christ Church College at Oxford, and then began his career as a mathematician, lecturing and tutoring at Oxford.

Dodgson was not the type of mathematician who often made breakthroughs and discoveries. However, he did find some interesting things in mathematics and logic. One of them, I actually talked about in a previous post, but never mentioned that Dodgson was the mind behind it. Click here to see it.

He is also known for a method of election he developed. In America, we just vote for the candidate of our choosing, which is called simple plurality. Yet, we run into problems in elections such as the Bush-Gore-Nader election of 2000. In this election, most Nader supporters preferred Gore to Bush and most Bush supporters preferred Gore to Nader, even if it was just by a little bit. So, Gore was never the least favorite of anybody, while Bush was the least favorite of most Nader supporters and probably many Gore supporters. Using a ranking method rather than a plurality method, we can find a winner that the most people are satisfied with.

Yet, this method can fail in elections such as the Obama-Romney election of 2012. The leading third party candidate was Gary Johnson, but was supported by only a small percentage of the population. But, Obama fans would rank Johnson as their second favorite in order to give as little support as possible to Romney, and vice versa. This may have led to a win for Gary Johnson, which very few people would be satisfied with.

So, mathematicians in the field of game theory are always struggling to find a perfect election method. There are dozens of ideas out there, one of which is created by Charles Dodgson. His method makes use of finding a Condorcet winner. If on every ballot, a certain candidate is ranked higher than another one, then this beaten candidate is eliminated. For example, if all Americans preferred Barack Obama to Jill Stein, the Green Party candidate, then Stein would be eliminated. If you can eliminate every candidate in this way, the candidate still standing is the Condorcet winner.

Since it is near impossible to have a Condorcet winner, this method is impractical. However, Dodgson extended it by saying that after eliminating everyone possible, you begin swapping rankings on people’s ballots until you are able to have a Condorcet winner. The candidate that requires the least swaps wins. This method will eliminate weak third party candidates, but still take them into strong consideration.

On a different note, Dodgson was also known for his ability to write. He wrote dozens of famous math textbooks, and compiled texts for undergraduate students. But he also enjoyed implementing his mathematical knowledge into fantasy writing. As a man who loved children, he would tell stories that had some mathematics and logic infused in them.

One of his favorite children was named Alice Liddell, whose father was the Dean of the school where Dodgson taught. He began telling her stories about a girl named Alice, which were always continued every time they saw each other. Years later, he gave her a written manuscript of the story for Christmas.

In the story, there were many mentions of mathematics. For instance, there is one point in the story where Alice is three inches tall. She finds out that she must eat from a mushroom to grow back to normal, but half of the mushroom stretches her neck and half shrinks her torso. She must find the correct proportions to grow properly.

This is the foundation of Algebra. In fact, the word “Algebra” comes from an Arabic algebra book whose title translates to “Restoration and Reduction.” And she must use the concepts in this book, by finding how much of each side of the mushroom is necessary to make her neck proportions equivalent to her torso proportions. This resembles an algebraic equation.

Dodgson even used his ability to play with words in the story, by describing the branches of arithmetic as ambition, distraction, uglification, and derision.

These manuscripts ended up getting published, and eventually found their way to the queen of England: Queen Victoria. After reading them, she demanded that she receive every book written by this man. To her surprise, she ended up with a huge stack of mathematics textbooks.

This book that the queen loved, that has many mathematical references, that is written by a mathematician from Oxford, is titled Alice and Wonderland. And you might know Charles Dodgson better by his pen name, Lewis Carroll.

Bonus: Lots of these historical mathematicians have a few funny stories in their successful careers as well. Here is one about Charles Dodgson.

Because of his friendliness to children, Dodgson was a popular guest at parties. One time when he was invited to a party in London, he decided to crawl into the room as a surprise to the kids. However, he crawled into the wrong household, where a group of adults were also having a party.

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