Travis Deitrich History of Mathematics Minor Paper 2
Augustus De Morgan
The beginning of the 19th century was an exciting time in the United States and around the world. The White House was built, Lewis and Clark embarked on and completed their journey across what is now the Western United States, and Napoleon declared himself emperor of France. In 1806, Noah Webster published his first dictionary on the English language, a book that is still being updated and published today under the same name. This was the same year that Augustus De Morgan was born.
Augustus was born in 1806 in when his father, a lieutenant, was stationed with the
English Army in India. Due to a birth defect, Augustus lost vision in his right eye shortly after birth. His father returned to England with his family before turning a year old. As a child, he struggled in school. This seems to be fairly common among future mathematicians as we have learned this was the case for others, including Newton, at a young age. De Morgan did not fit in with the other young boys at school. He could not join when they were playing sports due to his partial blindness. This resulted in him becoming the target of jokes by other students, which likely lead to his education struggles (O’Connor, Robertson, n.d.).
Fortunately, his struggles did not last, and he found pleasure and success in mathematics. It was then at a young age of 16 that he was off to Trinity College Cambridge in
1823 to study math. He was mentored by professors, and future lifetime friends, George
Peacock and William Whewell. At the age of 21, after earning his BA, De Morgan applied for the chair of mathematics at a new university, University College London. Despite having never published, he was granted the chair and became the first professor of mathematics at this new college. He would hold his position for a total of 33 years, despite being forced to resign twice. Travis Deitrich History of Mathematics Minor Paper 2
It was during these years that De Morgan began making his impact on the history of mathematics (O’Connor, Robertson, n.d.).
I first encountered Augustus De Morgan in the same way that I am sure most other students do. It was early on in my first college probability class that we discussed De Morgan’s
Laws. These laws seem very trivial to me now after a few courses in probability that I forget that they weren’t always known and had to be stated and proved. These laws state that (E∪F)`
= E`∩F’ and (E∩F)` = E`∪F` (Richards, n.d.).
In words, (E∪F)` = E`∩F’, can be explained as the complement of the union of two sets is equal to the intersection of the complement of each set. Also, (E∩F)` = E`∪F` can be explained as the complement of the union of two sets is equal to the union of the complement of each individual set. These laws are perhaps better understood visually at first through the use of
Venn Diagrams. If Venn Diagrams had been invented earlier, it is very possible that these laws would have been discovered earlier and would be known by a different mathematician’s name today.
Although I knew and understood these laws before researching Augustus De Morgan, I wanted to learn more. I have an interest in probability so I figured he would be a good choice.
During my research process, I was shocked to learn that he made an even bigger contribution to mathematics than his laws. De Morgan was the man who recognized the power of, and formalized mathematical induction. I don’t ever remember hearing his name when I took my first course in mathematical proofs. Of course, we learned and used induction, but that was it.
We never talked about who formalized it as a key method of proving mathematical statements and theorems. Even in the two semesters so far following this course, I have had numerous Travis Deitrich History of Mathematics Minor Paper 2 classes where mathematical induction was an important tool, and still no mention of De
Morgan.
Mathematical induction is a method of proving propositions and theorems. There are two types of induction, strong and weak, but they work similarly. Generalizing induction into one category, this is the idea behind how it works. When given a statement, you prove that it is true for some base case. Typically, this is done by allowing n to be a natural number and setting it equal to 1. After you have shown the statement is true for the base case, you assume that the statement is true for n equal to k, where k is any natural number. Finally, you use this assumption to show that the statement is also true for n equal to k + 1. As k is arbitrary, this then shows that the statement holds true for all natural numbers.
That is how I was first taught induction, and it is typically how I have used induction since. However, this semester I have experienced it in a slightly different way. I have used induction to prove theorems in graph theory. When using induction in this sense, we are typically removing a vertex or an edge and stating the result. We then continue to remove vertices and edges until we get to the desired conclusion, hence proving the original theorem or statement true. While we don’t use induction over the natural numbers, the same ideology and principles still hold true.
De Morgan first defined and used the term mathematical induction in an article titled
Induction (Mathematics) (O’Connor, Robertson, n.d.). This article was one of 712 articles he would eventually write for the Penny Cyclopedia. He defined successive induction as “a collection of general truth from a demonstration which implies the examination of every particular case” (O’Connor, Robertson, n.d.). He then went on to show how this method in Travis Deitrich History of Mathematics Minor Paper 2 action by proving that the sum of any number of successive odd numbers beginning with 1 is a square number (Kolpas, n.d.).
Another field that De Morgan studied was complex numbers. He published
Trigonometry and Double Algebra in 1849. In this book he gave a geometric interpretation of complex numbers. This work led him to understand that there were other forms of algebra outside of ordinary algebra. He realized that algebra had a purely symbolic nature which is why he worked with Charles Babbage and Lady Lovelace. Lovelace of course is said to have wrote the first computer program for Babbage. They all worked well together because of their understanding that algebra was symbolic and had other forms (O’Connor, Robertson, n.d.).
One question that De Morgan posed that I found particularly interesting is the four color problem. Essentially, De Morgan claimed that given any map that shows the borders of any number of countries, you can color them without two countries bordering each other being the same color using just four colors. He made this proposition in 1852 and despite being fairly obvious just by drawing and coloring any map, it wasn’t proved to be true until 1976 (Richards, n.d.).
Before I began the research process, I only knew Augustus De Morgan in the context of probability. Because of this, I assumed that De Morgan was primarily interested in probability and statistics. However, I have now learned that is not true. De Morgan was primarily a logician, it just so happens that his laws are used in probability. These laws also apply to set theory, which is more along the lines of their original use by De Morgan. He also used his interest in logic to formalize mathematical induction, which in my opinion is where we should really learn his name as induction is far more important to a math student then De Morgan’s laws. I will Travis Deitrich History of Mathematics Minor Paper 2 leave you with this “fun fact” as De Morgan, much like myself, was always interested in random numerical facts. In the year 1849 De Morgan turned age 43, which as he himself wrote, gave him the distinction of being x years old in the year 푥2.
Travis Deitrich History of Mathematics Minor Paper 2
References:
Kolpas, S. (n.d.). Mathematical Treasure: Augustus De Morgan and Mathematical Induction.
Retrieved October 17, 2019, from https://www.maa.org/press/periodicals/convergence/mathematical-treasure-augustus-de- morgan-and-mathematical-induction.
O'Connor, J., & Robertson, E. (n.d.). Augustus De Morgan. Retrieved October 17, 2019, from https://www-history.mcs.st-andrews.ac.uk/Biographies/De_Morgan.html.
Richards, Joan L. “‘This Compendious Language’: Mathematics in the World of Augustus De
Morgan.” Retrieved October 17, 2019, from www.jstor.org/stable/10.1086/661624.