Travis Deitrich History of Mathematics Minor Paper: 1

Pierre-Simone Laplace

The French and Indian War, a war between the British Colonies in North America and

France, began in 1754. This event is the early beginning of the American Revolution, and ultimately a changing world. Pierre-Simone Laplace was born in France in 1749, and thus lived through the heart of the American Revolution. Interestingly, many males of Laplace’s age in

France had a career in the military, and should he have gone that route, he likely would have had some sort of an impact in the American Revolution. Fortunately for the history of mathematics and science, Laplace was not a military man.

Pierre-Simone Laplace was a mathematician in the 16th and 17th centuries. He was born in France in 1749 and died and in 1827. Laplace was born into an uneducated farming family, but despite this the family was well off. They were neither peasants nor poor. His early educational path had him lined up to work in the church. This career path changed at the age of

16 when he discovered his passion of math. It was then that he moved away from his home to go to Paris and pursue his mathematical career (O’Connor, Robertson, 1999).

Remarkably, as a young man, Laplace was able to impress Jean d’Alembert, who was a famous mathematician of his time (Sanford, 1935). We encountered the name d’Alembert in our weekly readings where I discovered he was very involved in the Paris Academy and helped

King Frederick reestablish the Berlin Academy. Due to this connection, he became a professor at a young age and was able to teach math as a source of income which allowed him to live in

France and pursue his mathematical career. It is in this time in Paris that Laplace quickly became regarded as an extremely talented mathematician. Travis Deitrich History of Mathematics Minor Paper: 1

In 1771, Laplace attempted to get elected to Academie des Sciences, or Academy of

Sciences, in Paris. This academy was founded in 1666 and hosted several prestigious mathematicians as members. However, he was denied election in 1771 and 1772 before finally gaining election in 1773 (O’Connor, Robertson, 1999). In his early years as a member of the

Academy of Sciences, Laplace read many papers on a number of different topics. Because of this, he quickly gained recognition and fame both in and outside of France.

As mentioned above, Laplace read papers to the Academy on many different topics.

These topics include maxima and minima, difference equations, integral calculus, astronomy, differential equations, probability theory, and many more (O’Connor, Robertson, 1999). In this paper, I want to focus on differential equations and probability theory. These will be my focus as I have encountered Laplace Transforms in differential equations and I have an interest in the field of probability and statistics.

Interestingly, the original use of what is now called the Laplace Transform was actually used in probability theory by Laplace. However, today we most often encounter this method while solving differential equations. If you have an ordinary differential equation with constant coefficients, you can use the Laplace Transform method to turn the differential equation into an algebraic equation. Then, upon solving the algebraic equation, you use a table to find the solution to the original equation (Widder, 1945). Laplace’s work on differential equations can be found in one of his many great works, Mecanique Celeste (Richeson, 1942).

When I first encountered the Laplace Transform, it was difficult and seemed very abstract. However, it didn’t take long to make sense of it and it turned out to be quite Travis Deitrich History of Mathematics Minor Paper: 1 fascinating to me. When I encounter methods that seem so strange at first, I often wonder

“who came up with that and why/how did they do it”. Now I have a chance to find out.

Of course, the Laplace Transform was not originally used in the way I have used it.

Laplace came up with the ideas of this method while studying probability theory. This is the field in which he would produce one of his greatest works, Theorie Analytique des Probabilities.

This work gave us the method of least squares, which is used to generate a line of best fit that can help explain the relationship between dependent and independent variables (Sanford,

1935). He also used his knowledge of probability and statistics to positively impact the world he lived in. He investigated hospitals and determine which one’s had the highest mortality rates and why they had them (O’Connor, Robertson, 1999). I have always found mathematical formulas and topics more interesting when they have a clear application to everyday life, so it was fascinating to find this fact entangled in all of the technical readings and formulas.

Theorie Analytique des Probabilities is where Laplace first introduced his idea of generating functions. A generating function is an infinite sequence that is composed of two individual functions. The second function is composed of the coefficients of the first function and is called the generating function (Richeson, 1942). These generating functions are closely related to probability and moment generating functions. Laplace also included Bayes rule, which wasn’t named this until many years later, in this work (O’Connor, Robertson, 1999).

I chose to mention Laplace’s work in Probability theory because it partially lines up with my desired career path. My goal is to one day become an actuary and just this past summer I managed to pass my first actuarial certification exam, exam Probability. Probability and moment generating functions were one of the many topics covered on this exam. These Travis Deitrich History of Mathematics Minor Paper: 1 functions were used to find the expected value, variance, and standard deviation of functions which are all very important in probability and statistics. Bayes rule was also very important on the exam as it is a method used to solve problems concerning conditional probabilities. The formula essentially takes the desired probability and flips the condition. Then it is multiplied by the probability of the unconditioned event. Then you take this result and add it to undesired probability times the undesired condition. Finally, you divide the first result by the second and arrive at your final answer. This was one of the most used and most important formulas on the exam.

One of the most interesting facts I came across when researching Laplace is how he continually conformed his ideas to please the politicians of his time. It is unfortunate that great scientists had to agree with politicians and the church. If this weren’t the case, Laplace and others likely would have made numerous more contributions to their respective fields and to the societies they lived in. The reason I find this so interesting is because of past research I have done on Galileo. It is well known that Galileo had to recant his claims in astronomy due to the church. Galileo was born nearly 200 years before Laplace but yet Laplace had to deal with the same issues.

Can you imagine a world where your beliefs had to align with the religion of the ruler of the time or else you risk being imprisoned or even executed? That blows my mind, but that is what Laplace and many others had to do. Laplace lived through the French revolution and the

Reign of Terror, which followed the execution of King Louis XVI. It was during this time that individuals who did not agree with the revolution were guillotined. Because of this, Laplace left

Paris and managed to survive the Revolution unlike some other well-known mathematicians Travis Deitrich History of Mathematics Minor Paper: 1 who stayed in Paris (O’Connor, Robertson, 1999). Fortunately, Laplace survived the Reign of

Terror as a lot of his work, including his work in probability, came after this time.

Amazingly, in so many years of studying math, I only ever heard the name Laplace when using the Laplace transform. Before this paper, I never would have known that he was so influential in the field of probability which I have studied in depth. It would be interesting to see what other contributions to society Laplace could have made if he didn’t have to conform to certain beliefs or briefly leave Paris for his safety. Despite these obstacles, Laplace became one of the great mathematicians of his time and continues to impact the world today.

Travis Deitrich History of Mathematics Minor Paper: 1

Bibliography

O'Connor, J., & Robertson, E. (1999, January). Pierre-Simon Laplace. Retrieved from https://www-history.mcs.st-andrews.ac.uk/Biographies/Laplace.html

Richeson, A. (1942). Laplace's Contributions to Pure Mathematics. National Mathematics

Magazine, 17(2), 73-78. Retrieved from http://www.jstor.org/stable/3028283.

Sanford, V. (1935). Pierre-Simon Laplace: Born at Beaumont en Auge, March 23, 1749 Died in

Paris, March 5, 1827. The Mathematics Teacher, 28(2), 111-113. Retrieved from http://www.jstor.org/stable/27951763

Widder, D. (1945). What is the Laplace Transform? The American Mathematical Monthly, 52(8),

419-425. Retrieved from http://www.jstor.org/stable/2305640.