Calculus: God's Own Language - I
13th June 2014
[Note: This abridged article is excerpted from the upcoming book Quantitative Finance: The Long Road Through Physics by Rahul Bhattacharya.]
Before we embark upon this wonderful journey to understand how Physics has influenced the study of Quantitative Finance and how ultimately "physics-like" theories have failed to explain the true dynamics of the financial markets we need to understand the language in which physics is written because it is the same language that is used to explain quantitative finance. That language is the language of Calculus, a branch of mathematics, which is more technically known as, Mathematical Analysis, which arose in the 17th century due to the work of Fermat, Newton and Leibniz. The study of mathematical analysis is called "calculus" because it is concerned with a body of calculational techniques that renders conceptually difficult problems to be solved "automatically" and easily.
Almost all of quantitative finance is written in the language of calculus: calculus of real, complex and random numbers. But calculus posits a deep philosophical conundrum. In physics, it seeks to explain the motion and interaction of matter, which is finitely divisible by formulating laws and rules that assume that space and time, in which all matter resides and moves, is infinitely divisible. We are certainly not going to talk about theory and rules of calculus here, for that is a subject matter that will require many advanced textbooks, but f we have to understand the involvement of physics in quantitative finance we need to at least have an a priori knowledge of a couple of key assumptions which define that subject matter and which is responsible for generating the quandary that I just referred to.
Calculus is the language in which God talks to us.
In more understandable terms, it is the study of how quantities change locally, at the tiniest levels, and globally, in a total sense, and how the two kinds of changes are related. There are two ingredients of calculus, differentiation and integration, giving rise to two different branches of calculus, the differential and integral calculus respectively. Differentiation is about capturing the change in the value of a certain quantity when another quantity, on which the value of the former depends, changes in the "tiniest neighourhoods of single points". It is all about slopes and curvatures of curves and surfaces which is carried over to physics to explain velocity and acceleration of bodies. Integration - which simply means summation - is about measures of totality
Many decades ago, a Pulitzer Prize winning American novelist and a man of letters went to see a group of physicists at California Institute of Technology to learn more about the atomic bomb. The novelist was writing a novel about World War II and wanted to get more information on - and, deeper insight into - the workings of the Manhattan project which had developed the atomic bomb. Amongst that group at CalTech was a famous, Nobel Prize winning physicist, who had played a key role in the Manhattan project and whose ideas on quantum mechanics had a lot of do with the development of theories of nuclear physics that made the atomic bomb possible.
The physicist with his long hair and a "sharply humorous countenance" which reminded the novelist of a bust of the philosopher, Voltaire, which he had once seen, was more than forthcoming with respect to his experience in the project. At the end of the meeting while they were parting the physicist asked the novelist if he knew calculus. The novelist replied that he didn't. Upon hearing this, the physicist, in a somewhat admonishing tone, retorted. "You had better learn it...It's the language in which God Talks".
The physicist was Richard Feynman and the novelist, this man of letters, who had gone to CalTech to do research for his novels and to talk to the physicist, was Herman Wouk.
Herman Wouk, who knew only a little bit of algebra learnt during his high school years, and who was totally clueless about the subject of Calculus, was in his forties when he met up with Feynman at CalTech. For the better part of the next two decades of his life, he remained absorbed in writing his two masterpieces, Winds of War and War and Remembrance, about World War II, all along wondering if he would ever get a glimpse of the language in which God talks. Finally, in his late sixties he embarked upon the task to learn calculus, all by himself. He tried his hand at everything, from skimming the pages of a college freshman year textbooks on calculus to titles like "Calculus Made Easy". He engaged a private tutor and even ended up auditing a high school course in calculus and hanging out with teenagers to learn the subject. In the end, he gave up.
Learning the language of God still remains a distant dream for this 98 year old literary genius of the 20th Century. But for those in the field of physics and finance - and also in all areas of engineering - this language remains the only language with which to comprehend the world and how things unfold here.
- Wouk, Herman, The Language God Talks, Back Bay Books, 2010
- Bhattacharya, Rahul, Quantitative Finance: The Long Road Through Physics, CFE School Publishing, 2014
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