Mathematics and its application to social sciences
Mathematics has played a fundamental role in understanding and forecasting nature's events. For example, mathematical models can generate more accurate weather forecasting than experienced human experts. Today, weather forecasting is made by computed results from mathematical models and not by committees of experts.
However, this is not the case in the social field. Rarely, important decisions on economic policy, population policy and other social policies are based on computed results of mathematical models. Mathematical models mostly are generated to give an aura of sophistication in social science. A notable exception is the field of financial engineering. In this field,mathematical models are built on observable quantities. They usually generate more precise quantitative results than qualitative thinking. Today, many trading and risk management decisions in the financial institutions are based on mathematical models. Mathematicians have made great contributions in this field. Ed Thorp, a math professor for many years, is often called the father of quantitative investment. Fischer Black, a Ph.D. in mathematics, was the main founder of financial engineering. Many math graduates find their talents highly valued in the financial industry. Financial mathematics programs are established in many math departments to broaden the career opportunities for math students.
There is a criticism that these mathematical models, while enriching financial institutions, cause great damages to the general society. But this criticism reveals that these mathematical models do provide benefit to the financial industry, sometimes at the cost of broader society. Can the benefits from the mathematical theories of financial engineering be extended to broader community?
From the early days of the financial engineering, there is an expectation that the theory developed in this field could be a part of the general advance in economic theory. Fischer Black, the main founder of financial engineering, once stated,
I like the beauty and symmetry in Mr. Treynor's equilibrium models so much that I started designing them myself. I worked on models in several areas:
Monetary theory Business cycles Options and warrants
For 20 years, I have been struggling to show people the beauty in these models to pass on knowledge I received from Mr. Treynor.
In monetary theory  the theory of how money is related to economic activity  I am still struggling. In business cycle theory  the theory of fluctuation in the economy  I am still struggling. In options and warrants, though, people see the beauty.
Black's comments show that he, as well as many others,sensed the close relations among different economic and financial problems. His comments also show that it is not always easy to extend an idea from one field to another field, even this might look straight forward with the benefit of hindsight.
I tried to understand life systems from the thermodynamic laws since I was an undergraduate student. I hope to develop a mathematical theory of life systems parallel to classical mechanics as a mathematical theory of general systems. I read about many existing theories, such as Prigogine's theory. However, these theories do not model life processes directly. For a long time, I had little idea how to develop such a theory. I only knew that thermodynamic processes are represented by partial differential equations. So I stick to the theory of partial differential equations, hoping something will turn up some day.
It was after many years before I bumped into the BlackScholes equation. The BlackScholes equation was originated in financial economics. From my perspective, this equation is a mapping from lognormal processes. Lognormal processes can be understood as the representation of extracting low entropy to compensate for dissipation, which is the essence of life processes. I sensed that the BlackScholes theory could lead me further in developing a mathematical theory of living systems.
I started to think about the BlackScholes theory in 1995,when I was teaching mathematics in Hong Kong. In 1997, I joined an investmentbank. There, I learned to associate mathematical theories with investment decisions. Abstract symbols become concrete. A year later, I returned to academia , this time as a finance professor in Singapore. After several years, I worked out a theory of economics that provide an analytical relation among major factors in economics: such as fixed cost, variable cost, investment horizon, discount rate and uncertainty. It provides a simple and consistent understanding on broad range of problems in economic and biological systems. More detailed discussion can be found from books and papers written by me and others.
When financial crises or economic downturn occur, they are often blamed on "unintended consequences" from economic policies. But from our theory, we can solve the equations to obtain quantitative results of long term consequences of those policies. It turns out that the so called unintended consequences are simply long term consequences of the economic policies. Currently, economic policies are mainly measured from their short term impacts. We hope that the introduction of a theory on long term impacts of economic policies will stimulate more active discussion.
Since most prominent economists have a stake in the dominant theory, they are reluctant to discuss rival theories. But mathematicians don't have such concerns. This gives opportunities for mathematicians and other outsiders to make fundamental breakthroughs in economic theories. Neoclassical economics, the current dominant economic theory, was developed around 1870, mainly by Jevons and Walras. Both Jevons and Walras were trained as a scientist and an engineer, not as an economist.
Mathematics has played a substantial role in deepening our understanding of the world. Calculus, stochastic calculus, Maxwell equations Schrodinger equation, BlackScholes equation are just a few examples. But the applications of mathematics in the vast field of social sciences are still largely cosmetic. By actively engaging in the field of social sciences, mathematicians may crucially impact the future of science and human society. In the process, we can work on more exciting research that are more relevant to the real world.
References
Systematic discussion about the new economic theory can be found in my two books.
The Physical Foundation of Economics: An AnalyticalThermodynamic Theory, World Scientific, Hackensack,NJ (2005)
The Unity of Science and Economics: A New Foundation ofEconomic Theory, (2016), Springer
James Galbraith's book, The End of Normal: The Great Crisis and the Future of Growth, (2015), Simon & Schuster,discussed many of my ideas in great clarity.
More information, including all my papers, can be found from my website http://web.unbc.ca/~chenj/
