Mathematics and its application tosocial sciences

Mathematics has played a fundamental role in understanding and forecastingnature's events. For example, mathematical models can generate more accurateweather forecasting than experienced human experts. Today, weather forecastingis made by computed results from mathematical models and not by committees ofexperts.

However, this is not the case in the social field. Rarely, important decisionson economic policy, population policy and other social policies are based oncomputed results of mathematical models. Mathematical models mostly aregenerated 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 generatemore precise quantitative results than qualitative thinking. Today, manytrading and risk management decisions in the financial institutions are basedon mathematical models. Mathematicians have made great contributions in thisfield. Ed Thorp, a math professor for many years, is often called the father ofquantitative investment. Fischer Black, a Ph.D. in mathematics, was the mainfounder of financial engineering. Many math graduates find their talents highlyvalued in the financial industry. Financial mathematics programs areestablished in many math departments to broaden the career opportunities formath students.


There is a criticism that these mathematical models, while enriching financialinstitutions, cause great damages to the general society. But this criticismreveals that these mathematical models do provide benefit to the financialindustry, sometimes at the cost of broader society. Can the benefits from themathematical theories of financial engineering be extended to broadercommunity?

From the early days of the financial engineering, there is an expectation thatthe theory developed in this field could be a part of the general advance ineconomic theory. Fischer Black, the main founder of financial engineering, oncestated,

I like thebeauty and symmetry in Mr. Treynor's equilibrium models so much that I starteddesigning them myself. I worked on models in several areas:

Monetarytheory

Businesscycles

Optionsand warrants

For 20years, I have been struggling to show people the beauty in these models to passon knowledge I received from Mr. Treynor.

In monetarytheory --- the theory of how money is related to economic activity --- I amstill struggling. In business cycle theory --- the theory of fluctuation in theeconomy --- I am still struggling. In options and warrants, though, people seethe beauty.

Black's comments show that he, as well as many others,sensed the close relations among different economic and financial problems. Hiscomments also show that it is not always easy to extend an idea from one fieldto another field, even this might look straight forward with the benefit ofhindsight.

I tried to understand life systemsfrom the thermodynamic laws since I was an undergraduate student. I hope todevelop a mathematical theory of life systems parallel to classical mechanicsas a mathematical theory of general systems. I read about many existingtheories, such as Prigogine's theory. However, these theories do not model lifeprocesses directly. For a long time, I had little idea how to develop such atheory. I only knew that thermodynamic processes are represented by partialdifferential equations. So I stick to the theory of partialdifferential equations, hoping something will turn up some day.

It was after many years before I bumped into the Black-Scholes equation.The Black-Scholes equation was originated in financial economics. From myperspective, this equation is a mapping from lognormal processes. Lognormalprocesses can be understood as the representation of extracting low entropy tocompensate for dissipation, which is the essence of life processes. I sensedthat the Black-Scholes theory could lead me further in developing amathematical theory of living systems.

I started to think about the Black-Scholes 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 investmentdecisions. Abstract symbols become concrete.  A year later, I returned toacademia , this time as a finance professor in Singapore. After severalyears, I worked out a theory of economics that provide an analyticalrelation among major factors in economics: such as fixed cost, variable cost,investment horizon, discount rate and uncertainty. It provides a simple andconsistent understanding on broad range of problems in economic and biologicalsystems. More detailed discussion can be found from books and papers written byme and others.

When financial crises or economic downturn occur, they areoften blamed on  "unintended consequences" from economicpolicies. But from our theory, we can solve the equations to obtainquantitative results of long term consequences of those policies. It turns outthat the so called unintended consequences are simply long term consequences ofthe economic policies. Currently, economic policies are mainly measured fromtheir short term impacts. We hope that the introduction of a theory on long termimpacts of economic policies will stimulate more active discussion.

Since most prominent economists have a stake in the dominanttheory, they are reluctant to discuss rival theories.  But mathematiciansdon't have such concerns. This gives opportunities for mathematicians andother outsiders to make fundamental breakthroughs in economictheories. Neoclassical economics, the current dominant economic theory,was developed around 1870, mainly by Jevons and Walras. Both Jevons and Walraswere trained as a scientist and an engineer, not as an economist.

Mathematics has played a substantial role in deepening ourunderstanding of the world. Calculus, stochastic calculus, Maxwell equations Schrodingerequation, Black-Scholes equation are just a few examples. But the applicationsof mathematics in the vast field of social sciences are still largely cosmetic.By actively engaging in the field of social sciences, mathematicians maycrucially impact the future of science and human society.  In the process,we can work on more exciting research that are more relevant to the realworld.

References

Systematic discussion about the new economictheory 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, TheEnd of Normal: The Great Crisis and the Future of Growth, (2015), Simon & Schuster,discussed many of my ideas in great clarity.

Moreinformation, including all my papers, can be found from my website

http://web.unbc.ca/~chenj/