For centuries, mathematics has proven incredibly effective in describing, explaining and even predicting phenomena in the physical universe.
Newton's mathematical theory of gravity was used to predict the existence of Neptune — a never-before-seen planet. Mathematical theories in high-energy physics correctly predicted the existence and properties of never-before-detected subatomic particles (such as the W and Z). Einstein's celebrated theory of general relativity predicted the existence of the extremely compact objects we call black holes, and the list goes on and on.
Why aren't mathematical models equally successful in the prediction of major economic crises? Typically, we hear elaborate explanations for yesterday's market crash, but the predictions for what the markets will do tomorrow are often rather vague. The relative ineffectiveness of mathematics in economics is not due to lack of trying. Mathematical constructs have contributed significantly to the theory of financial markets today (admittedly, not always successfully). The role of mathematics in economics is perhaps best exemplified by the fact that seven of the 13 Nobel prizes in economics awarded between 1969 and 1981 were given for works in applied mathematics.
What, then, is the problem?
Well, as the saying goes: "It is always difficult to predict — especially the future."
One major reason for the difficulty in making predictions in economics is the fact that many variables of the world of economics — the psychology of the masses, to name one — do not naturally lend themselves to quantitative analysis. Consequently, some crucial aspects cannot be, at least at present, adequately represented in any model.
A second problem arises from the fact that the predictive value of any theory relies on the constancy of the underlying relationships among the different variables. In other words, one needs some assurance that under repeated, completely specified states of, say, consumers, employers, banks, trade unions and so on, the same probability for a given outcome is guaranteed to follow. In the absence of such guarantees, as one critic of mathematical economics has put it, "resembling a science is different from being a science."
Does this mean that we should give up on mathematical economics? In my very humble opinion, absolutely not. Recall that physics, also, was not considered mathematical in Aristotle's time. Yet physics advanced to the point where mathematics is at its very core. The fact that at the moment success in economic forecasts is limited should not impede research in mathematical economics any more than the failure to predict the precise number of spots on the skin of a person with measles should limit medical research into vaccines.
Mathematical modeling can provide valuable insights into trends, potential instabilities and the operation of various mechanisms. The criticism that mathematical theories of economics are "unrealistic" is also not valid. This is a young science in which theories, by construction, represent abstractions that attempt to single out only the most crucial variables and interrelationships, so that the crux of the problem can be identified. Furthermore, as the list of Nobel prizes in economics indicates, some mathematical discoveries were actually motivated by problems in economics.
Can we be sure that those variables that currently do not allow for quantitative modeling will ever be captured by a mathematical theory? No, we cannot. In fact, our experience with chaotic systems (systems that are extremely sensitive to the initial conditions) has demonstrated the limitations of some mathematical theories in producing predictions. However, everyone will agree that while we cannot predict the weather at the level of knowing what each atom in the atmosphere will do, weather forecasts are reasonably accurate.
Finally, recall that when on the public radio program Marketplace they want to summarize the economic activity of the day, they do it by saying: "Let's do the numbers."
Dr. Mario Livio is an astrophysicist and head of the Office of Public Outreach at the Space Telescope Science Institute. He has published over 400 scientific papers and four popular books on science and mathematics, the most recent of which is titled Is God a Mathematician?