What problems does it solve?
Mathematics has always been used as a tool for organizing and understanding the physical sciences. Today mathematics is also applied to other disciplines such as biology, medicine, management, linguistics, and the social sciences.
Initially the area of mathematics that was of primary importance outside the physical sciences was classical statistics, used in the collection and analysis of data. More recently there has been a growing interest in the exploration of other areas of mathematics for the construction of non-statistical models. These endeavors coexist and sometimes overlap, but are considered to be different aspects of their respective disciplines. The professional nomenclature reflects this difference. Economists who specialize in the application of statistics to their field are called econometricians; biologists, biometricians; psychologists, psychometricians, and so on. Economists who are primarily interested in non-statistical modeling are called mathematical economists. Similarly there are mathematical biologists, mathematical psychologists, and so on. The growing importance of these professions provides an opportunity to combine mathematical training with a serious interest in another discipline.
The role of statistics in applications of mathematics to the social sciences, biology and medicine is extremely influential, especially, as has been noted above, in a major category of interdisciplinary research. The mathematics used in non-statistical modeling varies with the kind of problem under consideration. The construction of a mathematical model entails the formulation of laws or axioms which describe in mathematical terms the (necessarily idealized) underlying structure of a system. Examples of systems range from free competitive economics to neural networks. Since we are discussing such a variety of fields and diversity of approaches within each field it is hardly possible to enumerate all the branches of mathematics used. Furthermore it must be remembered that many of these efforts are still young; the number of mathematical tools drawn upon and their level of sophistication are continually increasing. We offer below a sampling of the kinds of problems treated and the kinds of mathematics used.
Mathematical economics is the oldest, and probably the best developed of these interdisciplinary pursuits. The first Nobel Prize in Economics went to a principal founder of mathematical economics, and a more recent Nobel Prize in Economics was awarded for the mathematical study of derivative securities pricing, the Black-Scholes equation. One topic which has been the subject of research in this area is the existence of equilibrium in a competitive economy. The problem simply stated is this: given a free market in which prices respond to the law of supply and demand and a set of assumptions about the behavior of consumers and producers, will prices eventually regulate themselves to values at which supply and demand exactly balance? Other topics which occur concern individual behavior, stability of equilibria, oligopolic systems and the economics of the welfare state. Sociologists and political scientists have adopted some of the techniques of mathematical economics to study social and political issues. Linear algebra and real analysis are heavily used, as well as differential and difference equations, topology, set theory, logic, combinatorial mathematics, and game theory.
One of the earliest uses of mathematics in biology was in the study of population growth. If we assume that the growth of a population of organisms is not affected by pressure of resources, then we arrive quickly at the conclusion that the number of organisms existing after a given period of time is a constant multiple of an exponential function of the time period elapsed. However, as we take into account additional factors such as availability of resources, the model becomes more complicated and the mathematics more sophisticated. Other areas in biology and medicine which are studied by means of mathematical models are immunology, epidemiology, ion transfer across membranes, and cell differentiation. Neurophysiology is closely associated with psychology in the study of models of perception and learning. Frequently used mathematical tools are ordinary and partial differential equations, difference equations, dynamical systems, control theory, optimization theory, stochastic processes, and computer science, as well as some topology. The last decade has seen a significant growth in applications of mathematics to biological problems, to such an extent that every national mathematics meeting seems to have special sections devoted to mathematical biology.
In psychology, one finds mathematical modeling closely associated with experimentation. For example, consider the "simple learning'' model. A subject is placed in a repetitive choice situation in which different responses carry different rewards. As the reward pattern reveals itself to the subject, the subject's responses slowly change. The problem is to explain the laws governing the evaluation of the choice pattern within the framework of the experiment. More complex learning situations are studied, as well as problems in stimulus response, reaction time, preference behavior, and social interaction. Computer modeling is used to simulate the organization of the nervous system. Another kind of problem that arises in mathematical psychology occurs in the theory of measurement and scaling. The categories of mathematics which have been heavily used are probability and stochastic processes, ordinary and partial differential equations, computer science, combinatorial mathematics, set theory, and some analysis.
Mathematical linguistics has become a major force in the study of linguistics, the science of languages. It has some relationship with mathematical psychology since it is concerned with the range of humanly possible linguistic structures rather than with the particular qualities of any given language. This area makes use primarily of set theory, logic, algebra, automata theory, and computer mathematics.
What should one study in college?
There is no well-defined educational path for students wishing to enter these interdisciplinary areas. A sampling of those now engaged in each of the various fields would show considerable diversity in patterns of formal education, although it can be safely said that there is little opportunity without a doctoral degree. A strong undergraduate mathematical education with a double major would be the ideal start. Short of that ambitious program, a major in mathematics with considerable course work in the other field would be a good beginning. It does appear to be important that the mathematical training be started early, and preferably that it include some work in statistics and computer science. There is no prescription for graduate study. This depends very much on finding an individual or group working in the area one would like to pursue. Mathematical biologists and psychologists may be found in some departments of biology and psychology, respectively, but often are based in departments of mathematics or applied mathematics. A student who is interested in entering one of these interdisciplinary fields, or any of the others involving social sciences would do well to engage in some preliminary research to locate an appropriate graduate department.