Mathematical Psychology Research Paper

Sample Mathematical Psychology Research Paper. Browse other research paper examples and check the list of research paper topics for more inspiration. If you need a research paper written according to all the academic standards, you can always turn to our experienced writers for help. This is how your paper can get an A! Feel free to contact our research paper writing service for professional assistance. We offer high-quality assignments for reasonable rates.

‘Mathematical psychology’ constitutes the uses of mathematics in extracting information from psychological data and in constructing formal models of behavioral and cognitive phenomena. Its evolution is traced from faltering beginnings in the early nineteenth century to a flowering in the mid-1900s.

1. What Do Mathematical Psychologists Do?

During the history of psychology, the role of mathematics in research and theory construction has assumed many different forms. Even before psychology had gained recognition as a science independent of philosophy, efforts were underway to formulate quantitative laws describing unitary relations between indicators of mental processes and temporal or other physical variables. This activity, conducted in the hope of emulating the achievements of physical scientists like Kepler and Galileo has yielded laws describing, for example, the growth of sensation as a function of intensity of a light or sound, the fading of memories over time, and the rate at which people or animals work as a function of the rate at which the environment makes rewards available.

As psychological research expanded in scope and complexity during the early decades of the twentieth century, the search for simple laws largely gave way to the formulation of more elaborate models in which relations between behavioral measures and multiple causal variables are mediated by connections to hypothesized underlying processes. By mid-century, this approach had matured to the point of contributing substantially to the interpretation of many aspects of human cognition and behavior, including the encoding of stimulus representations during perception, searching of memory during recognition and recall, and making decisions in situations involving uncertainty and risk.

The focus of activity of the new specialty broadened as it gained a secure footing in the fabric of psychological sciences. An early phase in which the role of mathematical psychologists was limited to adapting familiar mathematical tools to existing problems of data analysis and theory construction gradually gave way to a new phase in which the goal was to anticipate new problems. Thus, to an increasing extent during the later 1900s, mathematical psychologists have devoted intensive efforts to the investigation of mathematical structures and systems that might offer potentialities for future application in psychological theory construction. Examples include studies of nonEuclidean geometries, algebraic structures, functional equations, graphs, and networks. Results of these investigations, originally conducted at an abstract level, have already found important applications to aspects of visual perception, memory, complex learning, psychological measurement, and decision making. (For reviews of these developments, see Dowling et al. 1998, Geissler et al. 1992, Suppes et al. 1994.)

2. The Route to Mathematical Psychology

Although precursors can be identified as far back as the early 1700s, mathematical psychology in the sense of a member of the array of psychological sciences took form with remarkable abruptness shortly after the middle of the twentieth century. In the next two sections, the course of development is traced in terms of notable events bearing on both the professional and the scientific aspects of the discipline.

2.1 Professional Milestones

A scientific discipline is defined not only by the common research focus of its continually changing membership but by the means it has developed to cope with the central problem of communication. Before 1960, investigators with common interests in the conjunction of mathematics and psychology had no publications devoted to communication and preservation of research results, no conventions to provide for face-to-face exchange of ideas, no textbook listings for mathematical psychology courses in university departments. The suddenness with which the situation changed can be seen in Table 1, which traces the emergence of an academic specialty of mathematical psychology in terms of some professional, or organizational, milestones.

 Tab. 2

A quantitatively oriented journal, Psychometrika, founded in 1936 by the Psychometric Society, has continued publication to the present, but both the journal and the society have specialized in the limited domain of theory and methods pertaining to psychological tests. On the appearance in 1963 of the threevolume Handbook of Mathematical Psychology, both psychometricians and psychologists at large must have been surprised at the volume of work extending across general experimental and applied psychology being produced by a heterogeneous but as yet little known post-World War II cohort of mathematical psychologists. Some of the more senior investigators in this growing specialty, among them R. C. Atkinson, R. R. Bush, C. Coombs, W. K. Estes, R. D. Luce, and P. Suppes, had even before publication of the Handbook begun informal discussions of ways of meeting the need for a new publication outlet, and these led to the founding of the Journal of Mathematical Psychology in 1964. Clearly the train of events was paralleled in countries other than the US, for the British Journal of Mathematical and Statistical Psychology (replacing the British Journal of Statistical Psychology) began publication the following year. By the middle of the 1970s, the discipline of mathematical psychology, now somewhat more visible, was being well served by its own societies, journals, and textbooks.

2.2 Scientific Milestones

In terms of theoretical and methodological advances, progress along the path from the earliest precursors to the dramatic expansion of mathematical psychology in the 1950s is marked by a long sequence of scientific milestones, from which a small sample is listed in Table 2 and reviewed in this section.

2.2.1 Measurement of risk.

Choice of a starting point for the sequence is somewhat arbitrary, but Miller (1964) suggests for this role a monumental achievement of the probabilist D. Bernoulli. In the early eighteenth century as in the late twentieth, coping with risk was a central concern of many people, both rich and poor. Thus there was, and continues to be, need for principled guidance for making wise choices, not only in games of chance, but in everyday economic activities like buying insurance. Bernoulli’s contribution was to show how one could, beginning with a few simple but compelling assumptions about choice situations, derive by pure mathematical reasoning the likelihood of success or failure for various alternative tactics or strategies even for situations that had seemed intractably complex.

However, addressing the highly pertinent question of why people generally fail to use the results of Bernoulli’s mathematical analyses to make only wise choices had to wait until a later era when mathematical reasoning would be coupled with experimental research.

2.2.2 Modeling processes of thought.

The next milestone in Table 2 marks the effort by one of the founders of scientific pedagogy, J. Herbart (also discussed by Miller 1964), to build a mathematical model for human thought in terms of a competitive interplay of ideas. Mathematically, Herbart’s model resembles one formulated a century and a half later by the ‘behavior theorist’ C. L. Hull for the oscillation of excitatory and inhibitory tendencies during generation of a reaction to a stimulus. However, Herbart’s preconception that mathematics and experiment are antithetical kept his theory on the sidelines of early psychological research and cost him his chance at greatness.

2.2.3 Weber’s law.

No such preconception hindered the work of physicists and physiologists who, contemporaneously with Herbart, were attacking the ancient philosophical problem of the connection between mental and physical worlds by means of experiments on relations between stimuli and sensations. Like many others, no doubt, the German physiologist E. Weber noted that the smallest change in a stimulus one can detect depends on its magnitude—a bright light or a loud sound, for example, requiring a larger change than a dim light or a weak sound. Weber went beyond this observation and, by systematic testing of observers in the laboratory with several sensory modalities, determined that the just noticeable change in a stimulus required for detection is a constant fraction of its magnitude, that is,

 ∆M/M = K (1)

where ∆M is the just noticeble change in stimulus magnitude M, and K is a constant. Equation (1) which became famous as Weber’s law, only holds strictly for a limited range of magnitudes in any sensory modality, but it applies so widely to a useful degree of approximation that it appears ubiquitously in human engineering applications.

2.2.4 Fechner’s law.

The Weber function has entered into many theoretical developments in the realm of sensation and perception. Among these is the work of the physicist G. Fechner, famous for his formulation of a quantitative relation between a sensation and the magnitude of the stimulus giving rise to it. Fechner showed that for all stimulus dimensions to which Weber’s law applies, the relation is logarithmic, in its simplest form expressible as

 S = Klog (R) (2)

where S denotes the measured magnitude of sensation produced by a stimulus of physical magnitude R, K is a constant, and log is the logarithmic function. Investigators in later years questioned whether Fechner’s formula is the best choice for scaling the relation between stimulus magnitude and sensation, but none have questioned the importance of the work of Weber and Fechner in producing the first glimmerings of the power of mathematics and experimental observation in combination for analyzing psychological phenomena.

2.2.5 Factor analysis.

The next contribution to merit the term ‘milestone,’ the founding of factor analysis by C. Spearman about 1904, requires a digression from the chain of research developments listed in Table 2. The data addressed by Spearman constituted not measures of response to stimulating situations but coefficients of correlation (a measure of degree of association) among assessments of academic achievement or talent and scores on psychological tests obtained for a group of school children. An orderly pattern characterizing a matrix of these correlations suggested that all of the abilities tapped depended on a general mental ability factor coupled with some minor factors specific to individual tests. Spearman identified the general factor, termed g, with general intelligence, an interpretation that has been vigorously debated both in the technical literature and the popular press up to the present day. Generalization of Spearman’s approach gave rise to a family of methods of ‘factor analysis’ that for some sets of test data yielded other structures, the most popular being a small set of factors with generality intermediate between Spearman’s g and his specific factors. Factor analysis continues to be a major focus of research in psychometrics, but it has remained outside the mainstream of mathematical psychology.

2.2.6 Theory of comparati e judgment.

Returning to the main line of development represented in Table 2, the direct consequence of the pioneering accomplishments of Weber and Fechner was the shaping of psychophysics, a quantitative approach to elementary phenomena of sensation and perception that continues a vigorous existence in experimental and engineering psychology. To psychologists at large, psychophysics may seem a narrow specialty, but Fechner, its principal founder, was by no means a narrow specialist. Though he pursued his research in simple experimental situations chosen for their tractability, he foresaw broad applicability for the conception of psychological measurement that emerged from it. In the vein of a proposition put forward by the great French mathematician LaPlace much earlier, Fechner noted that physical goods or commodities, including money, are significant to people only because they generate within us representations on an internalized scale of ‘psychic value.’ Weber’s and Fechner’s laws should be applicable to the relation between objectively measurable commodities and psychic values just as between physical stimuli and sensation. Developing formal machinery for operating with the internal scale of value was no trivial matter, however, and had to wait until another intellectual giant, L. L. Thurstone came on the scene nearly three-quarters of a century later.

Thurstone began with the observation that when people make judgments concerning almost any kind of stimulus quality, whether as simple as shades of gray or as complex as attractiveness of pictures or efficiency of workers’ performance, the judgments for any given stimulus vary somewhat over repeated tests but that, on average, the judgments exhibit an orderly pattern, as though based on a single attribute that might correspond to an internalized scale of measurement. Thurstone took a measure of the variability of repeated judgments (the ‘standard deviation’ in statistical parlance) as the unit of measurement on the hypothesized internal (‘subjective,’ or ‘psychological’) scale and derived characterizations of types of scales that arise under various conditions. Although the Thurstonian scales were not definable by reference to a physical device like a ruler or a thermometer, they had some of the useful properties of familiar physical scales, as, for example, that different pairs of pictures separated by the same amount on a scale of attractiveness would be judged equally similar on this attribute. The body of scaling theory flowing from Thurstone’s work has had almost innumerable applications in ability testing and assessment of performance.

2.2.7 Game theory.

The title of Von Neumann and Morgenstern’s Theory of Games and Economic Behavior (1944) scarcely hints at its epoch-making role in setting the framework within which theories of human decision making have developed. The scope of the theory includes not only social interactions that involve conflicts of interest but all situations in which individuals or groups make choices among actions or strategies whose outcomes involve uncertainty and risk. von Neumann and Morgenstern converted the intuitive notion of a behavioral strategy, prevalent under the rubic hypothesisin early learning theories, to a formal characterization of a decision space, and, with their treatment of psychological value, or utility, forged a close and enduring association between behavioral decision theory and psychological measurement.

3. Mathematical psychology in the 1950s

The pace with which important developments in mathematical psychology appeared during the preceding two centuries could not have prepared one to foresee the burst of activity in the 1950s that provided most of the material for a multivolume handbook by the end of the decade. Major research initiatives that sprang up and flourished during these years included the first mathematical theories of learning and memory, graph models of group structure, models of signal detection and risky choice, computer simulation of logical thinking and reasoning, and modeling of the human operator as a component of a control system.

What could account for this quantum leap in the evolution of mathematical psychology? A major factor appears to be a convergence of influences from new developments in disciplines outside of psychology:

(a) The advent of stored-program digital computers and, with a slight lag, the branching off from computer science of the specialty of artificial intelligence.

(b) Analyses of the foundations of physical measurement by P. W. Bridgman and N. R. Campbell and explication by G. G. Stevens (1946) of their implications for psychological measurement.

(c) The appearance of N. Wiener’s Cybernetics (1948) and treatments of the mathematics of control systems.

(d) The flowering of communication theory (Shannon and Weaver 1949) and exploration by Attneave (1959) of the uses of communication and information theory in psychology.

(e) N. Chomsky’s first formal models of grammar.

(f) The work of engineers on detection of signals occurring against noisy backgrounds in communi- cation systems, leading to the theory of ideal observers.

These events contributed directly to the expansion of mathematical psychology by supplying concepts and methods that could enter directly into the formulation of models for psychological phenomena. But perhaps at least as important, they created a milieu in which there was heightened motivation for students to acquire the knowledge of mathematics that would allow them to participate in the exciting lines of research opened up for the fledgling discipline of mathematical psychology.

Bibliography:

1. Atkinson R C, Bower G H, Crothers E J 1965 An Introduction to Mathematical Learning Theory. Wiley, New York
2. Attneave F 1959 Applications of Information Theory to Psychology: A Summary of Basic Concepts, Methods, and Results. Holt, New York
3. Coombs C H, Dawes R M, Tversky A 1976 Mathematical Psychology. Prentice-Hall, Englewood Cliffs, NJ
4. Dowling C E, Roberts F S, Theuns P 1998 Recent Progress in Mathematical Psychology. L. Erlbaum, Mahwah, NJ
5. Geissler H-G, Link S W, Townsend J T (eds.) 1992 Cognition, Information Processing, and Psychophysics: Basic Issues. L. Erlbaum, Hillsdale, NJ
6. Luce R D, Bush R R, Galanter E 1963 Handbook of Mathematical Psychology. Wiley, New York
7. Miller G A 1964 Mathematics and Psychology. Wiley, New York
8. Restle F, Greeno J G 1970 Introduction to Mathematical Psychology. Addison-Wesley, Reading, MA
9. Shannon C E, Weaver W 1949 The Mathematical Theory of Communication. University of Illinois Press, Urbana, IL
10. Spearman C 1904 General intelligence objectively determined and measured. American Journal of Psychology 15: 201–93
11. Stevens S S 1951 Mathematics, measurement and psychophysics. In: Stevens S S (ed.) Handbook of Experimental Psychology. Wiley, New York, pp. 1–49
12. Suppes P, Pavel M, Falmagne J-Cl 1994 Representations and models in psychology. Annual Re iew of Psychology 45: 517–44
13. Von Neumann J, Morgenstern O 1944 Theory of Games and Economic Beha ior. Princeton University Press, Princeton, NJ
14. Wiener N 1948 Cybernetics. Wiley, New York