Mathematical Psychology
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C HAPTER 20 MATHEMATICAL PSYCHOLOGY Trisha Van Zandt and James T. Townsend ASSOCIATION Mathematical psychology is not, per se, a distinct study. Some domains, such as vision, learning and branch of psychology. Indeed, mathematical psy- memory, and judgment and decision making, which chologists can be found in any area of psychology. frequently measure easily quantifiable performance Rather, mathematical psychology characterizes the variables like accuracy and response time, exhibit a approach that mathematical psychologists take in greater penetration of mathematical reasoning and a their substantive domains. Mathematical psycholo- higher proportion of mathematical psychologists than gists are concerned primarily with developing theo- other domains.PSYCHOLOGICAL Processes such as the behavior of indi- ries and models of behavior that permit quantitative vidual neurons, information flow through visual path- prediction of behavioral change under varying ways, evidence accumulation in decision making, and experimental conditions. There are as many mathe- language production or development have all been matical approaches within psychology as there are subjected to a great deal of mathematical modeling. substantive psychological domains. As with most However, even problems like the dynamics of mental theorists of any variety, the mathematical psycholo-AMERICANillness, problems falling in the domains of social or gist will typically start by considering the psycholog- clinical psychology, have benefited from a mathemati- ical phenomena and underlying structures THEor cal modeling approach (e.g., see the special issue on processes that she wishes to model. BY modeling in clinical science in the Journal of Mathe- A mathematical model or theory (and we do not matical Psychology [Townsend & Neufeld, 2010]). distinguish between them here) is2012 a set of mathemati- The power of the mathematical approach arises cal structures, including a set ©of linkage statements. when unrealized implications of particular model These statements relate variables, equations, and so on structures become obvious after the mathematical with components of the psychological process of inter- representation of the model has been written down. est and possibly also aspects of the stimuli or environ- By contrast, although verbal models might possess ment. RegardlessPROOFS of the domain, then, the first step in a logical structure, the inability to interpret concepts mathematical approach is to quantify the variables, in a mathematical fashion means that we cannot both independent and dependent, measured to study a derive their logical implications. The ability to make psychological process. Quantification permits variables such derivations for mathematical representations to be represented as parameters in a mathematical leads to better testability of theories, improved equation or statistical expression, the goal and defining experimental designs targeting specific model pre- feature of the mathematical psychology enterprise. dictions, and better data analyses—such analyses UNPUBLISHEDMathematical psychologists, then, construct math- frequently being rooted in the statistical properties ematical and statistical models of the processes they of the model variables. This work was supported by National Science Foundation Grant BCS-0738059, National Institutes of Mental Health Grant 57717-04Al, and Air Force Office of Special Research Grant FA9550-07-1-0078. DOI: 10.1037/13620-020 APA Handbook of Research Methods in Psychology: Vol. 2. Quantitative, Qualitative, Neuropsychological, and Biological, H. Cooper (Editor-in-Chief) 369 Copyright © 2012 by the American Psychological Association. All rights reserved. APA-COOPER_V2-11-0601-020.indd 369 12/13/11 7:17:06 PM Van Zandt and Townsend Mathematical modeling is the foundation of Germany. Eighteenth-century astronomers were many of the physical sciences. In comparison to well aware of the “personal equation” that character- these, psychology is often described as a “young” ized variations in observers’ times to estimate when science; as Laming (1973) described several decades celestial objects moved past wires on a grid. These ago, psychologists are still often focused on the estimates were made with the assistance of a metro- questions of what is happening rather than why it is nome. Thus, the estimates depended on the time the happening. Mathematical psychologists, pointing to astronomer needed to refocus attention from the the role that mathematics has played in the advance- visual to the auditory modality. Clearly, the reliabil- ment of the physical sciences, have argued that ity of astronomical measurements were therefore advancement in psychology (and other social sci- heavily dependent on the degree to which observers ences) will depend on the extent to which mathe- differed from each other or, indeed, from one obser- matical theorizing is applied to psychological issues. vation to the next. ASSOCIATION A testament to this argument is the fact that Many astronomers were thus naturally con- although not all important psychological models are cerned about precisely measuring the personal equa- mathematical, a great many of them are. tion so that equipment could be appropriately Psychology differs from a physical science in recalibrated for different observers. Astronomer more than its age, and the use of mathematical mod- and mathematician Friedrich Bessel (1784–1846), els will not, on its own, carry psychology forward. however, was further interested in why such timing First, the systems studied by psychologists are far issues arose. He formulated a hypothesis that a sec- more complex than comparable systems in the phys- ond stimulus (whether the auditory click of the met- ical sciences; and second, relationships between psy- ronome or visualPSYCHOLOGICAL movement of the star) produced a chological variables are obscured by intrinsic disturbance in the perceptual system already pro- variability in these complex systems. Thus, progress cessing the first stimulus (the visual movement of in psychology is tied to progress in statistics as well the star or the auditory click of the metronome; as technological developments that improve our Duncombe, 1945). This was perhaps the first for- ability to measure behavior. Even the best mathe- AMERICANmalization of what was later to be known as the psy- matical tools may not improve our understanding of chological refractory period (Rabbitt, 1969) or the some quirk of human behavior if we are unableTHE to doctrine of prior entry (Shore & Spence, 2005). measure that behavior or discriminate between Psychophysics. Although the interesting question changes in that behavior and random fluctuations—BY of the personal equation focused on the speed with fluctuations in either our measurements or the cog- which people can perform a task, a different histori- nitive system we are studying. 2012 © cal branch began with how frequently people make The remainder of this chapter consists of three different kinds of responses. Physiologist Ernst sections. The first outlines the history of mathemati- Weber (1795–1878) asked people to make yes–no cal psychology. The second describes its influence in judgments about whether the perceived weights of modern experimental psychology (i.e., all those PROOFS two objects were different. Holding the mass of the empirically driven and nonapplied fields of psycho- first object constant, he gradually increased the mass logical study). The third discusses some ongoing of the second object until people said “yes” (“differ- issues in the field. ent”). He was then able to define thejust noticeable difference, the smallest increase in weight ΔI that HISTORY a person could detect, and found that it was not a constant but instead a function of the weight I of the Foundations UNPUBLISHED first object, or Mathematical psychology traces its roots to before the beginning of experimental psychology, the latter Ik= I. (1) usually dated from the 1879 establishment of Wilhelm Weber found that the value of k, which deter- Wundt’s (1832–1920) laboratory in Leipzig, mined the just noticeable difference, was a constant 370 APA-COOPER_V2-11-0601-020.indd 370 12/13/11 7:17:06 PM Mathematical Psychology for most values of I, establishing what we now refer function prediction of Fechner’s law. It was not until to as Weber’s law. This law holds for a wide range of much later that Stevens (1957, 1961) tried and intensities I and across different stimulus modalities. tested other formulas for the relation of sensation to Gustav Theodor Fechner (1801–1887), founder stimulation. of the field of psychophysics and the first true math- ematical psychologist, was inspired by Weber’s work Mental chronometry. While Weber and Fechner (Fechner, 1860/1889). Although trained as a physi- were laying the foundations of psychological cist, Fechner yearned to solve one of philosophy’s measurement and psychophysics, Hermann von central and longstanding puzzles, namely, the rela- Helmholtz (1821–1894) was busy measuring the tionship of the mind to the outside world and the speed of nerve conduction in a frog’s leg (Helmholtz, physiological body itself. This giant of philosophical 1850). The realization that neural events take mea- enigmas is known as the mind–body problem, which sureable time spurred F. C. Donders (1818–1889)ASSOCIATION continues even now to attract attention from philos- to develop a system for mental chronometry, the ophers and cognitive scientists. Fechner tried to measurement