# Three questions about mathematical psychology

We are investigating the intellectual roots of mathematical psychology and its relationships to adjacent areas such as psychophysics, cognitive science, computational neuroscience, and formal measurement theory. Our approach is one of integrated history and philosophy of science, meaning that we seek not just to answer questions about the past, but use this history to gain clarity on outstanding philosophical issues for the field and in psychology and science more generally. To help situate mathematical psychology we are gathering primary sources and secondary materials, such as histories and textbooks, that help us to contextualize it within the much longer arc that begins with the emergence of scientific psychology in the 19th Century (recently covered historically by Murray & Link, 2021). We have begun to trace out the lineages of some key early figures, and started to interview those who are still alive, their students, and other active researchers about their conceptions of the field, motivations for identifying with it, and aspirations for its future. This preliminary work has helped to identify three related questions that we believe are central to understanding mathematical psychology. 1. What (if anything) makes its use of mathematics reasonably effective? This question explicitly echoes physicist Eugene Wigner’s widely-read paper “The unreasonable effectiveness of mathematics in the natural sciences.” It prompts us to address how mathematical psychology arose in the context of the use of mathematics in other sciences. Our interviews have already uncovered interesting points of contrast between mathematical psychology and work in other areas, such as the physics-inspired mathematical biology of Nicolas Rashevsky, the symbolic computational cognitive science of Herbert Simon, and connectionism. We have also heard contrasting views on the significance of the foundational work on measurement theory by Krantz, Luce, Suppes, and Tversky. 2. What makes its use of mathematics different from other branches of psychology? Navarro (2021) has recently argued that “If mathematical psychology did not exist we might have to invent it” but what contrasts between mathematical psychology and other subfields of psychology make this true? Here we are interested in the relationship of mathematical psychology to contemporaneous subfields such as psychophysics and psychometrics, as well as to nearby areas of cognitive science. In interviews we have already been told of the importance placed by early mathematical psychologists on constructing models responsive to experimental data. We seek to understand the other objectives constraining the types of models favored by mathematical psychologists. 3. How should we think about mathematical psychology in relation to cognitive science? In our interviews already we have learned about differences of opinion concerning the strategy of aligning mathematical psychology with cognitive science. Tracing the academic heritage of key figures such as Estes also reveals their proximity to animal learning and behaviorism, suggesting a possible source of skepticism about cognitive science. Interviewees have also stressed the importance of having access to powerful computers in the early days of mathematical psychology. Their use of computers to process relatively large amounts of experimental data stands in contrast to the inspiration computers have provided to the “computationalist” theory of mind which serves as a central dogma of many cognitive scientists. In future work we pursue questions about the interplay between mathematical psychology and cognitive neuroscience, as well as contrasts to rational-analysis approaches such as Bayesian cognitive science. We do not expect to find consensus among mathematical psychologists on the answers to these questions. But we intend that our investigation will help provide a roadmap to the field, past, present and future. In presenting this work at what is still a preliminary stage, we thus hope to engage MathPsych attendees not just in looking backwards, but in contributing to understanding the philosophical foundations of mathematical psychology in a way that can perhaps help shape its future.

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