Mathematical Psychology

This project investigates mathematical psychology's historical and philosophical foundations to clarify its distinguishing characteristics and relationships to adjacent fields. Through gathering primary sources, histories, and interviews with researchers, author Prof. Colin Allen - University of Pittsburgh [1, 2, 3] and his students Osman Attah, Brendan Fleig-Goldstein, Mara McGuire, and Dzintra Ullis have identified three central questions:

What makes the use of mathematics in mathematical psychology reasonably effective, in contrast to other sciences like physics-inspired mathematical biology or symbolic cognitive science?
How does the mathematical approach in mathematical psychology differ from other branches of psychology, like psychophysics and psychometrics?
What is the appropriate relationship of mathematical psychology to cognitive science, given diverging perspectives on aligning with this field?

Preliminary findings emphasize data-driven modeling, skepticism of cognitive science alignments, and early reliance on computation. They will further probe the interplay with cognitive neuroscience and contrast rational-analysis approaches. By elucidating the motivating perspectives and objectives of different eras in mathematical psychology's development, they aim to understand its past and inform constructive dialogue on its philosophical foundations and future directions. This project intends to provide a conceptual roadmap for the field through integrated history and philosophy of science.

The Project: Integrating History and Philosophy of Mathematical Psychology

This project aims to integrate historical and philosophical perspectives to elucidate the foundations of mathematical psychology. As Norwood Hanson stated, history without philosophy is blind, while philosophy without history is empty. The goal is to find a middle ground between the contextual focus of history and the conceptual focus of philosophy.

The team acknowledges that all historical accounts are imperfect, but some can provide valuable insights. The history of mathematical psychology is difficult to tell without centering on the influential Stanford group. Tracing academic lineages and key events includes part of the picture, but more context is needed to fully understand the field's development.

The project draws on diverse sources, including research interviews, retrospective articles, formal histories, and online materials. More interviews and research will further flesh out the historical and philosophical foundations. While incomplete, the current analysis aims to identify important themes, contrasts, and questions that shaped mathematical psychology's evolution. Ultimately, the goal is an integrated historical and conceptual roadmap to inform contemporary perspectives on the field's identity and future directions.

The Rise of Mathematical Psychology

The history of efforts to mathematize psychology traces back to the quantitative imperative stemming from the Galilean scientific revolution. This imprinted the notion that proper science requires mathematics, leading to "physics envy" in other disciplines like psychology.

Many early psychologists argued psychology needed to become mathematical to be scientific. However, mathematizing psychology faced complications absent in the physical sciences. Objects in psychology were not readily present as quantifiable, provoking heated debates on whether psychometric and psychophysical measurements were meaningful.

Nonetheless, the desire to develop mathematical psychology persisted. Different approaches grappled with determining the appropriate role of mathematics in relation to psychological experiments and data. For example, Herbart favored starting with mathematics to ensure accuracy, while Fechner insisted experiments must come first to ground mathematics.

Tensions remain between data-driven versus theory-driven mathematization of psychology. Contemporary perspectives range from psychometric and psychophysical stances that foreground data to measurement-theoretical and computational approaches that emphasize formal models.

Elucidating how psychologists negotiated to apply mathematical methods to an apparently resistant subject matter helps reveal the evolving role and place of mathematics in psychology. This historical interplay shaped the emergence of mathematical psychology as a field.

The Distinctive Mathematical Approach of Mathematical Psychology

What sets mathematical psychology apart from other branches of psychology in its use of mathematics?

Several key aspects stand out:

Advocating quantitative methods broadly. Mathematical psychology emerged partly to push psychology to embrace quantitative modeling and mathematics beyond basic statistics.
Drawing from diverse mathematical tools. With greater training in mathematics, mathematical psychologists utilize more advanced and varied mathematical techniques like topology and differential geometry.
Linking models and experiments. Mathematical psychologists emphasize tightly connecting experimental design and statistical analysis, with experiments created to test specific models.
Favoring theoretical models. Mathematical psychology incorporates "pure" mathematical results and prefers analytic, hand-fitted models over data-driven computer models.
Seeking general, cumulative theory. Unlike just describing data, mathematical psychology aspires to abstract, general theory supported across experiments, cumulative progress in models, and mathematical insight into psychological mechanisms.

So while not unique to mathematical psychology, these key elements help characterize how its use of mathematics diverges from adjacent fields like psychophysics and psychometrics. Mathematical psychology carved out an identity embracing quantitative methods but also theoretical depth and broad generalization.

Situating Mathematical Psychology Relative to Cognitive Science

What is the appropriate perspective on mathematical psychology's relationship to cognitive psychology and cognitive science? While connected historically and conceptually, essential distinctions exist.

Mathematical psychology draws from diverse disciplines that are also influential in cognitive science, like computer science, psychology, linguistics, and neuroscience. However, mathematical psychology appears more skeptical of alignments with cognitive science.

For example, cognitive science prominently adopted the computer as a model of the human mind, while mathematical psychology focused more narrowly on computers as modeling tools.

Additionally, mathematical psychology seems to take a more critical stance towards purely simulation-based modeling in cognitive science, instead emphasizing iterative modeling tightly linked to experimentation.

Overall, mathematical psychology exhibits significant overlap with cognitive science but strongly asserts its distinct mathematical orientation and modeling perspectives. Elucidating this complex relationship remains an ongoing project, but preliminary analysis suggests mathematical psychology intentionally diverged from cognitive science in its formative development.

This establishes mathematical psychology's separate identity while retaining connections to adjacent disciplines at the intersection of mathematics, psychology, and computation.

Looking Ahead: Open Questions and Future Research

This historical and conceptual analysis of mathematical psychology's foundations has illuminated key themes, contrasts, and questions that shaped the field's development. Further research can build on these preliminary findings.

Additional work is needed to flesh out the fuller intellectual, social, and political context driving the evolution of mathematical psychology. Examining the influences and reactions of key figures will provide a richer picture.

Ongoing investigation can probe whether the identified tensions and contrasts represent historical artifacts or still animate contemporary debates. Do mathematical psychologists today grapple with similar questions on the role of mathematics and modeling?

Further analysis should also elucidate the nature of the purported bidirectional relationship between modeling and experimentation in mathematical psychology. As well, clarifying the diversity of perspectives on goals like generality, abstraction, and cumulative theory-building would be valuable.

Finally, this research aims to spur discussion on philosophical issues such as realism, pluralism, and progress in mathematical psychology models. Is the accuracy and truth value of models an important consideration or mainly beside the point? And where is the field headed - towards greater verisimilitude or an indefinite balancing of complexity and abstraction?

By spurring reflection on this conceptual foundation, this historical and integrative analysis hopes to provide a roadmap to inform constructive dialogue on mathematical psychology's identity and future trajectory.

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Below you can read an abridged version of the results of our VUCA poll “Fears“. The full version of the results is available for free in the FAQ section after login or registration.

Beldurrak

Charts Korrelazioa

VUCA

Korrelazio koefizientea balioa kritikoa

Banaketa normala, William Seally Gosset-en (Ikaslea) r = 0.0331

Banaketa normala ez da, Spearman-ek eginda r = 0.0013

Erakutsi emaitzak soilik > r = 0.0331

Banaketa	Normala ez	Normala ez	Normala ez	Ohiko	Ohiko	Ohiko	Ohiko	Ohiko
Galdera guztiak Galdera guztiak Nire beldurrik handiena da
Nire beldurrik handiena da
Answer 1	-	Positibo ahula 0.0563	Positibo ahula 0.0317	Negatiboa ahula -0.0161	Positibo ahula 0.0907	Positibo ahula 0.0298	Negatiboa ahula -0.0126	Negatiboa ahula -0.1537
Answer 2	-	Positibo ahula 0.0216	Positibo ahula 0.0002	Negatiboa ahula -0.0458	Positibo ahula 0.0654	Positibo ahula 0.0445	Positibo ahula 0.0124	Negatiboa ahula -0.0937
Answer 3	-	Negatiboa ahula -0.0035	Negatiboa ahula -0.0111	Negatiboa ahula -0.0421	Negatiboa ahula -0.0456	Positibo ahula 0.0466	Positibo ahula 0.0786	Negatiboa ahula -0.0201
Answer 4	-	Positibo ahula 0.0435	Positibo ahula 0.0353	Negatiboa ahula -0.0181	Positibo ahula 0.0145	Positibo ahula 0.0301	Positibo ahula 0.0197	Negatiboa ahula -0.0979
Answer 5	-	Positibo ahula 0.0299	Positibo ahula 0.1279	Positibo ahula 0.0136	Positibo ahula 0.0730	Negatiboa ahula -0.0007	Negatiboa ahula -0.0207	Negatiboa ahula -0.1746
Answer 6	-	Negatiboa ahula -0.0004	Positibo ahula 0.0082	Negatiboa ahula -0.0629	Negatiboa ahula -0.0078	Positibo ahula 0.0193	Positibo ahula 0.0830	Negatiboa ahula -0.0318
Answer 7	-	Positibo ahula 0.0122	Positibo ahula 0.0381	Negatiboa ahula -0.0686	Negatiboa ahula -0.0242	Positibo ahula 0.0471	Positibo ahula 0.0636	Negatiboa ahula -0.0513
Answer 8	-	Positibo ahula 0.0698	Positibo ahula 0.0849	Negatiboa ahula -0.0321	Positibo ahula 0.0146	Positibo ahula 0.0345	Positibo ahula 0.0130	Negatiboa ahula -0.1368
Answer 9	-	Positibo ahula 0.0665	Positibo ahula 0.1674	Positibo ahula 0.0092	Positibo ahula 0.0691	Negatiboa ahula -0.0128	Negatiboa ahula -0.0528	Negatiboa ahula -0.1812
Answer 10	-	Positibo ahula 0.0778	Positibo ahula 0.0755	Negatiboa ahula -0.0180	Positibo ahula 0.0231	Positibo ahula 0.0346	Negatiboa ahula -0.0146	Negatiboa ahula -0.1298
Answer 11	-	Positibo ahula 0.0584	Positibo ahula 0.0524	Negatiboa ahula -0.0096	Positibo ahula 0.0081	Positibo ahula 0.0199	Positibo ahula 0.0318	Negatiboa ahula -0.1197
Answer 12	-	Positibo ahula 0.0380	Positibo ahula 0.1042	Negatiboa ahula -0.0352	Positibo ahula 0.0357	Positibo ahula 0.0254	Positibo ahula 0.0286	Negatiboa ahula -0.1515
Answer 13	-	Positibo ahula 0.0644	Positibo ahula 0.1057	Negatiboa ahula -0.0448	Positibo ahula 0.0268	Positibo ahula 0.0416	Positibo ahula 0.0169	Negatiboa ahula -0.1600
Answer 14	-	Positibo ahula 0.0717	Positibo ahula 0.1026	Negatiboa ahula -0.0006	Negatiboa ahula -0.0089	Negatiboa ahula -0.0012	Positibo ahula 0.0080	Negatiboa ahula -0.1168
Answer 15	-	Positibo ahula 0.0549	Positibo ahula 0.1375	Negatiboa ahula -0.0420	Positibo ahula 0.0178	Negatiboa ahula -0.0160	Positibo ahula 0.0216	Negatiboa ahula -0.1180
Answer 16	-	Positibo ahula 0.0591	Positibo ahula 0.0273	Negatiboa ahula -0.0386	Negatiboa ahula -0.0399	Positibo ahula 0.0653	Positibo ahula 0.0282	Negatiboa ahula -0.0708

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[1] https://twitter.com/wileyprof

[2] https://colinallen.dnsalias.org

[3] https://philpeople.org/profiles/colin-allen

2023.10.13

Valerii Kosenko

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