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.

The SDTEST^®

The SDTEST^® is a simple and fun tool to uncover our unique motivational values that use mathematical psychology of varying complexity.

The SDTEST^® helps us better understand ourselves and others on this lifelong path of self-discovery.

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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.

Pè

Charts Korelasyon

VUCA

Kritik valè de koyefisyan an korelasyon

Distribisyon nòmal, pa William Sealy Gosset (Elèv) r = 0.0335

Distribisyon ki pa nòmal, pa Spearman r = 0.0014

Montre rezilta sèlman > r = 0.0335

Distribisyon	Ki pa nòmal	Ki pa nòmal	Ki pa nòmal	Nòmal	Nòmal	Nòmal	Nòmal	Nòmal
Tout kesyon Tout kesyon Pi gran krent mwen se
Pi gran krent mwen se
Answer 1	-	Fèb pozitif 0.0521	Fèb pozitif 0.0294	Fèb negatif -0.0147	Fèb pozitif 0.0885	Fèb pozitif 0.0316	Fèb negatif -0.0110	Fèb negatif -0.1513
Answer 2	-	Fèb pozitif 0.0213	Fèb pozitif 0.0013	Fèb negatif -0.0432	Fèb pozitif 0.0618	Fèb pozitif 0.0453	Fèb pozitif 0.0103	Fèb negatif -0.0918
Answer 3	-	Fèb negatif -0.0042	Fèb negatif -0.0116	Fèb negatif -0.0406	Fèb negatif -0.0477	Fèb pozitif 0.0487	Fèb pozitif 0.0767	Fèb negatif -0.0191
Answer 4	-	Fèb pozitif 0.0421	Fèb pozitif 0.0350	Fèb negatif -0.0115	Fèb pozitif 0.0112	Fèb pozitif 0.0307	Fèb pozitif 0.0175	Fèb negatif -0.0980
Answer 5	-	Fèb pozitif 0.0288	Fèb pozitif 0.1272	Fèb pozitif 0.0146	Fèb pozitif 0.0697	Fèb pozitif 0.0037	Fèb negatif -0.0215	Fèb negatif -0.1746
Answer 6	-	Fèb negatif -0.0001	Fèb pozitif 0.0042	Fèb negatif -0.0607	Fèb negatif -0.0115	Fèb pozitif 0.0231	Fèb pozitif 0.0826	Fèb negatif -0.0309
Answer 7	-	Fèb pozitif 0.0117	Fèb pozitif 0.0372	Fèb negatif -0.0653	Fèb negatif -0.0283	Fèb pozitif 0.0495	Fèb pozitif 0.0626	Fèb negatif -0.0505
Answer 8	-	Fèb pozitif 0.0658	Fèb pozitif 0.0830	Fèb negatif -0.0310	Fèb pozitif 0.0139	Fèb pozitif 0.0334	Fèb pozitif 0.0134	Fèb negatif -0.1322
Answer 9	-	Fèb pozitif 0.0660	Fèb pozitif 0.1658	Fèb pozitif 0.0051	Fèb pozitif 0.0691	Fèb negatif -0.0093	Fèb negatif -0.0498	Fèb negatif -0.1820
Answer 10	-	Fèb pozitif 0.0758	Fèb pozitif 0.0724	Fèb negatif -0.0173	Fèb pozitif 0.0236	Fèb pozitif 0.0312	Fèb negatif -0.0115	Fèb negatif -0.1263
Answer 11	-	Fèb pozitif 0.0577	Fèb pozitif 0.0544	Fèb negatif -0.0075	Fèb pozitif 0.0082	Fèb pozitif 0.0185	Fèb pozitif 0.0293	Fèb negatif -0.1190
Answer 12	-	Fèb pozitif 0.0376	Fèb pozitif 0.1007	Fèb negatif -0.0342	Fèb pozitif 0.0296	Fèb pozitif 0.0273	Fèb pozitif 0.0341	Fèb negatif -0.1500
Answer 13	-	Fèb pozitif 0.0627	Fèb pozitif 0.1017	Fèb negatif -0.0443	Fèb pozitif 0.0248	Fèb pozitif 0.0434	Fèb pozitif 0.0189	Fèb negatif -0.1576
Answer 14	-	Fèb pozitif 0.0732	Fèb pozitif 0.1036	Fèb pozitif 0.0048	Fèb negatif -0.0105	Fèb negatif -0.0039	Fèb pozitif 0.0041	Fèb negatif -0.1157
Answer 15	-	Fèb pozitif 0.0539	Fèb pozitif 0.1381	Fèb negatif -0.0424	Fèb pozitif 0.0163	Fèb negatif -0.0147	Fèb pozitif 0.0216	Fèb negatif -0.1173
Answer 16	-	Fèb pozitif 0.0590	Fèb pozitif 0.0274	Fèb negatif -0.0375	Fèb negatif -0.0429	Fèb pozitif 0.0687	Fèb pozitif 0.0253	Fèb negatif -0.0698

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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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