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.

Here are reports of polls which SDTEST^® makes:

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Pè

Charts Korelasyon

VUCA

Kritik valè de koyefisyan an korelasyon

Distribisyon nòmal, pa William Sealy Gosset (Elèv) r = 0.033

Distribisyon ki pa nòmal, pa Spearman r = 0.0013

Montre rezilta sèlman > r = 0.033

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 0.0532	Fèb pozitif 0.0294	Fèb negatif -0.0180	Fèb pozitif 0.0922	Fèb pozitif 0.0300	Fèb negatif -0.0113	Fèb negatif -0.1522
Answer 2	-	Fèb pozitif 0.0207	Fèb negatif -0.0011	Fèb negatif -0.0438	Fèb pozitif 0.0644	Fèb pozitif 0.0447	Fèb pozitif 0.0131	Fèb negatif -0.0929
Answer 3	-	Fèb negatif -0.0053	Fèb negatif -0.0128	Fèb negatif -0.0410	Fèb negatif -0.0454	Fèb pozitif 0.0473	Fèb pozitif 0.0794	Fèb negatif -0.0203
Answer 4	-	Fèb pozitif 0.0426	Fèb pozitif 0.0329	Fèb negatif -0.0202	Fèb pozitif 0.0158	Fèb pozitif 0.0306	Fèb pozitif 0.0217	Fèb negatif -0.0980
Answer 5	-	Fèb pozitif 0.0255	Fèb pozitif 0.1256	Fèb pozitif 0.0141	Fèb pozitif 0.0733	Fèb negatif -0.0019	Fèb negatif -0.0196	Fèb negatif -0.1747
Answer 6	-	Fèb negatif -0.0027	Fèb pozitif 0.0073	Fèb negatif -0.0627	Fèb negatif -0.0075	Fèb pozitif 0.0199	Fèb pozitif 0.0834	Fèb negatif -0.0325
Answer 7	-	Fèb pozitif 0.0111	Fèb pozitif 0.0369	Fèb negatif -0.0684	Fèb negatif -0.0230	Fèb pozitif 0.0472	Fèb pozitif 0.0649	Fèb negatif -0.0524
Answer 8	-	Fèb pozitif 0.0694	Fèb pozitif 0.0824	Fèb negatif -0.0317	Fèb pozitif 0.0137	Fèb pozitif 0.0352	Fèb pozitif 0.0146	Fèb negatif -0.1370
Answer 9	-	Fèb pozitif 0.0644	Fèb pozitif 0.1658	Fèb pozitif 0.0085	Fèb pozitif 0.0697	Fèb negatif -0.0135	Fèb negatif -0.0514	Fèb negatif -0.1827
Answer 10	-	Fèb pozitif 0.0760	Fèb pozitif 0.0728	Fèb negatif -0.0214	Fèb pozitif 0.0252	Fèb pozitif 0.0319	Fèb negatif -0.0139	Fèb negatif -0.1319
Answer 11	-	Fèb pozitif 0.0570	Fèb pozitif 0.0518	Fèb negatif -0.0106	Fèb pozitif 0.0080	Fèb pozitif 0.0205	Fèb pozitif 0.0309	Fèb negatif -0.1210
Answer 12	-	Fèb pozitif 0.0373	Fèb pozitif 0.1012	Fèb negatif -0.0356	Fèb pozitif 0.0357	Fèb pozitif 0.0243	Fèb pozitif 0.0296	Fèb negatif -0.1524
Answer 13	-	Fèb pozitif 0.0620	Fèb pozitif 0.1041	Fèb negatif -0.0449	Fèb pozitif 0.0278	Fèb pozitif 0.0412	Fèb pozitif 0.0179	Fèb negatif -0.1607
Answer 14	-	Fèb pozitif 0.0703	Fèb pozitif 0.1005	Fèb pozitif 0.0005	Fèb negatif -0.0090	Fèb negatif -0.0010	Fèb pozitif 0.0083	Fèb negatif -0.1176
Answer 15	-	Fèb pozitif 0.0554	Fèb pozitif 0.1348	Fèb negatif -0.0414	Fèb pozitif 0.0178	Fèb negatif -0.0164	Fèb pozitif 0.0218	Fèb negatif -0.1182
Answer 16	-	Fèb pozitif 0.0580	Fèb pozitif 0.0256	Fèb negatif -0.0392	Fèb negatif -0.0405	Fèb pozitif 0.0653	Fèb pozitif 0.0283	Fèb negatif -0.0714

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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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