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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2) Ayyukan kamfanoni dangane da ma'aikata a watan da ya gabata (gaskiya a cikin%)

3) Tsoro

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10) Shin kana shirye ka karɓi ƙasa da biyan kuɗi don yin aiki tukuru?

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22) Malitocrate

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28) Tausasawa shine ...

29) Menene mahimmanci ga ƙwararrun ƙwararru wajen zabar tayin aiki?

30) Me yasa mutane ke tsayayya da canji (by siiyhán Mchale)

31) Taya zaka tsara motsin zuciyar ka? (by Nawal Mustafa M.A.)

32) 0 Kwarewar da ta biya ku har abada (by Irmiya Teo / 赵汉昇)

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34) Hanyoyi 12 don gyara aminci tare da wasu (ta Justin Wright)

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36) 10 makullin don motsa ƙungiyar ku

37) Algebra of Conscience (na Vladimir Lefebvre)

38) Rarraba Dama Uku Na Gaba (na Dr. Clare W. Graves)

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.

Tsoro

Charts Hulda

VUCA

M darajar da hulda coefficient

Rarraba al'ada, ta William Gubetes (Dalibi) r = 0.0331

Rarraba ba rarraba ba, da Spearman r = 0.0013

Nuna kawai sakamako > r = 0.0331

Rarrabuwa	Da ba al'ada ba	Da ba al'ada ba	Da ba al'ada ba	Na al'ada	Na al'ada	Na al'ada	Na al'ada	Na al'ada
Duk Tambayoyi Duk Tambayoyi Mafi girma tsoro shine
Mafi girma tsoro shine
Answer 1	-	Rauni kyau 0.0562	Rauni kyau 0.0311	Rauni korau -0.0164	Rauni kyau 0.0903	Rauni kyau 0.0301	Rauni korau -0.0120	Rauni korau -0.1534
Answer 2	-	Rauni kyau 0.0217	Rauni kyau 0.0011	Rauni korau -0.0455	Rauni kyau 0.0660	Rauni kyau 0.0440	Rauni kyau 0.0117	Rauni korau -0.0942
Answer 3	-	Rauni korau -0.0034	Rauni korau -0.0104	Rauni korau -0.0419	Rauni korau -0.0451	Rauni kyau 0.0462	Rauni kyau 0.0780	Rauni korau -0.0204
Answer 4	-	Rauni kyau 0.0436	Rauni kyau 0.0362	Rauni korau -0.0177	Rauni kyau 0.0150	Rauni kyau 0.0296	Rauni kyau 0.0189	Rauni korau -0.0984
Answer 5	-	Rauni kyau 0.0298	Rauni kyau 0.1270	Rauni kyau 0.0133	Rauni kyau 0.0724	Rauni korau -0.0002	Rauni korau -0.0199	Rauni korau -0.1742
Answer 6	-	Rauni korau -0.0003	Rauni kyau 0.0089	Rauni korau -0.0627	Rauni korau -0.0074	Rauni kyau 0.0190	Rauni kyau 0.0825	Rauni korau -0.0321
Answer 7	-	Rauni kyau 0.0123	Rauni kyau 0.0388	Rauni korau -0.0684	Rauni korau -0.0238	Rauni kyau 0.0468	Rauni kyau 0.0631	Rauni korau -0.0517
Answer 8	-	Rauni kyau 0.0699	Rauni kyau 0.0857	Rauni korau -0.0318	Rauni kyau 0.0150	Rauni kyau 0.0341	Rauni kyau 0.0125	Rauni korau -0.1372
Answer 9	-	Rauni kyau 0.0666	Rauni kyau 0.1681	Rauni kyau 0.0094	Rauni kyau 0.0694	Rauni korau -0.0131	Rauni korau -0.0533	Rauni korau -0.1815
Answer 10	-	Rauni kyau 0.0776	Rauni kyau 0.0744	Rauni korau -0.0185	Rauni kyau 0.0224	Rauni kyau 0.0352	Rauni korau -0.0135	Rauni korau -0.1293
Answer 11	-	Rauni kyau 0.0585	Rauni kyau 0.0531	Rauni korau -0.0094	Rauni kyau 0.0086	Rauni kyau 0.0195	Rauni kyau 0.0313	Rauni korau -0.1200
Answer 12	-	Rauni kyau 0.0378	Rauni kyau 0.1030	Rauni korau -0.0357	Rauni kyau 0.0350	Rauni kyau 0.0261	Rauni kyau 0.0297	Rauni korau -0.1510
Answer 13	-	Rauni kyau 0.0642	Rauni kyau 0.1044	Rauni korau -0.0454	Rauni kyau 0.0259	Rauni kyau 0.0424	Rauni kyau 0.0183	Rauni korau -0.1595
Answer 14	-	Rauni kyau 0.0718	Rauni kyau 0.1034	Rauni korau -0.0003	Rauni korau -0.0085	Rauni korau -0.0016	Rauni kyau 0.0074	Rauni korau -0.1172
Answer 15	-	Rauni kyau 0.0550	Rauni kyau 0.1382	Rauni korau -0.0418	Rauni kyau 0.0181	Rauni korau -0.0163	Rauni kyau 0.0211	Rauni korau -0.1183
Answer 16	-	Rauni kyau 0.0591	Rauni kyau 0.0276	Rauni korau -0.0384	Rauni korau -0.0397	Rauni kyau 0.0651	Rauni kyau 0.0280	Rauni korau -0.0710

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

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

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

2023.10.13

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