Draft:Mathematical Principles of Human Conceptual Behavior

Mathematical Principles of Human Conceptual Behavior: The Structural Nature of Conceptual Representation and Processing is a one-volume work, written by cognitive scientist and applied mathematician Ronaldo Vigo, introducing an original mathematical theory and set of mathematical principles for explaining and predicting conceptual behavior and cognition in general. Although the principles (and the theory and models derived from them) are based on the constructs of generalized invariance and symmetry, other key constructs such as complexity, similarity, pattern detection, dissimilarity, representation, information, and their unification, also play key roles in the work. The main novel theory introduced in the book is Generalized Invariance Structure Theory (GIST; Chapter 5). Its core invariance principles and models also serve as a basis for a new and general measure of information called Generalized Representational Information Theory (GRIT; Chapter 9). The general organization of the book is to provide a theoretical foundation in the first half (Chapters 1-6), followed by a report of experimental evidence across different areas involving human cognition in the second half (Chapters 7-12).

The book content has been recognized by experts in the field of Cognitive Science and has been incorporated into other academic books. Additionally, the 4A Framework (Accessibility, Active Engagement, Advocacy for Inclusion, and Accountability) for evaluating and helping teachers select online instructional materials was built using the principles of GIST. Finally, GRIT has been used by the European Commission’s Joint Research Center to help understand and define data and information in the context of digital economics and copyright law.

Qualitative/Conceptual Accomplishments

 * GIST is the first mathematical theory of concept learning and cognition in general based on the idea that observers are invariance pattern detectors and that they possess a prewired universal engine for this purpose that connects observers (their brains and minds) to the invariance and symmetries in environmental stimuli and between exemplars in the mind. This engine is responsible for concept formation and its output is the precursor to other concept representations (e.g., prototypes, rules, judgments). It also controls, informs, and integrates all other cognitive capacities (Chapter 5).
 * The importance of invariance and symmetry in the physical sciences is well-known. GIST is the first original systematic mathematical theory of human concept learning, categorization behavior, and cognition in general that is based on invariance and symmetry principles (Chapters 4, 5, and 6).
 * GIST is the first theory that defines the mathematical relationship between invariance and symmetry principles in nature and invariance detection principles in the observer. As such, it is the first theory that defines a cognitive mechanism for invariance detection in terms of attention shifting and partial similarity assessment (Chapters 5 and 6).
 * GIST provides a unification of Richard Herrnstein’s stimulus control theory based on such principles.
 * The development of a new universal notion of information referred to as representational information as a context-sensitive replacement for Shannon information (Chapter 9), and new characterizations of other constructs of universal science such as complexity, pattern, duality, similarity, dissimilarity, and representation.

Mathematical/Technical Accomplishments

 * An original mathematical measure of the overall degree of invariance of a category and its degree of learning difficulty (i.e., subjective structural complexity; Chapters 4 and 5).
 * The first direct mathematical geometrization of the degree of invariance of a category (Chapters 4 and 5).
 * The development of several original mathematical constructs to model cognitive phenomena: these include structural manifolds, logical manifolds, tau-(invariance) threshold, structural equilibrium, the precise connection between categorical invariance and sentential logic, and use of partial Boolean derivatives for characterizing categorical invariance (Chapters 4 and 5).
 * A generalization of categorical invariance to continuous and n-ary dimensional categories concerning human conceptualization, thereby offering a low-level and general mathematical connection between the observer and the physical world (Chapter 5).
 * The first theory that offers a systematic derivation of several original candidate cognitive laws from a set of invariance principles. (Chapters 5, 6, 7, 8, 9, and 11).
 * Accurate fits without free parameters or with one free parameter to data from key experiments in concept learning behavior (Chapter 7), perception (Chapter 8), and choice behavior (Chapter 11). The theory has been proven to be a precise theory of classification behavior (i.e., its models without free parameters or with a single parameter have, in many cases, accounted for over 90% of the variance in data from many experiments; Chapter 7).

Pre-publication and Post-publication Empirical Evidence
Although some of the models in the book are tested using historical empirical results and results from the recent relevant literature, further empirical evidence has emerged since the book’s publication that further corroborates its formal models. The following is a partial list of such studies, all published in peer-reviewed journals:

Pre-publication Empirical Support

 * 1) The law of invariance (LOI; aka the GISTM) accounts for the classic learning difficulty ordering with Boolean categories consisting of four objects and defined over separable-dimension stimuli.
 * 2) For 84 structure types (including the 76 tested in an earlier study), the non-parametric LOI (without free parameters; aka the GISTM-NP) accounted for approximately 88% of the variance in classification error rates, the LOI accounted for approximately 90% of the variance, and the LOI with structural equilibrium (aka the GISTM-SE) accounted for approximately 91% of the variance.
 * 3) The Information Model of Fixation (IMF), based in GRIT, accounted for up to 72% of the variance in fixation time among objects during category learning tasks.
 * 4) The Inverse of the non-parametric LOI, derived from the Conceptual-Choice Principle, accounted for nearly 90% of the variance in choice response times across structure types in two decision-making experiments. Participants selected which object they preferred from a set of related dimensionally-defined objects (category).
 * 5) In a study comparing classification performance between adults, adolescents without ADHD, and adolescents with ADHD, the LOI accounted for approximately 96% of the variance in adults, approximately 97% of the variance in adolescents without ADHD, and approximately 91% of the variance in adolescents with ADHD. The use of the single scaling parameter in the LOI allowed group differences in pattern discrimination capacity to be captured.
 * 6) GRIT accounts for one-dimensional unsupervised sorting behavior observed with free-sorting classification tasks involving two categories and separable-dimension stimuli.
 * 7) GRIT accounts for one-dimensional unsupervised sorting behavior observed with the match-to-standards (“building array” condition) and the match-from-array (“building array” condition) classification tasks using separable-dimension stimuli.
 * 8) GRIT accounts for family-resemblance unsupervised sorting behavior observed with the match-to-standards (“single-standard available” condition) and the match-from-array (“single-standard available” condition) classification tasks using separable-dimension stimuli.

Post-publication Empirical Support and Applications

 * 1) Researchers have shown correlations among brain regions associated with category induction and invariance detection as operationalized via GIST.
 * 2) The non-parametric LOI accurately predicts the learning difficulty ordering for both visual and auditory modal concepts.
 * 3) In a study that examined error rates and response times for classification tasks across multiple category learning sessions, the LOI with structural equilibrium accounted for approximately 91% of the variance in error rates and approximately 85% of the variance in response times when participants were exposed to structure types that were randomly sampled from all possible types in the study. When participants were repeatedly exposed to different category instances of the same structure type, the this model accounted for approximately 79% of the variance in error rates and 83% of the variance in response times.
 * 4) Across 41 structure types tested, GRIT accounted for approximately 58% to 60% of the variance in human informativeness judgments.  Human informativeness judgments were operationalized in terms of the extent to which each member of a category represents the category as a whole.
 * 5) The Dual Discrimination Invariance Model (DDIM), derived from the Law of Invariance at the heart of GIST, accounts for the learning difficulty orderings associated with Boolean categorical stimuli composed of objects with 3 integral dimensions. The learning difficulty orderings accounted for are associated with tasks that vary the presentation method of category members (sequential vs. simultaneous display) and whether corrective feedback is provided to aid learning.
 * 6) The GRIT-NPE, a non-parametric model derived from GRIT, accurately accounts for unsupervised one-dimensional, two-dimensional, and three-dimensional sorting behavior using separable-dimension stimuli. For both construction and deconstruction tasks, it accounts for the predominant behavior of one-dimensional rules, exclusive-or rules, and complex three-dimensional relations.
 * 7) The GRIT-NPE, a non-parametric model derived from GRIT, accurately accounts for unsupervised one-dimensional, two-dimensional, and three-dimensional sorting behavior using integral-dimension stimuli. For both construction and deconstruction tasks, it accounts for the predominant behavior of one-dimensional rules and family-resemblance relations.
 * 8) The Invariance-based Choice Response Time law (ICRT-NP), derived from the Conceptual-Choice Principle, accounted for approximately 87% of the variance in choice response times across 56 logically distinct three- and four-dimensional category structures.