Working title: The foundations of geometric cognition: From space to abstract concepts. General fields: .... Press, 2012) and Gärdenfors's theory of conceptual spaces (idem, The Geometry of. Meaning, MIT Press, 2014). Existing monographs ...
Book proposal Author: Mateusz Hohol, Ph.D. Working title: The foundations of geometric cognition: From space to abstract concepts General fields: cognitive science, psychology, interdisciplinary studies on cognitive foundations of mathematics Keywords: cognitive science of mathematics, geometric cognition, cognitive artifacts, diagrams, professional language, scaffolding, visuospatial capacities Type: research monograph Abstract: In this book, I explain how language—in particular, the formulaic language which employs repeated structures—and diagrams provide cognitive scaffolding for abstract geometric thinking. Although geometric cognition is founded on our basic spatial abilities—such as navigation based on purely geometric cues, discrimination of two-dimensional shapes or mental rotations of figures—abstract geometric concepts go beyond these core foundations. I claim that in the context of the Euclidean system, geometric cognition requires interactions between concrete spatial representations and abstract linguistic ones. I explore this theory in both developmental and historical contexts. By using the neo-mechanistic framework and the idea of interfield theories, I aim to unify data from various scientific disciplines such as psychology, comparative ethology, cognitive neuroscience and cognitive history of mathematics into one coherent framework. 1. A Statement of Aims The proposed book attempts to develop an empirically inspired theory of geometric cognition. In order to achieve this goal, it is necessary to answer the question of how spatial proto-geometric cognition based on so-called “core knowledge systems” is transformed—by the use of language and cultural inventions such as diagrams or maps—into fully developed geometric skills manifested by the human ability to understand and develop Euclidean geometry. Empirical studies in the field of mathematical cognition have been conducted since the 1960s, and in recent decades a new research field has been established, known as the cognitive science of mathematics. The findings of this discipline include the claim that mathematical cognition is based on the so-called “number sense” (rudimentary capacity to process numerosities). According to the cognitive science of mathematics, symbolic mathematics is the outcome of the processes of the cumulative cultural evolution. It is also assumed that the link between the number sense and symbolic mathematics is constituted by embodied factors, such as finger counting. By contrast, Euclidean plane geometry—one of the key areas of mathematics—has been somewhat neglected by cognitive scientists interested in the naturalistic explanation of mathematical skills. This is manifested in the fact that the problem of geometry is essentially absent in the fundamental monographs in the field of mathematical cognition. In this book, I defend the claim that human geometric cognition is funded on phylogenetically ancient and ontogenetically early, core cognitive systems. The
mental representations generated by these systems are originally used in the domain of spatial navigation, recognition of shapes and spatial transformations (e.g., mental rotations). I emphasize the fact that full-blooded mathematical geometry goes beyond these representations. Furthermore, the Euclidean plane geometry has several special features: geometric conceptual structures are abstract; these structures exhibit high precision—geometric concepts are well-defined and clear; geometry exhibits high conceptual stability. By engaging in the discussion on the embodiment of abstract concepts, I reject both purely amodal theories of cognition and the idea of radically embodied cognition. I defend a moderate theory of embodiment of concepts inspired by the Paivian dual-coding theory. I argue that the acquisition of language provides access to a syntactically re-combinable, productive and disembodied system of symbols. This system extends our core cognition. Features of natural language— which are similar to properties of amodal symbol systems—form the “scaffolding” for abstract concepts. The technical language of geometry, which is a kind of formulaic language, enables the precise transmission of abstract geometric concepts in various discourses. I use this account in two senses: ontogenetic (“how is it possible that human beings enter into the world of Euclidean geometry?”), and historical (“where does geometry come from?”). Regarding the ontogenetic sense, I point to the fact that the acquisition of spatial language and other artifacts (e.g., maps) enables the productive combination of spatial representations, generated by core cognitive systems. In turn, this enables the acquisition of the principles of Euclidean geometry. Regarding the historical sense, I emphasize that the cultural invention of language which uses repeated structures played an important role in the origin and early evolution of the Greek geometry systematized by Euclid. I point to the fact that although external visualizations, called “diagrams,” refer to concrete objects, linguistic constructions associated with them generalize their properties. Finally, I take methodological issues into consideration. By using the neomechanistic framework associated with the idea of interfield theories, I aim to integrate data from various scientific disciplines such as comparative ethology, developmental psychology, cognitive neuroscience and cognitive history of mathematics into one coherent theory of geometric cognition. 2. A detailed synopsis Chapter 1. Geometry, the paradise of abstraction In this chapter not only do I answer the questions: “what does belong to the domain of Euclidean geometry?”, and “why geometry is so important?”, but I also point to the fact that the scientific understanding of geometric cognition should cover the abstract nature of mathematical concepts. For most people, Euclidean plane geometry is the prototypical system of mathematical geometry. It utilizes both graphical representations, such as diagrams, and linguistic entities. I distinguish and introduce “a full-blooded Euclidean geometry” and “hardwired geometry.” The latter is connected with our ordinary spatial intuitions and navigational capacities. In contrast, full-blooded Euclidean geometry has several special features: geometric conceptual structures are abstract; they are well-defined and clear; they exhibit high precision. Furthermore, geometric reasonings are stable, compelling and general. I introduce these properties recalling prototypical examples from geometry and also from the history of this branch of mathematics.
Chapter 2. The hardwired foundations of geometric cognition In this chapter I will attempt to identify and characterize the hardwired cognitive abilities that are necessary to engage in Euclidean geometry. Using the Tinbergenian strategy of explanation, I introduce empirical findings on the foundations of the mental representation of geometry. Firstly, following the path delineated by Elizabeth Spelke and her collaborators, I describe the core systems of layout geometry and object geometry. Secondly, according to biological data as well as findings of cognitive robotics, I discuss an evolutionary adaptiveness of spatial navigation based on the geometry of the environment. Thirdly, using discoveries of comparative cognitive ethology, I reconstruct the phylogeny of these mechanisms. Finally, I reconstruct an ontogeny of basic geometric abilities. I stress that mental representations generated by these systems may be productively combined— according to the acquisition of spatial language and the use of cultural inventions, e.g., simple maps—to create a new system of full-blooded Euclidean geometric representations. Chapter 3: Embodiment and geometric concepts I start this chapter by briefly discussing theories of concepts proposed in cognitive science. A classical (e.g., Fodorian) approach presupposes the existence of the amodal conceptual system. This system manipulates representations based on their purely syntactical properties. Nonetheless, advocates of various theories of embodied cognition claim that concepts are encoded in a sensorimotor apparatus and conceptual processing involves re-enacting (and manipulating) some sensorimotor states. At this point, taking into account Machery’s “heterogeneity hypothesis”, I discuss current controversies over this embodied, or neo-empiristic, approach to concepts. I claim that abstract geometric concepts are a serious challenge for the neo-empiristic theories. Following Paivio and Dove, I operationalize abstract geometrical concepts in terms of imageability. Next, I argue against both the radical theories of embodiment of abstract concepts and the purely amodal approach. Then, I turn to the moderate version of neo-empiricism inspired by the Paivian dual coding theory. I propose the claim that although geometric cognition involves both kinds of processing, i.e., visual and verbal, only linguistic representations can capture abstract geometric ideas. I defend the claim, arguing that the acquisition of a natural language provides access to the new conceptual system which extends our cognitive abilities and creates “scaffolding” for abstract geometric entities. Finally, I discuss this issue in the context of existing studies on psychological mechanisms and neural networks involved in geometric cognition Chapter 4. Cognitive artifacts and geometry: Visualization, language and culture In this chapter, I further develop idea that cognitive artifacts, such as maps and language—spatial language in particular—together with proto-geometric core systems for processing space and shapes, provide the necessary background for the construction of a new system of geometric representations. I turn to the history of mathematics, conducting case studies from Euclidean Elements, exploring interactions between diagrams and textual discourse. I describe the role of diagrams in geometric cognition. Subsequently, I turn to the language of geometry. The current scientific understanding supports the claim that the features of geometry mentioned in Chapter 1 are associated with the use of language—not ordinary
language, rather a technical one. Following Reviel Netz, I argue that the language of geometry is formulaic (it employs repeated linguistic structures), which facilitates the following: the acquisition of the geometric conceptual framework, easy reconstruction of the logical structure of geometric proofs, the transfer of the results from one geometric problem to another, and understanding the structure of discourse. In this sense, the formulaic language constitutes the vehicle for abstract geometric and general thinking. Chapter 5. Integrating geometric cognition In the final chapter, I discuss results described earlier from a more philosophical point of view. I argue that the claim that geometric cognition is based on interactions of visual, spatial and linguistic representations explains—in the neo-mechanistic sense—phenomena reported in previous chapters. I show that the fact that acquisition of spatial language precedes the acquisition of the rules of Euclidean geometry is not merely coincidental. Spatial language gives us access to the syntactically re-combinable, disembodied system of symbols, extending our embodied cognitive abilities. Then I show that the embodied version of the dualcoding theory, introduced in Chapter 3, plays the role of an “interfield theory” and therefore integrates the research—related to geometric cognition—conducted in various disciplines such as developmental psychology, comparative ethology, cognitive neuroscience and the cognitive history of mathematics. Furthermore, I sketch further perspectives of cognitive studies on geometry, which should go beyond Euclidean geometry. Table of contents: 0. Preface, outline, and acknowledgements 1. Geometry, the paradise of abstraction 1.1. Introduction and synopsis of the chapter 1.2. Euclid and his legacy 1.3. Geometry in everyday life 1.4. Euclidean geometry and beyond 1.5. Features of geometric concepts 1.6. Summary 2. The hardwired foundations of geometric cognition 2.1. Introduction and synopsis of the chapter 2.2. Hardwired sensitivity to geometry 2.3. In search for the theoretical framework 2.4. Causal factors: Core systems of geometry 2.5. Evolutionary adaptiveness of core geometry 2.6. Phylogeny of core geometry 2.7. Ontogeny: Toward a new representational system 2.8. Summary 3. Embodiment and geometric concepts 3.1. Introduction and synopsis of the chapter 3.2. Classic cognitive science of concepts 3.3. Amodal systems vs. neo-empiricism 3.4. Embodied theories of abstract concepts 3.5. Are mathematical concepts really embodied? 3.6. Disembodied cognition and geometric concepts
3.7. Summary 4. Cognitive artifacts and geometry: Visualization, language and culture 4.1. Introduction and synopsis of the chapter 4.2. Diagram as cognitive artifact 4.3. Linguistic formulae: Another cognitive artifact 4.4. Necessity and generality in Euclid’s geometry 4.5. Toward theory of geometric reasoning 4.6. Diagrams and professional language today 4.7. Summary 5. Integrating geometric cognition 5.1. Introduction and synopsis of the chapter 5.2. New mechanism and geometric cognition 5.3. Interfield theories and geometry 5.4. From mechanism sketches to complete explanations 5.5. Toward the integrated cognitive theory of Euclidean geometry 5.6. Beyond Euclid: Further perspectives from cognitive science 5.7. Summary 3. A Description of the Target Market As the issue of geometric cognition is foundational to the cognitive science of mathematics and related to other disciplines such a visuospatial cognition, philosophy of mathematics or cognitive science of abstract concepts, the book should be of interest to cognitive scientists, psychologists, historians of science, mathematicians, as well as math educators. To make the book accessible not only to professionals or graduate students with academic backgrounds in only one of these fields, I discuss technical details only when necessary. 4. Main Competing Titles No other book on the market gives a comprehensive account of the cognitive science of geometry. Cognitive scientists generally do not perceive geometry as a phenomenon to be explained, or explanandum, but they use certain geometrical or topological structures for modeling processes and cognitive representations. Geometry is used to explain phenomena, or plays a role in the explanans for example, in Churchland’s theory of conceptual maps (idem, Plato’s Camera, MIT Press, 2012) and Gärdenfors’s theory of conceptual spaces (idem, The Geometry of Meaning, MIT Press, 2014). Existing monographs such as Butterworth’s The Mathematical Brain (Macmillan, 1999), Dehaene’s The Number Sense (Oxford University Press, 2011), Continuous Issues in Numerical Cognition edited by Henik (Academic Press, 2016), Lakoff’s and Núñez’s Where Mathematics Comes From (Basic Books, 2000), Development of Mathematical Cognition edited by Berch, Geary and Koepke, (Academic Press, 2015), The Handbook of Mathematical Cognition edited by Campbell (Psychology Press, 2004), The Oxford Handbook of Numerical Cognition edited by Cohen Kadosh and Dowker (Oxford University Press 2015), The Nature and Development of Mathematics edited by Adams, Barmby and Mesoudi (Psychology Press 2017), Saxe’s Cultural Development of Mathematical Ideas (Cambridge University Press, 2014), as well as most recent An Introduction to Mathematical Cognition by Gilmore, Göbel and Inglis (Routledge, 2018) and Naturalizing Logico-Mathematical Knowledge edited by Bangu (Routledge, 2018) are limited to the issues of processing numerical structures. In other words, in these
books geometry is ignored. Furthermore, in Space, Time and Number in the Brain edited by Dehaene and Brannon (Academic Press, 2011) or Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts edited by Geary, Berch, Ochsendorf and Koepke (Academic Press, 2017) only a few chapters concern geometry. It should be emphasized that geometry is basically absent even in Lakoff’s and Núñez’s book, which aspires to be the most comprehensive account of mathematical cognition. The situation in which mathematical knowledge is reduced by psychologists and cognitive scientists to numerical cognition is not descriptively adequate vis-à-vis a large part of contemporary mathematics, in which geometric thinking plays an important role. This is also a serious flaw in the light of advanced contemporary philosophical research on the epistemology of geometry (e.g. M. Giaquinto, Visual Thinking in Mathematics, Oxford University Press, 2007) and historical studies on the emergence of the discipline (e.g. R. Netz, The Shaping of Deduction in Greek Mathematics, Cambridge University Press, 1999). 5. Format, timeline and aditional information • The preparation of the manuscript is financed by a Polish Ministry of Science and Higher Education research grant entitled Mechanisms of Geometric Cognition (2015/19/B/HS1/03310). The research grant will cover the costs of proofreading the manuscript by a professional science editor before submitting it to the publisher as a final draft. • I am going to submit the full manuscript at the beginning of 2019. • The planned length of the book is about 70,000 words, including footnotes, references and indexes. • No unusual features are planned. Any necessary figures (e.g. diagrams) and tables will be prepared by the author. • The materials have not been published before. • The proposal has not been submitted to other publishers.
