Mathematics in Mind – serie

This book looks at classic puzzles from the perspective of their structures and what they tell us about the brain. It uses the work on the neuroscience of mathematics from Dehaene, Butterworth, Lakoff, Núñez, and many others as a lens to understand the ways in which puzzles reflect imaginative processes blended with rational ones. The book is not about recreational or puzzle-based mathematics in and of itself but rather about what the classic puzzles tell us about the mathematical imagination and its impact on the discipline. It delves into the history of classic math puzzles, deconstructing their raison d’être and describing their psychological features, so that their nature can be fleshed out in order to help understand the mathematical mind.This volume is the first monographic treatment of the psychological nature of puzzles in mathematics. With its user-friendly technical level of discussion, it is of interest to both general readers and those who engage in the disciplines of mathematics, psychology, neuroscience, and/or anthropology. It is also ideal as a textbook source for courses in recreational mathematics, or as reference material in introductory college math courses.

This volume examines mathematics as a product of the human mind and analyzes the language of "pure mathematics" from various advanced-level sources. Through analysis of the foundational texts of mathematics, it is demonstrated that math is a complex literary creation, containing objects, actors, actions, projection, prediction, planning, explanation, evaluation, roles, image schemas, metonymy, conceptual blending, and, of course, (natural) language. The book follows the narrative of mathematics in a typical order of presentation for a standard university-level algebra course, beginning with analysis of set theory and mappings and continuing along a path of increasing complexity. At each stage, primary concepts, axioms, definitions, and proofs will be examined in an effort to unfold the tell-tale traces of the basic human cognitive patterns of story and conceptual blending. This book will be of interest to mathematicians, teachers of mathematics, cognitive scientists, cognitive linguists, and anyone interested in the engaging question of how mathematics works and why it works so well.

This is an anthology of contemporary studies from various disciplinary perspectives written by some of the world's most renowned experts in each of the areas of mathematics, neuroscience, psychology, linguistics, semiotics, education, and more. Its purpose is not to add merely to the accumulation of studies, but to show that math cognition is best approached from various disciplinary angles, with the goal of broadening the general understanding of mathematical cognition through the different theoretical threads that can be woven into an overall understanding.This volume will be of interest to mathematicians, cognitive scientists, educators of mathematics, philosophers of mathematics, semioticians, psychologists, linguists, anthropologists, and all other kinds of scholars who are interested in the nature, origin, and development of mathematical cognition.

This is an anthology of contemporary studies from various disciplinary perspectives written by some of the world's most renowned experts in each of the areas of mathematics, neuroscience, psychology, linguistics, semiotics, education, and more. Its purpose is not to add merely to the accumulation of studies, but to show that math cognition is best approached from various disciplinary angles, with the goal of broadening the general understanding of mathematical cognition through the different theoretical threads that can be woven into an overall understanding.This volume will be of interest to mathematicians, cognitive scientists, educators of mathematics, philosophers of mathematics, semioticians, psychologists, linguists, anthropologists, and all other kinds of scholars who are interested in the nature, origin, and development of mathematical cognition.

This monograph uses the concept and category of “event” in the study of mathematics as it emerges from an interaction between levels of cognition, from the bodily experiences to symbolism. It is subdivided into three parts.The first moves from a general characterization of the classical approach to mathematical cognition and mind toward laying the foundations for a view on the mathematical mind that differs from going approaches in placing primacy on events.The second articulates some common phenomena–mathematical thought, mathematical sign, mathematical form, mathematical reason and its development, and affect in mathematics–in new ways that are based on the previously developed ontology of events. The final part has more encompassing phenomena as its content, most prominently the thinking body of mathematics, the experience in and of mathematics, and the relationship between experience and mind. The volume is well-suited for anyone with a broad interest in educationaltheory and/or social development, or with a broad background in psychology.

This monograph uses the concept and category of “event” in the study of mathematics as it emerges from an interaction between levels of cognition, from the bodily experiences to symbolism. It is subdivided into three parts.The first moves from a general characterization of the classical approach to mathematical cognition and mind toward laying the foundations for a view on the mathematical mind that differs from going approaches in placing primacy on events.The second articulates some common phenomena–mathematical thought, mathematical sign, mathematical form, mathematical reason and its development, and affect in mathematics–in new ways that are based on the previously developed ontology of events. The final part has more encompassing phenomena as its content, most prominently the thinking body of mathematics, the experience in and of mathematics, and the relationship between experience and mind. The volume is well-suited for anyone with a broad interest in educationaltheory and/or social development, or with a broad background in psychology.

This book brings together ideas from experts in cognitive science, mathematics, and mathematics education to discuss these issues and to present research on how mathematics and its learning and teaching are evolving in the Information Age. Given the ever-broadening trends in Artificial Intelligence and the processing of information generally, the aim is to assess their implications for how math is evolving and how math should now be taught to a generation that has been reared in the Information Age. It will also look at the ever-spreading assumption that human intelligence may not be unique—an idea that dovetails with current philosophies of mind such as posthumanism and transhumanism. The role of technology in human evolution has become critical in the contemporary world. Therefore, a subgoal of this book is to illuminate how humans now use their sophisticated technologies to chart cognitive and social progress.Given the interdisciplinary nature of the chapters, this will be of interest to all kinds of readers, from mathematicians themselves working increasingly with computer scientists, to cognitive scientists who carry out research on mathematics cognition and teachers of mathematics in a classroom.

This book brings together ideas from experts in cognitive science, mathematics, and mathematics education to discuss these issues and to present research on how mathematics and its learning and teaching are evolving in the Information Age. Given the ever-broadening trends in Artificial Intelligence and the processing of information generally, the aim is to assess their implications for how math is evolving and how math should now be taught to a generation that has been reared in the Information Age. It will also look at the ever-spreading assumption that human intelligence may not be unique—an idea that dovetails with current philosophies of mind such as posthumanism and transhumanism. The role of technology in human evolution has become critical in the contemporary world. Therefore, a subgoal of this book is to illuminate how humans now use their sophisticated technologies to chart cognitive and social progress.Given the interdisciplinary nature of the chapters, this will be of interest to all kinds of readers, from mathematicians themselves working increasingly with computer scientists, to cognitive scientists who carry out research on mathematics cognition and teachers of mathematics in a classroom.

This book explores the universal mathematical properties underlying big language data and possible reasons why such properties exist, revealing how we may be unconsciously mathematical in our language use. These properties are statistical and thus different from linguistic universals that contribute to describing the variation of human languages, and they can only be identified over a large accumulation of usages. The book provides an overview of state-of-the art findings on these statistical universals and reconsiders the nature of language accordingly, with Zipf's law as a well-known example.The main focus of the book further lies in explaining the property of long memory, which was discovered and studied more recently by borrowing concepts from complex systems theory. The statistical universals not only possibly lie as the precursor of language system formation, but they also highlight the qualities of language that remain weak points in today's machine learning.In summary, this book provides an overview of language's global properties. It will be of interest to anyone engaged in fields related to language and computing or statistical analysis methods, with an emphasis on researchers and students in computational linguistics and natural language processing. While the book does apply mathematical concepts, all possible effort has been made to speak to a non-mathematical audience as well by communicating mathematical content intuitively, with concise examples taken from real texts.

This book explores the universal mathematical properties underlying big language data and possible reasons why such properties exist, revealing how we may be unconsciously mathematical in our language use. These properties are statistical and thus different from linguistic universals that contribute to describing the variation of human languages, and they can only be identified over a large accumulation of usages. The book provides an overview of state-of-the art findings on these statistical universals and reconsiders the nature of language accordingly, with Zipf's law as a well-known example.The main focus of the book further lies in explaining the property of long memory, which was discovered and studied more recently by borrowing concepts from complex systems theory. The statistical universals not only possibly lie as the precursor of language system formation, but they also highlight the qualities of language that remain weak points in today's machine learning.In summary, this book provides an overview of language's global properties. It will be of interest to anyone engaged in fields related to language and computing or statistical analysis methods, with an emphasis on researchers and students in computational linguistics and natural language processing. While the book does apply mathematical concepts, all possible effort has been made to speak to a non-mathematical audience as well by communicating mathematical content intuitively, with concise examples taken from real texts.

This volume covers many diverse topics related in varying degrees to mathematics in mind including the mathematical and topological structures of thought and communication. It examines mathematics in mind from the perspective of the spiral, cyclic and hyperlinked structures of the human mind in terms of its language, its thoughts and its various modes of communication in science, philosophy, literature and the arts including a chapter devoted to the spiral structure of the thought of Marshall McLuhan. In it, the authors examine the topological structures of hypertext, hyperlinking, and hypermedia made possible by the Internet and the hyperlinked structures that existed before its emergence. It also explores the cognitive origins of mathematical thinking of the human mind and its relation to the emergence of spoken language, and studies the emergence of mathematical notation and its impact on education. Topics addressed include:•The historical context of any topic that involves how mathematical thinking emerged, focusing on archaeological and philological evidence. • Connection between math cognition and symbolism, annotation and other semiotic processes.• Interrelationships between mathematical discovery and cultural processes, including technological systems that guide the thrust of cognitive and social evolution.• Whether mathematics is an innate faculty or forged in cultural-historical context• What, if any, structures are shared between mathematics and language

This volume covers many diverse topics related in varying degrees to mathematics in mind including the mathematical and topological structures of thought and communication. It examines mathematics in mind from the perspective of the spiral, cyclic and hyperlinked structures of the human mind in terms of its language, its thoughts and its various modes of communication in science, philosophy, literature and the arts including a chapter devoted to the spiral structure of the thought of Marshall McLuhan. In it, the authors examine the topological structures of hypertext, hyperlinking, and hypermedia made possible by the Internet and the hyperlinked structures that existed before its emergence. It also explores the cognitive origins of mathematical thinking of the human mind and its relation to the emergence of spoken language, and studies the emergence of mathematical notation and its impact on education. Topics addressed include:•The historical context of any topic that involves how mathematical thinking emerged, focusing on archaeological and philological evidence. • Connection between math cognition and symbolism, annotation and other semiotic processes.• Interrelationships between mathematical discovery and cultural processes, including technological systems that guide the thrust of cognitive and social evolution.• Whether mathematics is an innate faculty or forged in cultural-historical context• What, if any, structures are shared between mathematics and language

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning itself. Thus conceived, one's capacity to diagram their thought reveals a set of characteristics common to ordinary language, visual perception, and necessary mathematical reasoning. The book offers an original synthetic approach that allows tracing the roots of Peirce’s conception of a diagram in certain patterns of interrelation between his semiotics, his pragmaticist philosophy, his logical andmathematical ideas, bits and pieces of his biography, his personal intellectual predispositions, and his scientific practice as an applied mathematician.

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning itself. Thus conceived, one's capacity to diagram their thought reveals a set of characteristics common to ordinary language, visual perception, and necessary mathematical reasoning. The book offers an original synthetic approach that allows tracing the roots of Peirce’s conception of a diagram in certain patterns of interrelation between his semiotics, his pragmaticist philosophy, his logical andmathematical ideas, bits and pieces of his biography, his personal intellectual predispositions, and his scientific practice as an applied mathematician.

This book treats eighteenth-century Italian philosopher Giambattista Vico’s theory of poetic logic for the first time as the originating force in mathematics, transforming instinctive counting and spatial perception into poetic (metaphorical) symbolism that dovetails with the origin of language. It looks at current work on mathematical cognition (from Lakoff and Núñez to Butterworth, Dehaene, and beyond), matching it against the poetic logic paradigm. In a sense, it continues from where Kasner and Newman left off, connecting contemporary research on the mathematical mind to the idea that the products of early mathematics were virtually identical to the first forms of poetic language. As such, this book informs the current research on mathematical cognition from a different angle, by looking back at a still relatively unknown philosopher within mathematics.The aim of this volume is to look broadly at what constitutes the mathematical mind through the Vichian lens of poetic logic. Vico was among the first to suggest that the essential nature of mind could be unraveled indirectly by reconstructing the sources of its “modifications” (his term for “creations”); that is, by examining the creation and function of symbols, words, and all the other uniquely human artifacts—including mathematics—the mind has allowed humans to establish “the world of civil society,” Vico’s term for culture and civilization.The book is of interest to cognitive scientists working on math cognition. It presents the theory of poetic logic as Vico articulated it in his book The New Science, examining its main premises and then applying it to an interpretation of the ongoing work in math cognition. It will also be of interest to the general public, since it presents a history of early mathematics through the lens of an idea that has borne fruit in understanding the origin of language and symbols more broadly.

This book treats eighteenth-century Italian philosopher Giambattista Vico’s theory of poetic logic for the first time as the originating force in mathematics, transforming instinctive counting and spatial perception into poetic (metaphorical) symbolism that dovetails with the origin of language. It looks at current work on mathematical cognition (from Lakoff and Núñez to Butterworth, Dehaene, and beyond), matching it against the poetic logic paradigm. In a sense, it continues from where Kasner and Newman left off, connecting contemporary research on the mathematical mind to the idea that the products of early mathematics were virtually identical to the first forms of poetic language. As such, this book informs the current research on mathematical cognition from a different angle, by looking back at a still relatively unknown philosopher within mathematics.The aim of this volume is to look broadly at what constitutes the mathematical mind through the Vichian lens of poetic logic. Vico was among the first to suggest that the essential nature of mind could be unraveled indirectly by reconstructing the sources of its “modifications” (his term for “creations”); that is, by examining the creation and function of symbols, words, and all the other uniquely human artifacts—including mathematics—the mind has allowed humans to establish “the world of civil society,” Vico’s term for culture and civilization.The book is of interest to cognitive scientists working on math cognition. It presents the theory of poetic logic as Vico articulated it in his book The New Science, examining its main premises and then applying it to an interpretation of the ongoing work in math cognition. It will also be of interest to the general public, since it presents a history of early mathematics through the lens of an idea that has borne fruit in understanding the origin of language and symbols more broadly.

This book examines the correspondence between international relations (IR) theories of structural realism and constructivism and paintings, notably the artwork of Mark Rothko and Jackson Pollock, in a game theory setting. This interdisciplinary approach, through the lens of game theory and semiotics, permits different, enriched interpretations of structural realism and constructivism. These theories constitute an axis of debate between social and systemic approaches to international politics, as well as an axis of differentiation between scientific realism and positivism as philosophies of science. As such, the interpretations explored in this book contribute to what we know about international relations, how semiotics intersect with strategic uncertainty, and explains these interactions in the proposed games model.The book’s use of game theory and semiotics generate ‘visual semiotic games’ (VSGs) that shed light on the debate axes through strategic uncertainty, interactions,and players’ interactive belief systems. VSGs will contribute to literature on experimental semiotics in the sense of players’ coordination behavior, beliefs, and artistic evaluations. The equilibria, interpreted through branches of philosophy of mind and theories of explanation, will reveal possibilities of agreement among players about which artwork representing the theory at hand is the best, opening innovative research perspectives for the discipline of IR theory.

This book examines the correspondence between international relations (IR) theories of structural realism and constructivism and paintings, notably the artwork of Mark Rothko and Jackson Pollock, in a game theory setting. This interdisciplinary approach, through the lens of game theory and semiotics, permits different, enriched interpretations of structural realism and constructivism. These theories constitute an axis of debate between social and systemic approaches to international politics, as well as an axis of differentiation between scientific realism and positivism as philosophies of science. As such, the interpretations explored in this book contribute to what we know about international relations, how semiotics intersect with strategic uncertainty, and explains these interactions in the proposed games model.The book’s use of game theory and semiotics generate ‘visual semiotic games’ (VSGs) that shed light on the debate axes through strategic uncertainty, interactions,and players’ interactive belief systems. VSGs will contribute to literature on experimental semiotics in the sense of players’ coordination behavior, beliefs, and artistic evaluations. The equilibria, interpreted through branches of philosophy of mind and theories of explanation, will reveal possibilities of agreement among players about which artwork representing the theory at hand is the best, opening innovative research perspectives for the discipline of IR theory.

This book addresses interconnections between contemporary advances in mathematics, especially algebra, with applications in the social sciences and the arts. It promotes the idea that knowledge cannot remain in disciplinary silos, rather, it belongs to all people, and is dedicated to associative relationships in a variety of mathematics applications, from sociology to linguistics, including anthropology, semiotics, education, and cognitive science. Contributions illuminate some of the ways in which algebra is developed, learned, understood, communicated, and applied in the social sciences and the humanities.The content in this book has its origins in a conference, Mathematics for Social Sciences and Arts–Algebraic Modelling, virtually hosted by the Faculty of Mechanical Engineering, University of Niš, Serbia, from the 24th to the 26th of May, 2021. This conference brought together scholars from various disciplines and geographic regions and focused on thejuxtaposition of algebra and social science applications. The conference organizers, higher education institutes from four different continents, invited leading scientists in social sciences and algebra, to contribute to this volume.The first part of this book addresses algebraic and mathematical thinking, specifically learning and practicing mathematics from a cognitive science perspective, as well as illustrative applications to distinctly human concerns, like education and semiotics. The second part focuses on algebraic semigroups and some of their generalizations. This book and the conference that engendered it provide an example of a fruitful collaboration, in which the skills and deep knowledge of algebraic structures, modeling, social sciences and arts brought in by the authors from different countries and continents merge in harmonious ways. These expositions are a rich resource and are of interest both to mathematicians and non-mathematicians.

This book addresses interconnections between contemporary advances in mathematics, especially algebra, with applications in the social sciences and the arts. It promotes the idea that knowledge cannot remain in disciplinary silos, rather, it belongs to all people, and is dedicated to associative relationships in a variety of mathematics applications, from sociology to linguistics, including anthropology, semiotics, education, and cognitive science. Contributions illuminate some of the ways in which algebra is developed, learned, understood, communicated, and applied in the social sciences and the humanities.The content in this book has its origins in a conference, Mathematics for Social Sciences and Arts–Algebraic Modelling, virtually hosted by the Faculty of Mechanical Engineering, University of Niš, Serbia, from the 24th to the 26th of May, 2021. This conference brought together scholars from various disciplines and geographic regions and focused on thejuxtaposition of algebra and social science applications. The conference organizers, higher education institutes from four different continents, invited leading scientists in social sciences and algebra, to contribute to this volume.The first part of this book addresses algebraic and mathematical thinking, specifically learning and practicing mathematics from a cognitive science perspective, as well as illustrative applications to distinctly human concerns, like education and semiotics. The second part focuses on algebraic semigroups and some of their generalizations. This book and the conference that engendered it provide an example of a fruitful collaboration, in which the skills and deep knowledge of algebraic structures, modeling, social sciences and arts brought in by the authors from different countries and continents merge in harmonious ways. These expositions are a rich resource and are of interest both to mathematicians and non-mathematicians.

Mindmatics invites readers into a captivating exploration where the boundaries between mind and mathematics dissolve. Professor Neuman delves into the profound connections between cognitive processes and mathematical expression in this groundbreaking work. From how children grasp abstract concepts to symmetry's role in art and mathematics, this book uncovers the hidden structures that shape our understanding of the world. With insightful discussions on the relationship between poetry and mathematics and the essential role of the unconscious in fostering mathematical imagination, Mindmatics offers a unique perspective on the interplay of thought, creativity, and logic. This book is a must-read for anyone curious about the deeper links between the human mind and the mathematical universe.

This book focuses on the potential contributions of Artificial Intelligence (AI) for enhancing mathematics education. It includes rationales for an AI-oriented pedagogical model, such as interdisciplinarity and even sensitivity to crucial world issues, such as climate change.The chapters in this book highlight what the new age of mathematics education entails concretely, covering themes from the utilization of AI directly into classroom pedagogy and the semiotic consequences of what this entails, to how mathematics training can be tailored to get students to relate concretely to problems of climate change, and to understand the relevance of the differences between symmetry and asymmetry as psychological constructs. The overall picture we can glean from these chapters is not mere eclecticism, but an integration of disciplinary perspectives into a holistic framework that has great relevance and resonance for mathematics education in the age of AI.

This book focuses on the potential contributions of Artificial Intelligence (AI) for enhancing mathematics education. It includes rationales for an AI-oriented pedagogical model, such as interdisciplinarity and even sensitivity to crucial world issues, such as climate change.The chapters in this book highlight what the new age of mathematics education entails concretely, covering themes from the utilization of AI directly into classroom pedagogy and the semiotic consequences of what this entails, to how mathematics training can be tailored to get students to relate concretely to problems of climate change, and to understand the relevance of the differences between symmetry and asymmetry as psychological constructs. The overall picture we can glean from these chapters is not mere eclecticism, but an integration of disciplinary perspectives into a holistic framework that has great relevance and resonance for mathematics education in the age of AI.

This book uses blending theory in math cognition, and assesses the main aspect of this theory, called image schema theory. Applied work in math pedagogy has used this theory, but no work has assessed its validity. This book provides an overall assessment of the theory with regard to its validity in the study of math cognition.Overall, this book presents image schema theory as it has evolved today to mathematicians, cognitive scientists, and math educators, deriving from it any concrete implications for modeling math in computer science and, on the other side, for making math pedagogy more effective.

This book addresses the well-known capability and flexibility of classical and constructive semigroups (inherited from algebraic structures), to model, solve problems in extremely diverse situations, and develop interesting new algebraic ideas with many applications and connections to other areas of mathematics (logic, biomathematics, analysis, geometry, etc.), natural sciences, engineering and life sciences, interconnections between semigroups, cognitive sciences, social sciences, arts and humanities. The book promotes the idea that algebra came at the core of interdisciplinarity, belongs to all life disciplines, and serves in a variety of mathematics applications. It focuses on recent developments in classical and constructive semigroups, and other basic algebraic structures as well as on some of their potential applications in other fields. Further, it helps shed light on ways in which classical and constructive semigroups have been developing and applying in various domains, and extended with other sciences. The content is based on contributions of an international team of renowned scientists with expertise in different disciplines of mathematics, classical and constructive semigroups, other algebraic structures and their applications in logic, cognitive sciences, linguistics, biology, machine learning, and collective phenomena.

The book presents the history and forms of geometric and arithmetic matchstick puzzles, giving special attention to the different editions of the first book on matchstick puzzles and games, published in 1889 by Sophus Tromholt. Many of the puzzles and solutions from this book appear still today in the same or slightly different forms. One particular puzzle with an erroneous solution is especially interesting. It appeared in a few of the first editions, but later Tromholt excluded it. Nevertheless, posterior book authors continue to repeat that erroneous solution until today.The book also presents which matchstick puzzles appear in books written by authors who used matchstick puzzles before and after Tromholt and famous authors of recreational mathematics.A section of the book is dedicated to commenting on how matchstick puzzles are used in different research fields, from learning and teaching problem solving to creative thinking.The book critically analyses the past and actual usage of matchstick puzzles with multiple solutions. These puzzles, if properly used, have a great potential to increase cognitive mathematics skills of puzzle solvers of every age. These skills include:Strategic analysis and solution planningMathematical creativityCritical thinkingAvoiding self-imposed cognitive biases Space-visual intelligenceCombinatory thinkingUnfortunately, when only one solution is given to puzzle solvers, as it is common, the chance for increasing cognitive mathematics skills is lost. As such, this book takes care to highlight the controversial theoretical issue of what are different solutions of a matchstick puzzle.It considers the value of solutions obtained by rotation or mirror reflection for increasing the visual intelligence of puzzle solvers.This book is of interest to researchers in cognitive psychology and educational mathematics who are interested in using matchstick puzzles to explore the relationships between different cognitive skills and levels of performance in solving matchstick puzzles. It is also of interest to mathematics teachers at various educational levels interested in using matchstick puzzles to foster development of cognitive mathematics skills and puzzle posing abilities in their students.

This text, structured in two parts, presents a distinctive pedagogical approach to the foundations of mathematics by treating the subject as a formal language to be learned. Its central thesis is that mastery of modern mathematics, logic, and computer science requires fluency in a precise and structured mode of thinking—an “architecture of thought” grounded in a rigorous and expressive language. The book develops a method for learning mathematical language in a way aligned with contemporary foreign language pedagogy, enabling readers to interpret mathematical texts with clarity and precision while gaining insight into their underlying patterns, structures, capabilities, and limitations. In doing so, it bridges the gap between intuitive understanding and formal rigor, complementing standard courses in logic, set theory, algebra, and philosophy.The text includes approximately 190 exercises of varying difficulty, each accompanied by detailed solutions to support independent study and self-assessment. It also explicitly connects the formal language of mathematics with the operational logic of artificial intelligence. Undergraduate students in mathematics, computer science, philosophy, and linguistics—as well as graduate students seeking a clear and structured review of foundational concepts—will find the book a valuable contribution to their understanding of mathematical formalism. Those interested in the logical principles underlying computer science and AI will likewise find it an engaging resource for deepening their grasp of formal structures.Part I, The Ascent, serves as a conceptual and intuitive guide, using the metaphor of language acquisition levels (A1 to C1) to develop understanding from the basic syntax of mathematical expressions (the “Matryoshka principle”) to advanced ideas in formal logic and set theory, enriched by analogies, historical context, and philosophical motivation. Part II, Proof-Theoretical Basis, provides the rigorous definitions and proofs underpinning these concepts, completing the transition from intuition to formal precision.

This book uncovers mathematical structures underlying natural intelligence and applies category theory as a modeling language for understanding human cognition, giving readers new insights into the nature of human thought. In this context, the book explores various topics and questions, such as the human representation of the number system, why our counting ability is different from that which is evident among non-human organisms, and why the idea of zero is so difficult to grasp.The book is organized into three parts: the first introduces the general reason for studying general structures underlying the human mind; the second part introduces category theory as a modeling language and use it for exposing the deep and fascinating structures underlying human cognition; and the third applies the general principles and ideas of the first two parts to reaching a better understanding of challenging aspects of the human mind such as our understanding of thenumber system, the metaphorical nature of our thinking and the logic of our unconscious dynamics.

This volume examines mathematics as a product of the human mind and analyzes the language of "pure mathematics" from various advanced-level sources. Through analysis of the foundational texts of mathematics, it is demonstrated that math is a complex literary creation, containing objects, actors, actions, projection, prediction, planning, explanation, evaluation, roles, image schemas, metonymy, conceptual blending, and, of course, (natural) language. The book follows the narrative of mathematics in a typical order of presentation for a standard university-level algebra course, beginning with analysis of set theory and mappings and continuing along a path of increasing complexity. At each stage, primary concepts, axioms, definitions, and proofs will be examined in an effort to unfold the tell-tale traces of the basic human cognitive patterns of story and conceptual blending. This book will be of interest to mathematicians, teachers of mathematics, cognitive scientists, cognitive linguists, and anyone interested in the engaging question of how mathematics works and why it works so well.