University of Isfahan Journal of Researches in Linguistics 2322-3413 18 2 2026 06 22 A descriptive study of mathematical infinity and limit based on cognitive semantics A descriptive study of mathematical infinity and limit based on cognitive semantics 33 48 29544 10.22108/jrl.2025.143506.1882 FA Naser Hafezi Motlagh Ph.D. Candidate of Cognitive Linguistics, Department of Linguistics, Faculty of Letters and Humanities, Ferdowsi University of Mashhad, Mashhad, Iran Mohammadreza Pahlavannezhad Professor, Department of Linguistics, Faculty of Letters and Humanities, Ferdowsi University of Mashhad, Mashhad, Iran Journal Article 2024 11 25 . Abstract The hypothesis of embodied mathematics grounded in cognitive semantics posits that the foundation and origin of mathematical concepts stem from human embodiment. Consequently, opposing theories, such as mathematical Platonism, which assert the existence of mathematics independent of human cognition, are not supported by recent findings in cognitive sciences. This research employed a descriptive-analytical method to explore the cognitive origins of concepts, such as mathematical infinity, limits, and related ideas like transfinite numbers, derivatives, and integrals, all through the lens of embodied mathematics. In addition to detailing the concepts of numbers and sets, this study examined infinity through the foundational metaphor of infinity, a type of conceptual metaphor. Building on this framework, the analysis included infinitesimals, the concept of limits, transfinite numbers, and the principles of differential and integral calculus, focusing on the derivative and integral. Furthermore, it elucidated the role of image schemas and conceptual structures, such as metonymy, metaphor, and blending, in the formation of basic arithmetic concepts, drawing on linguistic intuition and introspective insights. This research provided a descriptive study of the origins of fundamental arithmetic concepts based on embodiment and the conceptual structures derived from the second generation of cognitive sciences. Keywords: Embodied Mathematics, Conceptual Metaphor, Conceptual Blending, Infinity, Limit. Introduction According to the second generation of cognitive sciences, embodied perception, which arises from human embodiment, alongside the formation of image schemas as the abstract foundation of our thinking and cognition, facilitates conceptual mappings and projections, such as conceptual metaphors and blending. These insights not only illuminate the reflections of human thought, but also pave new avenues in the epistemology of various other sciences. Building on this foundation, the hypothesis of embodied mathematics proposed by Lakoff and Núñez seeks to identify the origins of mathematics and its role within human cognitive faculties. This hypothesis posits that mathematics, as we understand it, is not a transcendent or external entity but rather a physical and internal construct. As such, traces of conceptual structures as discussed in cognitive sciences and cognitive linguistics can be observed within mathematics. Consequently, the influence of structures that shape the conceptualization process—such as image schemas, metaphors, and conceptual blends—plays a crucial role in embodied mathematics. Arithmetic, traditionally considered the oldest branch of mathematics, has been integral to human interaction with numbers and calculations since ancient times, predating other mathematical disciplines. Therefore, it is argued that embodiment is more pronounced in arithmetic compared to other branches of mathematics. The evolution of arithmetic, which has permeated various other mathematical fields, has further developed this embodiment, leading to the creation and innovation of new and significant concepts. The research question explored in this article is: How are mathematical concepts and their formations influenced by embodiment, conceptual blending, and metaphor? In contrast to the opposing hypothesis of Mathematical Platonism, which posited that mathematical concepts are universal realities existing independently of human cognition—implying that our relationship with them is purely one of discovery—we considered the alternative hypothesis of embodied mathematics. This perspective rejected the notion that these concepts are pre-existing, mental, or objective entities that exist independently of humans. Given this framework, how could we describe the formation and emergence of mathematical concepts through the lens of embodiment and the conceptual structures derived from it? Among the numerous fundamental mathematical concepts selected for analysis, some were more foundational and took precedence. Notably, certain elementary concepts of arithmetic played a particularly significant role. To address the research question related to the framework for describing basic concepts of arithmetic, this study examined concepts like number, infinity, infinitesimals, limits, and transfinite numbers. These concepts were analyzed through the lens of conceptual structures, including blending and metaphor. Materials & Methods Conceptual structures that emerge from embodiment, particularly at an abstract level through image schemas, play a crucial role in meaning-making. Among the most significant conceptual structures are conceptual metaphor, conceptual metonymy, and conceptual blending. A conceptual metaphor involves mapping from one domain to another. In essence, a metaphor serves as a linguistic and cognitive figure that allows one concept to refer to or illuminate another related concept. The key distinction between a conceptual metaphor and conceptual metonymy lies in their mapping processes: metonymy connects elements within the same domain, while metaphor establishes connections across different domains. Thus, conceptual metaphors are formed by mapping one conceptual domain onto another and their implications can be traced within the hypothesis of embodied mathematics. Conceptual blending refers to the combination of two distinct cognitive structures that maintain fixed correspondences. This blending process generates new entities as the selective properties of two concepts combine to form a third concept. In the theory of conceptual blending, a dynamic process facilitates the integration of various domains within mental space, leading to the emergence of new conceptual structures in real-time. These conceptual frameworks can be effectively applied to describe arithmetic concepts, illustrating how the principles of embodied mathematics manifest through these cognitive structures. This research employed a descriptive-analytical approach grounded in linguistic intuition and introspective analysis. The study focused on arithmetic concepts as the primary data for investigation. Discussion of Results & Conclusion The embodiment of arithmetic, traditionally recognized as the oldest branch of mathematics, is both evident and easily observable. For instance, the decimal counting system is derived from the number of fingers and toes. Additionally, natural numbers represent whole entities through metonymy and other sets of numbers can be derived from these foundations. Arithmetic operators, which emerge from the basic addition operator, are rooted in image schemas. Concepts like infinity, infinitesimals, and transfinite numbers arise from conceptual metaphors. The concept of a limit is informed by the "basic metaphor of infinity", while differential and integral calculus represents a generalization of this metaphor, as well as the concept of limit. This leads to the development of foundational concepts, such as "derivative" (differential) and "integral". Overall, these insights highlight the interconnectedness of embodiment and arithmetic, demonstrating how fundamental mathematical concepts are shaped by our cognitive structures. . Abstract The hypothesis of embodied mathematics grounded in cognitive semantics posits that the foundation and origin of mathematical concepts stem from human embodiment. Consequently, opposing theories, such as mathematical Platonism, which assert the existence of mathematics independent of human cognition, are not supported by recent findings in cognitive sciences. This research employed a descriptive-analytical method to explore the cognitive origins of concepts, such as mathematical infinity, limits, and related ideas like transfinite numbers, derivatives, and integrals, all through the lens of embodied mathematics. In addition to detailing the concepts of numbers and sets, this study examined infinity through the foundational metaphor of infinity, a type of conceptual metaphor. Building on this framework, the analysis included infinitesimals, the concept of limits, transfinite numbers, and the principles of differential and integral calculus, focusing on the derivative and integral. Furthermore, it elucidated the role of image schemas and conceptual structures, such as metonymy, metaphor, and blending, in the formation of basic arithmetic concepts, drawing on linguistic intuition and introspective insights. This research provided a descriptive study of the origins of fundamental arithmetic concepts based on embodiment and the conceptual structures derived from the second generation of cognitive sciences. Keywords: Embodied Mathematics, Conceptual Metaphor, Conceptual Blending, Infinity, Limit. Introduction According to the second generation of cognitive sciences, embodied perception, which arises from human embodiment, alongside the formation of image schemas as the abstract foundation of our thinking and cognition, facilitates conceptual mappings and projections, such as conceptual metaphors and blending. These insights not only illuminate the reflections of human thought, but also pave new avenues in the epistemology of various other sciences. Building on this foundation, the hypothesis of embodied mathematics proposed by Lakoff and Núñez seeks to identify the origins of mathematics and its role within human cognitive faculties. This hypothesis posits that mathematics, as we understand it, is not a transcendent or external entity but rather a physical and internal construct. As such, traces of conceptual structures as discussed in cognitive sciences and cognitive linguistics can be observed within mathematics. Consequently, the influence of structures that shape the conceptualization process—such as image schemas, metaphors, and conceptual blends—plays a crucial role in embodied mathematics. Arithmetic, traditionally considered the oldest branch of mathematics, has been integral to human interaction with numbers and calculations since ancient times, predating other mathematical disciplines. Therefore, it is argued that embodiment is more pronounced in arithmetic compared to other branches of mathematics. The evolution of arithmetic, which has permeated various other mathematical fields, has further developed this embodiment, leading to the creation and innovation of new and significant concepts. The research question explored in this article is: How are mathematical concepts and their formations influenced by embodiment, conceptual blending, and metaphor? In contrast to the opposing hypothesis of Mathematical Platonism, which posited that mathematical concepts are universal realities existing independently of human cognition—implying that our relationship with them is purely one of discovery—we considered the alternative hypothesis of embodied mathematics. This perspective rejected the notion that these concepts are pre-existing, mental, or objective entities that exist independently of humans. Given this framework, how could we describe the formation and emergence of mathematical concepts through the lens of embodiment and the conceptual structures derived from it? Among the numerous fundamental mathematical concepts selected for analysis, some were more foundational and took precedence. Notably, certain elementary concepts of arithmetic played a particularly significant role. To address the research question related to the framework for describing basic concepts of arithmetic, this study examined concepts like number, infinity, infinitesimals, limits, and transfinite numbers. These concepts were analyzed through the lens of conceptual structures, including blending and metaphor. Materials & Methods Conceptual structures that emerge from embodiment, particularly at an abstract level through image schemas, play a crucial role in meaning-making. Among the most significant conceptual structures are conceptual metaphor, conceptual metonymy, and conceptual blending. A conceptual metaphor involves mapping from one domain to another. In essence, a metaphor serves as a linguistic and cognitive figure that allows one concept to refer to or illuminate another related concept. The key distinction between a conceptual metaphor and conceptual metonymy lies in their mapping processes: metonymy connects elements within the same domain, while metaphor establishes connections across different domains. Thus, conceptual metaphors are formed by mapping one conceptual domain onto another and their implications can be traced within the hypothesis of embodied mathematics. Conceptual blending refers to the combination of two distinct cognitive structures that maintain fixed correspondences. This blending process generates new entities as the selective properties of two concepts combine to form a third concept. In the theory of conceptual blending, a dynamic process facilitates the integration of various domains within mental space, leading to the emergence of new conceptual structures in real-time. These conceptual frameworks can be effectively applied to describe arithmetic concepts, illustrating how the principles of embodied mathematics manifest through these cognitive structures. This research employed a descriptive-analytical approach grounded in linguistic intuition and introspective analysis. The study focused on arithmetic concepts as the primary data for investigation. Discussion of Results & Conclusion The embodiment of arithmetic, traditionally recognized as the oldest branch of mathematics, is both evident and easily observable. For instance, the decimal counting system is derived from the number of fingers and toes. Additionally, natural numbers represent whole entities through metonymy and other sets of numbers can be derived from these foundations. Arithmetic operators, which emerge from the basic addition operator, are rooted in image schemas. Concepts like infinity, infinitesimals, and transfinite numbers arise from conceptual metaphors. The concept of a limit is informed by the "basic metaphor of infinity", while differential and integral calculus represents a generalization of this metaphor, as well as the concept of limit. This leads to the development of foundational concepts, such as "derivative" (differential) and "integral". Overall, these insights highlight the interconnectedness of embodiment and arithmetic, demonstrating how fundamental mathematical concepts are shaped by our cognitive structures.

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