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\n1 Faculty of Social Informatics, Zen University  \n2 Data Science and AI Innovation Research Promotion Center, Shiga University  \n3 Center for Advanced Intelligence Project, RIKEN  \nMay 20, 2026  \nAbstract  \nThis paper examines the connections among various approaches to understanding concepts in philosophy, cognitive science, and machine learning, with a particular focus on their mathematical nature. By categorizing these approaches into Abstractionism, the Similarity Approach, the Functional Approach, and the Invariance Approach, the study highlights how each framework provides a distinct mathematical perspective for modeling concepts. The synthesis of these approaches bridges philosophical theories and contemporary machine learning models, providing a comprehensive framework for future research. This work emphasizes the importance of interdisciplinary dialogue, aiming to enrich our understanding of the complex relationship between human cognition and artificial intelligence.  \n1 Introduction  \nWhat are concepts? This is arguably one of the fundamental questions in philosophy, with nearly as many answers as there are philosophers. Aristotle’s Categories, Locke’s compositional theory of ideas, Kant’s transcendental logic, Hegel’s concrete universals, Cassirer’s Funktionsbegriff, and Wittgenstein’s discussion of family resemblance are just a few notable examples of philosophical theories on concepts. Since the last century, philosophical inquiries into concepts have been further accelerated and enriched through interaction with cognitive science, which added an empirical dimension to the discussion. Various models of concepts, such as prototype theory, exemplar theory, and the so-called theorytheory, have been proposed, oftentimes in opposition to or drawing inspiration from existing philosophical theories, and brought to empirical scrutiny (Margolis and Laurence, 1999; Murphy, 2004) . At the same time, the discussions  \n∗ [junotk@gmail.com](junotk@gmail.com)  \nhighlighted the multifaceted nature of what we call concepts, prompting skepticism about the general theory of concepts that would unify all the empirical findings and the theoretical roles we expect for “concepts”(Machery, 2009) .  \nOn the other hand, the nature of concepts has also been a main topic in the machine learning and artificial intelligence literature since the first generation of AI scholars set out to seek appropriate knowledge representation to build expert systems (cf. McCarthy and Hayes, 1969) . Today, the main impetus of machine learning has shifted from rule-based algorithms to statistical inference powered by deep neural networks (DNNs), which spurs an interest on how these models are extracting appropriate representations from data and using them in various tasks including object classifications and texts or image generations, to name a few (e.g. Buckner, 2019, 2023) . The swift advancement in AI research has developed hand in hand with the mathematical modeling of representations (cf. , Goodfellow et al., 2016) . The “Good-Fashioned Old AI” or GOFAI employed a lattice-like structure to represent the knowledge base of expert systems. Contemporary natural language processing (NLP) and object recognition embed words and objects in some metric space (e.g. vector space) . Generative models (such as variational auto-encoder or VAE; Kingma and Welling 2013) are thought to map data onto manifolds (intuitively, curved hypersurfaces) in the latent space. Recent studies use group theory to formulate the invariant and compositional features of concepts (Cohen and Welling, 2016; Higgins et al. , 2018; Wang et al., 2024; Weiler, 2024) . These mathematical approaches aim to explicate the formal nature of machine representations from different aspects, with a view toward unlocking insights into the remarkable capabilities of DNNs.  \nThe aim of this paper is to put t","cbCaidwWpAE3QjbA","https://ap.wps.com/l/cbCaidwWpAE3QjbA","pdf",720621,25,"English","# Introduction\n## Four Approaches to Concepts\n## Abstractionism\n## Similarity Approach\n## Functional Approach\n## Invariance Approach","[{\"question\":\"What is the main goal of the paper?\",\"answer\":\"To integrate insights on concepts from philosophy, cognitive science, and machine learning and to connect these perspectives through a shared mathematical lens.\"},{\"question\":\"Which four approaches are used to categorize theories of concepts?\",\"answer\":\"The paper categorizes them into Abstractionism, the Similarity Approach, the Functional Approach, and the Invariance Approach.\"},{\"question\":\"How does the Invariance Approach relate to machine learning models?\",\"answer\":\"It ties concepts to invariance under transformations, which is reflected in invariant or disentangled representations and formulated using group theory.\"}]","What Machine Learning Tells Us About the Mathematical Structure of Concepts - Paper | PDF",63]