领域：学术论文 · 人文科学

The preceding sections have examined various theories on concepts and representations in philosophy, cognitive science, and machine learning, classifying them into four categories: abstractionist, similarity, functional, and invariance approaches. This section compares these views from a meta-perspective, exploring their connections and deriving implications for further studies. By identifying the intersections and differences among these approaches, this discussion section aims to provide a clearer framework for interdisciplinary studies of concepts and representations.
The first axis for comparison is the role of concepts. Arguably, concepts play various roles in classification, learning, communication, problem solving, and so on; but these roles can be further understood through the lens of two major functionalities: descriptive and inferential. In the descriptive use, concepts serve to characterize and summarize data in various formats. Among the approaches discussed so far, the abstractionist and similarity approaches are particularly motivated by these descriptive tasks. The major goal of abstractionists is to classify objects in the hierarchical manner, by constructing a conceptual lattice from a data table or “context” that lists items and their observed properties (Section 2). Given a similar (but possibly continuous) dataset, the similarity approach visualizes mutual relationships between items in a two or more dimensional space based on their attributes and delineates concepts as clustered points that are close to each other (Section 3). These outcomes—lattices or plots—reveal the inherent structure of data not visible in the original format.
Concepts also play a central role in inferential tasks, e.g., for the purpose of predicting future events or possible behaviors of objects. By identifying an object in front of me as a dog, I can anticipate what it will and will not do: it may bark, chase after balls, and sniff around, but probably not climb trees or purr like a cat. Though very simplistic, it is nonetheless a bona fide act of prediction: based on some observable cues like floppy ears, I derive its behavioral characteristics that I have not yet seen, at least with respect to this particular dog in front of me. Such inferential role of concepts is the primary focus of the functional perspective (Section 4). Inferences from one set of attributes to another exploit the information about internal constraints on the possible combinations of attributes. This information, encoded in the functional form, constitutes our understanding of concepts. Alternatively, the invariance approach captures inference from a different perspective, by identifying a concept as a set of trans- formation rules that govern possible changes in the object (Section 5). Such equivariant group actions highlight the inferential role of concepts that allows us to simulate how objects or their perceptions transform in accordance with changes in the environment or the perceiving subject.
The above discussion should not imply that models are inherently connected to or designed for particular roles. Indeed, there is no contradiction in taking a conceptual lattice as representing internal constraints of concepts or using the similarity space for the purpose of prediction, as is quite common in natural language processing. The descriptive/inferential contrast rather concerns the modeling purpose: namely whether one is interested in data themselves, or in some underlying structure of “uniformity of nature” from which data are sampled. In the former context, mathematical models are taken as a sort of descriptive statistics that serve for the economy of thought; while in the latter, they are interpreted as representing a generative mechanism that lies beyond any particular data (Otsuka, 2022). These two desiderata are often in conflict: the more one tries to adjusts one’s concepts as faithful as possible to given data, the more likely they are to overfit the data, resulting in a loss of predictive ability (Forster and Sober, 1994). A good descriptive model needs not be a good inferential model, and vice versa. Therefore, when evaluating a particular model, it is essential to clearly specify the criteria by which it is being evaluated. Our second point of consideration concerns the relationship between concepts and their defining features. In the philosophical and cognitive science literature, concepts have been defined or characterized in terms of familiar and explicit features we use in everyday life, such as color, size, shape, and so on. Accordingly, conceptual hierarchies or similarity maps have been build and evaluated on the basis of a pre-specified set of mostly ostensible features. This represents a “feature-first” approach, where features act as the raw material for thoughts, with concepts being formed by combining or mixing these features. In contrast, features in representation learning are not fixed beforehand, but rather are extracted from data as axes or dimensions that constitute the latent space. In addition, the meaning of these extracted features is not given a priori, but must be determined by unpacking the internal intricate structure of the trained neural network through detailed analysis. This can be seen as a “representation-first” approach, considering representations to be epistemologically prior to features. These two approach faces different challenges. The main challenge of the feature-first approach lies in its empirical adequacy. A common criticism against abstractionism or the classic view of concepts points out that it is simply im- possible to define concepts, such as game or human, through combinations of existing features or traits (Wittgenstein, 1953; Boyd, 1991). Moreover, it is not clear which features should be considered to define concepts. Outcomes of formal concept analysis heavily depend on the list of attributes used to characterize data (Section 2). And Feigenbaum’s bottleneck highlights the difficulty of determining the relevant attributes for the problem at hand. A similar issue arises for prototype and exemplar theorists in determining the appropriate at- tributes/dimensions and their relative importance in constructing the similarity space, which significantly affects similarity judgments (Murphy, 2004).
With its successful applications to a range of empirical problems, the representation- first approach seems to be overcoming all the above issues: deep learning models are able to learn robust representations (with some reservations discussed shortly) that automatically extract relevant features from data, capture complex patterns, and generalize well to new situations. The problem, however, is that these features generally resist intuitive interpretation. One thus needs to read off meanings from trained models, but to do so requires a clear understanding of what one is looking for—that is, what are meaningful features? The idea of disentangled representation discussed in Section 5 is one attempt to explicate what we consider meaningful features of a representation in terms of independent group actions. Understanding features is crucial not just for interpretability but also to enhance model performance, ensure robustness, and improve fair- ness in machine learning applications (Lipton, 2016). It is pointed out that the well-known phenomenon of adversarial attack, where deep learning models mis- classify objects due to the addition of small, often imperceptible perturbations to the input data, is a consequence of the model’s using complex, high-dimensional features that are not necessarily aligned with human-perceptible features (Ilyas et al., 2019). Also, understanding features allows data scientists to identify and mitigate biases in machine learning models by examining how different features influence predictions (Ribeiro et al., 2016). These efforts can be understood as attempts to identify the concepts (representations) used by machines and analyze them into components (features) amenable to human understanding.
The feature-first and representation-first approaches can thus be understood as akin to digging a tunnel from opposite sides. The former starts with a set of explicit features and builds complex concepts in a bottom-up way, while the latter aims to break down given representations into understandable pieces in a top-down fashion. The important challenge in contemporary concept research is to make these approaches meet halfway.
The final, but not least, point concerns the relationship among the four approaches discussed in this paper. In his influential book, Edouard Machery (2009) argued for the disunity of concepts, challenging the traditional view that concepts are a unified phenomenon within cognitive science. His view is that what are labeled “concepts” actually involve heterogeneous mental kinds with different functionalities, purposes, and empirical bases. Be that as it may, mathematical considerations naturally suggest the logical relationships between different conceptual models. Indeed, we have already seen some of such attempts in the present paper. The group-theoretic analysis of disentangled representations can be thought as an attempt to integrate the theoretic aspect of concepts (encoded by group operations) and their similarity-based aspect (represented by a manifold). In Section 3, we have seen some recent works in natural language processing that aim to encode the hierarchical structure of concepts into the word vector space by using non-Euclidean (hyperbolic) spaces (Nickel and Kiela, 2017) or representing words by boxes instead of vectors (Vilnis et al., 2018). If successful, this line of research will reconcile the abstractionist and similarity approaches, which have been considered rivals in both philosophy and cognitive science literature.
Underlying these studies is the overarching theme of the relationship be- tween geometry and algebra. Lattices and groups are algebraic in nature, while metric spaces and manifolds have a clear geometric character. Hence the four approaches discussed in this paper can each be seen as shedding light on the geometric or algebraic aspects of concepts. Kant (1999) was the first to make a clear distinction between, and propose a unification of, these two aspects of human cognition—namely sensibility equipped with a geometric form, and understanding that follows logical and algebraic principles. Over two centuries after Kant, the contemporary machine learning research is trying to integrate the both components to understand and improve the performance of neural networks, with the aid of much more advanced mathematical machinery than those available to Kant, including non-Euclidean geometry, topology, and group or gauge theory (Sanborn et al., 2024). Just as Kant was inspired by Newtonian physics in his time, these developments in machine learning will provide new insights into the philosophical understanding of concepts.


翻译考点：
考点1：“Sensibility”推荐译为“感性”
考点2：“unpacking”推荐译为“解析”，不可译为“拆解”
考点3：“The first axis for comparison is the role of concepts”中“axis”不可直译为“轴”，应为“维度”或“方面”
考点4：“read off meanings”推荐译为“解读含义”“阐释含义”，不可译为“读出含义”