From symbols to geometry: an indexical epistemology of generative AI

Generative AI challenges traditional philosophical accounts of knowledge by introducing a form of representation that is neither symbolic nor merely statistical. This paper argues that the distinctive feature of contemporary generative systems lies in their reliance on high-dimensional parametric representations, where concepts are encoded not as discrete symbols but as positions within structured geometric spaces. Building on recent work in high-dimensional geometry, we show how properties such as concentration of measure, near-orthogonality, exponential directional capacity, and manifold regularity create the structural conditions for a new mode of signification. To explain the epistemological significance of these structures, we develop an Indexical Epistemology of High-Dimensional Spaces, taking generative AI as its paradigmatic case. Drawing on Peirce’s semiotics, we argue that meaning in embedding spaces is primarily indexical rather than symbolic: concepts acquire significance through their positional relations within a semantic field rather than through explicit definitions. This indexical mode of signification provides the missing link between geometric organization and what we call navigational knowledge—the capacity to orient and operate within a structured space of meanings. The paper further considers early Judeo-Christian hermeneutics and the medieval Jewish PARDES tradition, suggesting as a programmatic direction—rather than an established historical claim—that the symbolic–indexical contrast developed here may open a fresh reading of these long-standing debates about the nature of textual meaning. We conclude that generative AI should be understood not merely as a technological innovation but as the emergence of a new epistemic regime, characterized by the transition from symbolic representation to parametric-indexical forms of meaning and knowledge.