Counterfactual Basis Extension and Representational Geometry: An MDL-Constrained Model of Conceptual Growth

TL;DR

This paper presents a geometric, MDL-based framework for understanding how conceptual representations can expand through structured, error-driven basis extension, clarifying the role of imagination in learning.

Contribution

It introduces a novel MDL-constrained geometric model of conceptual growth that explains how representations can selectively expand based on residual error.

Findings

Conceptual growth modeled as admissible basis extension under MDL.

Extensions are restricted to low-rank transformations within residual error.

The framework clarifies the limits of imagination and representational change.

Abstract

Concept learning becomes possible only when existing representations fail to account for experience. Most models of learning and inference, however, presuppose a fixed representational basis within which belief updating occurs. In this paper, I address a prior question: under what structural conditions can the representational basis itself expand in a principled and selective way? I propose a geometric framework in which conceptual growth is modeled as admissible basis extension evaluated under a Minimum Description Length (MDL) criterion. Experience, whether externally observed or internally simulated, is represented as vectors relative to a current conceptual subspace. Residual components capture systematic representational failure, and candidate conceptual extensions are restricted to low-rank, admissible transformations. I show that any MDL-accepted extension can be chosen so that its novel directions lie entirely within the residual span induced by experience, while extensions orthogonal to this span strictly increase description length and are therefore rejected. This yields a conservative account of imagination and conceptual innovation. Internally generated counterfactual representations contribute to learning only insofar as they expose or amplify structured residual error, and cannot introduce arbitrary novelty. I further distinguish representational counterfactuals--counterfactuals over an agent's conceptual basis--from causal or value-level counterfactuals, and show how MDL provides a normative selection principle governing representational change. Overall, the framework characterizes conceptual development as an error-driven, geometry-constrained process of basis extension, clarifying both the role and the limits of imagination in learning and theory change.