Abstraction as Constrained Interpretive Labor: A Formal Account of Why Theoretical Languages Feel Hard

A recurring feature of mathematics and other formal sciences is that their texts are often experienced as unusually abstract, difficult, and discontinuous to read. This paper proposes a structural account of that phenomenon. The central claim is that abstraction is not merely a property of objects, nor solely a property of logical depth, but arises when a text imposes a high precision requirement under low semantic redundancy, thereby restricting the reader’s ability to rely on ordinary contextual inference. In ordinary language, understanding is supported by broad semantic drift, pragmatic repair, and flexible generalization. In formal theoretical language, by contrast, interpretive freedom is sharply constrained: a symbol, definition, or theorem statement typically permits only a narrow admissible neighborhood of readings. When a reader does not yet possess the relevant local grammar, comprehension cannot proceed by smooth global guessing; it must proceed by incremental acquisition of the language itself. This creates a gating effect: reading becomes discontinuous, working memory is repeatedly reloaded, and the text subjectively appears abstract. We develop a simple model in which a text has a semantic redundancy parameter and a precision demand parameter. A text is difficult when the reader’s available inferential capacity falls below the ratio of precision demand to redundancy. We define an abstraction index and show that it increases monotonically with precision demand and decreases monotonically with semantic redundancy. We then prove several elementary propositions clarifying how examples, local context, and prior mastery reduce cognitive cost. The framework explains why mathematical texts often require “learning while reading,” why novice readers experience abrupt breaks in continuity, and why seemingly small changes in notation or exposition can drastically alter readability. The paper concludes by distinguishing the proposed account from standard descriptions in cognitive load theory, philosophy of language, and mathematics education.