Limited Math: Aligning Mathematical Semantics with Finite Computation

Abstract

Classical mathematical frameworks used in the semantics of computation and programming languages are traditionally grounded in idealized abstractions, including infinite-precision numbers, unbounded sets, and unrestricted operations. Concrete computation, however, is intrinsically finite, operating under explicit bounds on precision, memory, and structural resources. This foundational mismatch complicates semantic reasoning about numerical behavior, algebraic properties, and execution in realistic computational settings. This paper proposes Limited Math (LM), a bounded foundational framework intended to realign mathematical semantics with the realities of finite computation. Rather than presenting a complete formal theory or establishing meta-theoretical results, LM is introduced as a conceptual foundation in which finiteness is made explicit at the semantic level. Numeric magnitude, precision, and structural complexity are treated as primitive constraints, enforced through a finite numeric domain parameterized by a single bound $$M$$ and a deterministic value-mapping operator that makes quantization and boundary behavior explicit. Within representable bounds, LM coincides with classical arithmetic; beyond these bounds, deviations are explicit, deterministic, and open to analysis rather than implicit or idealized. By additionally bounding structural constructions such as set cardinality, the framework prevents infinitary assumptions from re-entering through indirect means. The resulting semantics induces finite-state models of computation, offering a principled basis for reasoning about arithmetic, structure, and execution under finite computational constraints and for guiding future formal and technical developments.