Mathematics is a language whose grammar is rigorous logic. Following Hilbert’s program, finite applications of rigorous logic introduce no new information: equivalence preserves content, and implication reduces it. New mathematical information arises only when infinite processes are admitted, historically through infinitesimals and proof by contradiction. The introduction of infinitesimals—quantities smaller than any positive value yet nonzero—forces continuity and eliminates discrete boundaries. Logical gaps in the number system are filled, yielding continuous mathematics and the real number line. As a consequence, atomic identity, scale separation, and indivisible units cannot be preserved; all objects become infinitely divisible representations. Modern physics is formulated within this continuous mathematical language. As a result, new structures appear that are linguistic consequences rather than empirical discoveries: empty vacuum as absence of particles and electromagnetic fields, complete cancellation of electromagnetic surface effects, smooth spacetime metrics, absence of interactions within the slit in interference experiments, Doppler-based cosmological redshift, and expanding-universe interpretations. This paper analyzes the information gained and the information lost when mathematics transitions from finite, discrete construction to infinite, continuous structure. It identifies which physical assumptions arise from mathematical continuity itself and prepares the ground for a discrete reconstruction that restores the information eliminated by continuous grammar. Using Young’s experiment as a diagnostic example, we show that continuous mathematics suppresses interaction inside physical boundaries, leading to idealized explanations of photon behavior, redshift, and light bending that rely on smooth geometry rather than local physical interaction.
dong zhang (Fri,) studied this question.