The Meaning of Mathematics(Part I): An Ontological Relationship among Cognition, Prime Numbers, the Riemann Hypothesis, and Goldbach's Conjecture

Starting from the cognitive operation of information collapse, this paper re-anchors the foundational position of mathematics—as the pure grammar of information-collapse operations. Based on a strict correspondence between the four arithmetic operations and the four basic collapse actions, it argues that subtraction is the more primordial first action than addition: extracting the first determinate object from chaos. This act of subtraction is itself the most primordial division; the boundary is the trace left by the division—the two occur simultaneously in the same cognitive instant, but division is the operation that makes the boundary possible. The first determinate object thus extracted is precisely "1"—not merely the number 1, but the boundary of all numbers. The divisive nature of 1 is simultaneously the origin of space and geometry: geometry is the unfolding of divisiveness in the spatial dimension, and by virtue of its necessary closure-range, it possesses a binding constraint that governs the infinite whole. The essence of prime numbers is the ineliminable surplus sifted out by the interweaving of object-aggregation and temporal segmentation, after subtraction has supplied the boundary for addition. On this basis, the paper demonstrates that the Riemann Hypothesis and Goldbach's Conjecture are a pair of dual propositions within the same cognitive structure: the former governs the complete cognitive cycle from indeterminacy to determinacy and back to indeterminacy, anchoring its position on the 1/2 line; the latter traces backward from determinacy to indeterminacy—whether every even number greater than 2 can be decomposed back into two primes. The even number 2, as the numerical starting point of "the sum of two primes," cannot be governed by its own origin and process—this is the signature that the divider is not within its own division-range. The self-adjoint operator is precisely the mathematical counterpart of the unity of the entire cognitive-operation chain that starts from subtraction. The self-adjoint operator exists structurally but is absent in the result—because it is precisely the person who is doing the calculation in reality, i.e., the divider itself. The divider is structurally also a manifestation of division, but in the result is not within its own division-range. This is the ultimate reason why the self-adjoint operator can only be designated but never realized, and it is also the ontological ground for the suspension of the Riemann Hypothesis and Goldbach's Conjecture. This paper does not attempt to prove these two conjectures in the mathematical sense; rather, starting from the underlying structure of cognitive operations, it provides an ontological necessity argument for their validity.