The Unity of the Discrete and the Continuous as a Structural Principle: The Infinium △₁ₓ₁ and the Theory of Motives

Abstract

This article is devoted to one of the oldest and deepest problems in mathematics—the gap between the discrete and the continuous. We show that this gap is not an accidental difficulty but a direct consequence of choosing the structureless point as the initial primitive. Within Δ‑ontology, a structural alternative is proposed: the right isosceles triangle △₁ₓ₁ (the infinium) with legs 1 and hypotenuse √2. In it, the discrete (countable legs) and the continuous (irrational hypotenuse) are for the first time given in indissoluble unity. The theory of motives, developed in application to the infinium, allows us to express this unity in the deepest algebraic‑geometric language: the motive of the infinium M(ℑ) = ℚ(0) ⊕ ℚ(1)[1] ⊕ ℚ(1)[√2] unites the point, the discrete line, and the continuous diagonal. The article sequentially examines historical attempts to reconcile the discrete and the continuous, shows how the great problems of mathematics (including the already proved Fermat’s theorem and the Poincaré conjecture) contain this dichotomy within themselves, and demonstrates how the infinium removes it at the level of the primitive. The key metaphor: the infinium is simultaneously a “brick” and a “road,” an object and a relation, an atom and a continuum. Important clarification: throughout the article we emphasize that the infinium is not merely a figure but a carrier of a dual nature—it is simultaneously an object (a geometric form) and a relation (the connection between the legs through the hypotenuse). It is precisely this duality that makes it the only candidate for the role of a primitive capable of generating all of mathematics without external crutches.