An Aphorism on Partitions Found in Riemann Integrals

In preparation for the construction of automatics, here lies a short or quick conceptualization of a particular nature or aspect of the partitions of Riemann integrals.

Suppose the elements a, b ∈ ℝ with a < b. A partition of the interval [a, b] is a finite list of the form x₀, x₁, ⋯, xₙ, where

a = x₀ < x₁ < ⋯ < xₙ = b.

The interval [a, b] as a union

[a, b] = [x₀, x₁] ⋃ [x₁, x₂] ⋃ ⋯ ⋃ [xₙ₋₁, xₙ]

possibly shows that the choice of xᵢ* in the interval [xᵢ₋₁, xᵢ] is arbitrary when calculating Riemann integrals’ Riemann sums

S = ∑xᵢ*∆xᵢ

over interval with partition. This is maybe less evident when the partition x₀, x₁, ⋯, xₙ is instead thought of as a set

P = {[x₀, x₁], [x₁, x₂], ⋯, [xₙ₋₁, xₙ]},

because of the lack of the set union denotation. The whatness of a and b need not be members of ℝ, as, ultimately, any determination of being or distinction requires synthesized understanding and not any understanding given a priori.—So the internal composition of [xᵢ₋₁, xᵢ], influenced by what [a, b] is, is possibly a dynamic and constantly in flux interaction between the choices of, now contingent, elements contained in the interval [xᵢ₋₁, xᵢ]. This aphorism is really, then, my official or so break or departure from traditional mathematical Platonism of the static a priori kind and a demonstration and exercise of my own particular brand of inverted Platonism. No more “givens” or static a prioris: a very dangerous renouncement of truth… May it and the ensuing chaos be the powerful inspiration of great fulfillment.

December 24, 2019