# 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 *x*₀, *x*₁, ⋯, *x*ₙ, 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 *x*ᵢ* in the interval [*x*ᵢ₋₁, *x*ᵢ] is arbitrary when calculating Riemann integrals’ Riemann sums

S = ∑xᵢ*∆xᵢ

over interval with partition. This is maybe less evident when the partition *x*₀, *x*₁, ⋯, *x*ₙ 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 [*x*ᵢ₋₁, *x*ᵢ], 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 [*x*ᵢ₋₁, *x*ᵢ]. 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 priori*s: a very dangerous renouncement of truth… May it and the ensuing chaos be the powerful inspiration of great fulfillment.