linearity ≅ whole=∑parts?

In general, the main condition or step in proving a set to be a linear space is to show that for any two members f, g of the set under consideration, and for any two reals α,β

a. the linear combination αf + βg is in the set
b. this satisfies closure, from whence follow the remaining axioms 
  if applicable.

As an illusration, consider an operator qcq , L: D → ℝ^ℝ , where D is some set of functions , and ℝ^ℝ is the set of all real valued functions of a real variable.

L is linear, ie, is a linear space if , ∀y∈D, ∀z∈D, ∀α∈ℝ , ∀β∈ℝ

L(αy+βz)= αL(y)+βL(z)

This suggests some meaning. Viz., that operations on / properties of the whole are equal to sums of operations on / props of the parts.

This property appears to be akin to that of self-similarity, suggesting a rather profound meaning for the character of natural organization.

The ubiquity of linearity in the abstract algebra by which we represent natural systems^** (as noted earlier) has such consequences as the integrable character of physical law (discussed elsewhere).

^** if theory were complete then one could say up to an isomorphism, ie, the mathematical formalism is in a one-to-one, onto and domain-covering image of the fields of the physical system being described.