math rant: on the definition of a relation versus its representation

let there be two sets X and Y, and some binary relation r ⊂ X × Y.

suppose ∃(x ∈ X ⋀ y ∈ Y) (x, y) ∈ r.

suppose there is also a schema to represent some (α, β) ∈ ρ ⊂ A × B in the form of (α, ρ, β). ergo we can represent the binary relation (x, y) ∈ r as a triple (x, r, y).

let μ = (x, y) ∈ r and ν = (x, r, y) ∈ {(α, ρ, β)} for the sake of being succinct. although the tuple ν can be read as "relation r holds for x and y", ν is indeed only a representation of μ and not an equivalence with μ. this is because ν ∈ {(α, ρ, β)} whereas μ ∉ {(α, ρ, β)}. that is, ν is structured according to a different schema {(α, ρ, β)} than that of μ, which is structured according to r ⊂ X × Y or r = {(x, y) : x ∈ X ⋀ y ∈ Y}.

i take this to mean that mathematical representation is no less self-referential than linguistic representation, that mathematics can be understood in terms of a language or a signifying system. we see this in that the representation of μ by ν is not equivalent with μ itself.

nevertheless, this opens up new ways to discuss how mathematics represents itself, and perhaps offers itself up as a basis to discuss the structure of signifying chains in general. the tuple (x, r, y) can be taken to represent a query as to whether indeed (x, y) ∈ r. for example, taking (x, r, y) as a query, let us define a new relation:

relationTrue = {((α, ρ, β), bool) : bool = true if (α, β) ∈ ρ, else bool = false}.

this relation lays bear the linguistic structure of mathematics, alongside that any n-ary tuple can be decomposed into nested tuples of smaller sizes (since ((α, ρ, β), bool) = (α, ρ, β, bool)).

finally, i'd like to suggest that there is an implicit relation underlying any explicit relation which grounds the significance of terms given. i will represent the reference to this relation as ▶.

the function below represents the meaning of a binary relation, for which i use "a + b" as an example meaning "add a to b". the two forms given below are equivalent representations of each other. keep in mind that these are not equivalent with a + b as such, but they are only representations of it combined with how one might read this representation in english.

signification((a, +, b)) = "add a to b"

signification((a, +, b), "add a to b")

signification(a, +, b, "add a to b")

the relation below is a representation of the signification function, but it is not an equivalence with it. valid is the set of all binary pairs which are valid according to a given binary relation. notice how here, significance appears as a term within the tuple and not as an external designator of the relation.

valid(signification, (a, +, b), "add a to b")

this relation can be read as "with respect to the signification relation, the pair (a, +, b) and 'add a to b' are valid". we can use this same function to describe the validity of (2, 2) with respect to another pairing.

valid(+, (2, 2), 4)

this can be read is "with respect to the + function, the pair (2, 2) and 4 are valid." it should be restated that the relation above only contains a representation of the function below, not an equivalence with it. (excuse the prefix math notation)

+(2, 2) = 4

+((2, 2), 4)

+(2, 2, 4)

of course, this sort of differentiation between definitions and representations can go on forever. i'd just like to say now that the valid relation can be replaced with any one measure of validity. let me just go up one more layer:

▶(valid, +, (2, 2), 4)

≠

valid(+, (2, 2), 4)

once more, the upper term is only a representation of the lower term, not an equivalence with it. the ▶ character signifies what i think is an apt cutting-off point, at which there is no use in trying to represent layers of relations further. it is basically what lacan calls the point de capiton, which grounds meaning in a chain of signifiers and prevents endless self-reference and representation. i claim that this relation grounds all the others and ensures that they are interpretable, or that they can be represented to us in our brain's language translator.

▶(valid, +, (2, 2), 4) means:

"That (2,2) and 4 is a valid pair with respect to + is true."

▶valid(signification, (a, +, b), "add a to b") means:

"That 'add a to b' and (a, +, b) is a valid pair with respect to signification is true."

▶(+, (2, 2), 4) means:

"That 2+2=4 is true."

the framing of "That [...] is true" does not have to necessarily be included, but that is just how i am trying to represent that the ▶ function returns meaning.

i hope this clarifies what i have been thinking about the difference between a function's definition and its relation, and how mathematics in this sense is structured like a language.