here are several categories of mathematical notation that are especially opaque or hard to track for someone from a computer science background, along with illustrative examples. these represent excellent test cases for stress-testing a referential, programming-inspired notation.

example:

f(x + y) = f(x) + f(y)

problems:

• + may mean real addition, group operation, vector addition, etc.
• function f might be a morphism, homomorphism, or arbitrary function.
• no types or domains are specified.

example:

∀ε > 0, ∃δ > 0, ∀x, |x - c| < δ ⇒ |f(x) - l| < ε

problems:

• no indication of types (x, c, f, l).
• scope nesting is unclear.
• logical structure is compressed and non-linear.

example:

∑_{i=1}^{n} a_{ij} b_{jk}

problems:

• index tracking is manual.
• variable binding is implicit.
• semantics are lost outside the matrix context.

example:

f: a -> b, g: b -> c, h = g ∘ f

problems:

• directional semantics depend on convention.
• composition order (g ∘ f) is opposite of evaluation order.
• diagrams are informal, often with no canonical syntax.

example:

(id × f) ∘ δ

problems:

• no variables, so the dataflow is opaque.
• requires fluency with combinators.
• difficult to evaluate step-by-step.

example:

a + b * c^d / e - f

problems:

• implicit precedence and associativity.
• no parentheses, types, or grouping indicators.
• meaning changes dramatically with minor edits.

example:

{x ∈ ℝ | x² < 2}

problems:

• semantics of | vs : varies by source.
• logical interpretation (x² < 2) is implicit.
• no binding or scoping shown explicitly.

example:

1 + 2 + ... + n

problems:

• no defined semantics for ....
• interpretation is context-sensitive.
• not machine-readable.

example:

let φ: g -> h be a group homomorphism

problems:

• no definition of φ, only type.
• all behavior is inferred from "homomorphism".
• mapping rules are abstracted away.

example:

ℒ(x, y), hom(x, y), ⟨x, y⟩, [x]_∼

problems:

• notation depends on domain-specific conventions.
• no universal syntax for defining these constructs.
• requires extensive background knowledge to interpret. would you like to prototype referential rephrasings for one or more of these cases?