apply

hypermath | formal-universality | dictionary

What it is

apply is the sole primitive operation of hypermath. Its signature is apply :: Form -> Form. It takes any Form and returns a Form. apply IS □ acting on a Form. Written apply(x) or □(x). There is no other primitive operation. All structure in the system derives from apply alone.

What it is not

• Not function application in the lambda calculus sense. Lambda calculus apply takes a function and an argument. Hypermath's apply takes one Form — it is a unary operation, not a binary one.
• Not composition. Composition of two operations requires two operations. apply is the only one. What looks like composition (apply(apply(x))) is iterated application of the same operation.
• Not evaluation or reduction. apply does not simplify. It generates. apply(ground) is not a reduced form of something more complex. It is a new structural entity distinct from ground.

What it clarifies

Clarifies why the entire system is generated from one operation: apply is the □-act. Everything that exists in hypermath is reachable from ground by iterated applications of apply. All of L0 is a product of apply and its consequences.

Concrete example

apply(ground)          -- first generated Form
                       -- ax-diff: struct-distinct(apply(ground), ground)
                       -- ax-sim:  struct-continues(apply(ground), ground)

apply(apply(ground))   -- second generated Form
                       -- ax-box: struct-orbits(apply(apply(ground)), ground)

Visualization

apply is a stamp. You have one stamp. You stamp anything you have and get a new thing. The new thing is different from what you stamped (ax-diff) but made of the same material (ax-sim). Stamp the stamp itself: you get something new. Stamp that: it comes back around (ax-box).

Speculation

Whether apply has a category-theoretic interpretation as an endofunctor on the category of Forms is not yet derived. If the deriver closes its ==-cycle at L3, the resulting structure may map directly onto the notion of a self-applying endofunctor — linking hypermath to enriched category theory and potentially to the foundations of the Language Calculus that follows. Commutativity of apply in any derived sense remains OPEN.