Functional programming is what people talk about when they "talk about FP". The historical story is that "functional programming" just meant purity, originally

Functional programming is what people talk about when they "talk about FP". The historical story is that "functional programming" just meant purity, originally. Now it seems to mean purity + higher order functions + other strong, fancy types (which I'd claim include algebraic structures such as monads). This current meaning, exemplified by the ML/Haskell tradition, spiritually comes from Peter Landin's ideas in his paper "The mechanical evaluation of expressions" and its supporters don't revere Backus. Sure, algebraic laws characterise monads etc., and his work was foundational—but the Turing Award paper is not a script that everyone is following. On the specific topic of deriving programs with algebra, there is the book "Algebra of Programming" by Bird and de Moor from the 90s, (https://themattchan.com/docs/algprog.pdf https://themattchan.com/docs/algprog.pdf) but it is difficult to look at the functional programming scene now and conclude that deriving your program algebraically is a more essential aspect of the activity than using referentially transparent expressions. It may well be or become an important idea, but it's hardly the bread and butter of functional programming in practice (while maybe in your headcanon it is functional programming). I think part of the problem here is complexity/irregularity: it's simply not practical or helpful to describe typical desired systems algebraically and then derive code from that description.