That's what I said. Only in the 20th century for something that might seem to an outsider to be fundamental to mathematics (after all, mathematicians sometimes compute, right?)

> So why avoid this kind of stuff? Because you want to pick a "cognitive abstraction", in your terms, which is most useful.

Oh, absolutely, but there are many ways and many forms of reconciling "equational" mathematics and computational mathematics, and PFP is just one of them. As I've shown in another comment, Prolog takes this much further than Haskell because it's truly equational (or, rather, relational), while Haskell gladly allows a direct (one-way) computation if it defines it as pure, i.e. in Haskell 2+2=>4 is still as different from 4=>2+2 as it is in Java. Other ways are the plain procedural style (which simply lets you name and parameterize a computation), and my favorite (although I don't know how practical), temporal logic used in, e.g. Esterel (and TLA+).

My beef is not with equational reasoning as a concept, and not even with Haskell's flavor of it. Certainly "equationality" in programming languages is a spectrum (as computation is inherently non-equational), and finding useful points on it is a matter for lots of empirical study. My "beef" (if that's what you want to call it) is with the claim that Haskell's particular design is somehow more fundamental or mathematical than other designs, and, to bring us back to the topic of the discussion, that (cognitive) monads are somehow essential. The reality is that there are many (good) justifications for Haskell's design, but they've been made by people to match certain human aesthetics and certain human goals[1] and they don't have any transcendent mathematical justification. They are just a point in the design space. Similarly, monads are only (pretty much) essential in Haskell given its design. Some people try to present Haskell (and monads) as the language (and abstraction) God, nay, better -- Math -- has given us, and that's just wrong. It's just a language like many others, with its own design choices. Besides, everybody knows God uses Lisp. My other "beef" is with the unjustified belief that a certain way of expressing an algorithm is automatically cognitively optimal just by virtue of it being equational (be it either in the Haskell or the Prolog sense).

[1]: One of them is a particular balance between correctness and specification effort, which, again, is also arbitrary and defined by the limitations of HM type inference.

P.S.

To clarify something I wrote in another comment, Haskell's definition of purity, while fundamentally arbitrary (from a computational theory point-of-view), makes a lot of sense if you've chosen that particular design.