>What variable or data was mutated within haskell?
The stdout buffer.
Just because mutation isn't explicit doesn't mean it isn't there. Programming languages are not syntax devoid of meaning: they have semantics. What happens at runtime is part of what a programming language does. (Arguably, that is the most important part of what they do.)
>What this means is that imperative primitives must exist for any programming to work.
That's completely untrue. Imperative languages can be implemented as a subset of functional ones[1] and vice versa. Again, they're just different models. No language can do anything if it isn't implemented in a machine. A machine isn't "imperative"[2], it's a pile of atoms that do what atoms do, without paradigm or instruction. You absolutely could implement a pure functional assembly language. The reason nobody has, is because it doesn't matter: any Turing complete language can be used to implement any other language[3].
Try to implement `volatile` in C without using another language. Does that mean C fails to model real hardware? No, because it has `volatile` to get volatile semantics! Just like Haskell has IO to get I/O side-effects. Or ST to get mutation semantics.
> Use your algebra to make it work. You'll find it's impossible.
Don't assert it, Prove it. Show me one computable function that cannot be computed using boolean algebra.
[1] https://www.microsoft.com/en-us/research/wp-content/uploads/...
[3] https://en.wikipedia.org/wiki/Lisp_machine
[2] https://en.wikipedia.org/wiki/Turing_completeness
The stdout buffer.
Just because mutation isn't explicit doesn't mean it isn't there. Programming languages are not syntax devoid of meaning: they have semantics. What happens at runtime is part of what a programming language does. (Arguably, that is the most important part of what they do.)
>What this means is that imperative primitives must exist for any programming to work.
That's completely untrue. Imperative languages can be implemented as a subset of functional ones[1] and vice versa. Again, they're just different models. No language can do anything if it isn't implemented in a machine. A machine isn't "imperative"[2], it's a pile of atoms that do what atoms do, without paradigm or instruction. You absolutely could implement a pure functional assembly language. The reason nobody has, is because it doesn't matter: any Turing complete language can be used to implement any other language[3].
Try to implement `volatile` in C without using another language. Does that mean C fails to model real hardware? No, because it has `volatile` to get volatile semantics! Just like Haskell has IO to get I/O side-effects. Or ST to get mutation semantics.
> Use your algebra to make it work. You'll find it's impossible.
Don't assert it, Prove it. Show me one computable function that cannot be computed using boolean algebra.
[1] https://www.microsoft.com/en-us/research/wp-content/uploads/... [3] https://en.wikipedia.org/wiki/Lisp_machine [2] https://en.wikipedia.org/wiki/Turing_completeness