The link you has its fallacy in the premise. It's defining "function" as "computable function", then complaining that many results that follow do not agree with the former definition! Either I'm really missing something or this is some poor attempt at trolling / bait-and-switch. Of course you have problems with that "computable function" definition when dealing with functions from Real -> Real, since many (in fact, almost all) real numbers are not computable!

Re mutating getter:

Monads for example are a standard way of formalising state in a pure-function universe. You seem to have some fundamental misconception about what "mathematics" is, since you keep repeating something about purity. Mathematics is merely the rigorous study of formal systems, of which your mystical "mutating getter" is one such system.

You have some deeply flawed (or entirely absent) philosophical upbringing. There are no "foundations" to anything - all definitions are arbitrary.

To assert that a premise is fallacious mandates that you have some prior notion of "fallaciousness".

You don't have an objective criterion for asserting whether one definition is better than another because you don't have a notion of "betterness" - it's all conventional.

It's precisely because I am an over-zealous formalist is why I see Mathematics for what it is - grammar, syntax and semantics.

It's just language.

Mutating getter? Sounds like a generator function that yields. Or a pseudo random number generator implemented with a monad (WorldState, Value). Or any tail-recursive non-void function that carry state without loss?

I'm definitely not an expert but aren't irrational numbers are the best examples to show that these types of constructs already exist in math? You can't calculate them in finite amount of time. You use an analytical formula or series expansions to get the nth term instead.

Assume that it is a black box to you, even though it is a white box to me.

Every time you call read() on it - you get a different result. It has side-effects.

This violates the “purity” of Mathematics.

I just explained three different models to represent side-effects without leaving the purity. I'm not sure if you are trolling, throwing random big words or simply have a really misguided impression about how computer science connects with math.

Your monad(WorldState, Value) example assumes shared global state. So not a networked system then?

That's not a train-smash for your argument if you can show me a Mathematical model for a global mutex.

When the discussion turns to control-flow you are inevitably in the land of well-ordered imperatives, not Mathematical, lazy-evaluated declaratives.

What is special about a mutating getter? It operates on the state of the object S and outputs a new state S' and some value x. Some programming languages allow you to write 'functions' without explicitly passing all input state into the function, but does that imply anything fundamental or interesting about mathematics or computation? Is that anything more than a type of syntactic sugar?

What is "special" is that the system has a peculiar property: f(x) != f(x).

If we claim to be subscribed to denotational (Mathematical) semantics then the above contradicts the identity axiom.

And it's not any deep and world-changing insight either - it's obvious to anybody who sees that the LHS and RHS are only evaluated at runtime - they don't have any inherent (denotational) meaning, which is why I keep harping on: programmers use different semantics to mathematicians. We care about things like interaction and control flow structures as first class citizens - those are precisely the things that have no mathematical equivalents. Timeouts, exceptions, retry loops.

It's not "syntactic sugar" - it's necessity for reifying control flow. In the words of the (late) Ed Nelson: The dwelling place of meaning is syntax; semantics is the home of illusion.

https://web.math.princeton.edu/~nelson/papers/rome.pdf

Of course it isn't. OP seems to have some kind of misconception about what "mathematics" means, since they keep fixating on a vague concept of purity of notation.

> In addition to the examples in the blog post linked above, you cannot express a mutating getter in Mathematics.

Neither can you express foobarish bamboozles...

I can express a mutating getter in any modern programming language much like you just expressed “foobarish bambozles” in English.

Q.E.D

Please don't use that style of quotation for non-code. It is awful on mobile.