I disagree completely on both examples. The first I have a technical nitpick to illustrate a deeper point and the second I have merely a stylistic point.
Firstly, I disagree completely that sum notation implies a hidden mutating variable. The most obvious way to attack this point is to note that sum notation is often applied when the summing domain is infinite and therefore if it means anything constructive (and it easily can!) then it certainly cannot be interpreted as a mutable loop.
Instead, it's best to merely have it represent the well-defined notion of a sum and note that it can be interpreted in many ways. This interpretation could involve using an algebraic identity (as one must with infinities) or, in the exact case you have an ordered finite domain, be interpreted either as a mutating iterative summing algorithm or a recursive immutable one.
The recursive bit applies exactly and only when the mutating algorithm one does.
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As for Djikstra's algorithm, Djikstra wrote it using the popular computer science notation of the time. Nobody denies that this formulation of the algorithm requires mutation to implement, but it's merely a formulation of it used for proof purposes. The idea of Djikstra's algorithm can be expressed without using a mutable language---it's trivial at its basest, merely replace mutation with a bunch of copies of your graph---and there's no particular reason to prefer either one!
As a further intensifying point, this is why Djikstra spent so long making sure he understood and communicating ALGOL and FORTRAN and the like so well. He wanted to explore new notations and needed to be absolutely clear how they worked.
This is similarly why Milner, Tofte, Harper, MacQueen completely specified the behavior of SML. [0]
Firstly, I disagree completely that sum notation implies a hidden mutating variable. The most obvious way to attack this point is to note that sum notation is often applied when the summing domain is infinite and therefore if it means anything constructive (and it easily can!) then it certainly cannot be interpreted as a mutable loop.
Instead, it's best to merely have it represent the well-defined notion of a sum and note that it can be interpreted in many ways. This interpretation could involve using an algebraic identity (as one must with infinities) or, in the exact case you have an ordered finite domain, be interpreted either as a mutating iterative summing algorithm or a recursive immutable one.
The recursive bit applies exactly and only when the mutating algorithm one does.
---
As for Djikstra's algorithm, Djikstra wrote it using the popular computer science notation of the time. Nobody denies that this formulation of the algorithm requires mutation to implement, but it's merely a formulation of it used for proof purposes. The idea of Djikstra's algorithm can be expressed without using a mutable language---it's trivial at its basest, merely replace mutation with a bunch of copies of your graph---and there's no particular reason to prefer either one!