> ... foundations of what I would call functional programming, Lambda Calculus as defined by Church ...

Ah! Now we’re getting to the nub of it. Yes, the lambda calculus, absolutely correct.

> ... almost exclusively uses callbacks ...

Whoops! This is absolutely wrong. The functions in the lambda calculus cannot have side effects (like saving a file or displaying a message to the user) and they can only depend on their inputs (e.g. they cannot return a number entered by a user, data from a socket, the current time, etc.). In the software and computer science world they are called "pure functions". Mathematicians (including Church) would simply call them "functions" because that, by definition, is what a mathematical function is (it is a subset of the Cartesian product of the domain and codomain, very much like a lookup table in computer programming except it can be infinite or even uncountably infinite).

These are very different from the callbacks being discussed in the original post. A callback - as the name suggests - is a function passed from a higher-level abstraction to a lower level abstraction to be notified when something happen. Those exist purely for their side effects - that's the whole point of a callback - and usually have no return value at all.

> Functional programming still computes heavily via function composition; the principle of passing callbacks around ... [e.g.] via currying "context" values, by passing a function to "act in a context" (hole-in-the-middle/delegator)

Sure, except you're passing (pure) functions around, not "callbacks" (at least for it to count as true functional programming).

> closing (closures) over mutable state

As you said, F# is a hybrid language, and mutable state is on the imperative side (admittedly a concession made even by stricter functional languages like LISP).

> This is generally a false statement. ... how lazy evaluation can share chunks (bits of compute)

It would be true in a pure functional language, and is only false to the extent that a real-world "functional" languages actually have some imperative features.

> > write a series of instructions that are executed top-to-bottom

> That's procedural programming, not OOP:

No, it's imperative programming, as I said in my previous comment. Procedural programming and OOP are both subtypes of imperative programming. On the Wikipedia page you linked to, look at the box on the right hand side, and note how imperative says "(contrast declarative)" next to it, and vice-versa. The box shows that declarative includes functional and dataflow programming.

> Assembly and even C also heavily use callbacks

Agreed, and they're a lot more similar to the type of callbacks being described in the original post than pure functions: they're totally imperative and allowed side effects. But they don't normally have included state - you usually get a void* for your extra data but no functions to copy and destroy that, unlike C++ lambdas and std::function (Python uses garbage collection to avoid needing those for state attached to function objects). So using them ends up with a slightly different feeling to the developer than callbacks in OOP languages, which is why the article specified callbacks in OOP languages rather than callbacks in general.

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There are other details I could pick apart in your original comment or other replies. But there is a broader point. I know this isn't terribly constructive, but I think it's important to address the elephant in the room: frankly, none of your comments make any sense. You misuse terminology all over the place, but then pedantically pick apart individual technical terms in other people’s comments, mostly because they've used the terms with their real meanings rather than your imagined meanings. The commenter _pmf_ chose to highlight your incorrect criticism of "callbacks" and I followed up on it, but almost everything else in your original post deserves the same treatment; that was just one example. I really think people downvoted your original comment rather than replying because they took a look at it and thought "this is a lost cause".

In the comment I'm replying to, you say "I'm very confident in saying that" and complete the sentence with something that is wrong, and several people have already told you so. But this attitude comes across implicitly elsewhere. I implore you, please stop trying to just push your understanding of things, and instead lean back and consider that the comments contradicting you could actually be right.

This especially applies to looking back at Kenton's replies, even if you ignore what I'm saying (fair enough). I realise he gave up replying to you eventually but his replies were very reasonable and clear explanations and you responded with just an explosion of more confusion that would take more and more effort to clear up - I think he has the patience of a saint for addressing as much of your comments as he did.

Bear in mind that no one else in the comments had trouble with either the original article or the introduction to Cap'n Proto in general. No one came along and agreed with anything you said, which does happen when bad articles are posted here. In general (of course there are exceptions), Hacker News commenters are clever and reasonable. If you find yourself surrounded by clever reasonable people and they're all wrong and you alone are right, it is time to consider if it is really the other way round.