The thing is that we use different (and incompatible) abstractions to describe processes in differing contexts.
The map is not the territory.
I mostly agree with you, but I would add that systems fit different models (aka abstractions) as they scale and that as we add water molecules to a system we'll start to see microfluidic, then fluid, then oceanographic,.. cosmological properties emerge. The molecules are the same and always behaved with all of those properties, but whether or not we can see them easily, or can use them as a modelling tool depends on the scale.
In this case, isn't this just a matter of the definition of a "heap of sand"? The article seems to sort of forget to define a heap and just proceeds to make a problem out of it and asking questions like "when does a heap become not a heap?" when it was never even defined why would we call something a heap in the first place.
So I think the real question there is just that why are you calling something a heap in the first place and when you have an answer to that, that same answer will also help you with figuring out when the thing you call a heap is not a heap anymore.
Cultural conditioning, technically, "just [the] reality", "you know what I mean", etc colloquially. It's a broadly "known" meaning, despite no definition, that's where the paradox comes from.
Lots of things are like this, the requirement(!) to fight wars for example - there's no objective definition for why we must do it, we just "know" that we must. And if you disagree with me, just ask anyone and see what they say (watch out though, quite often people will trick you and answer both yes and no to the same question, with complete sincerity).
Watch children playing house and listen carefully to how they talk, how they describe the details of their imaginary world, etc....its an innate feature of human consciousness, it never goes away, but it becomes cloaked by education, culture, "facts", etc.
So now what, does the definition of a heap have to account for the subject's and object's relative scales? Formalizing informal notions is not always straightforward.
More specifically, I think "emergence" is more about the blind spots we have as meat calculators than something magical, unless you ascribe to the notion of "magic" as "something sufficiently advanced or complex as to be difficult to understand", in which case I think actually yeah, emergence is magical behavior from that perspective.
I imagine they would agree with you that there's nothing magical about it! The map is not the territory but some maps are better than others.
It was about statistics and probabilities. I think I was talking with ChatGPT about superpositions or something and somehow we got to talking about how e.g it might be impossible to make a system which could predict everyone's favourite flavour of ice cream.
Interestingly enough though, it is perfectly possible to gather data about people's favourite ice cream flavours (and we could even go as far as to say that we could ask every single human on the planet) and make a statistical model which is able to answer what is most probably everyone's favourite ice cream flavour.
I find this really interesting. We could think of one person's flavour as essentially random and impossible to predict, but when we gather enough of these random data points, we are for some reason able to build a relatively accurate system to guess someone's favourite flavour. I don't think this is obvious at all.
Anyway, I didn't have any real point here. I just wanted to share one example of interesting thing that I think is an example of "emergent behaviour" and seemingly magical at that too.
> My take on this is that there is no "emergence" that appears when grouping elements together into a larger system.
You can, for example, go mathematically from describing individual behavior to describing macro behaviour and see the macro emerging in the limit (e.g., homogenization of PDEs).