Thanks for this interesting and clear question.

I will start by noting that a heap is a most simple form of structure: elements are randomly stacked upon one another, without strong connections between them. The structure is, therefore, very weak. It can be disposed of by just kicking at it. A heap is not much of a structure, it lies at the low end of structures, as far as complexity is concerned.

Is a heap a case of emergence? Yes, but it's a very, very simple case, simplistic really. A heap is very limited in what it can do. It cannot fly or swim, for instance. Yet it can do a few things that individual grains of sand can't do, like block a vehicle, and in that sense I suppose one could say things like "a heap of sand emerged next to the kid as he started to dig into the beach".

One could consider a heap as emergent, though it seems a borderline case because barely structured at all.

Does this means that any and all forms of emergence can be seen as particular cases or instances of the Sorites paradox? Not really, for the already mentioned reason that a heap is barely a structure at all, as interactions between elements remain very weak.

A better metaphor would be some bona fide structure, like a house, a car or a boat. So the question would become: at what point does a house stops being a house, when you take out one piece of it at a time?

Unlike heaps, real structures have a function, which one can use to define them. Considering that a house primarily functions as a shelter, I would say that once you take out the roof, it ceases to offer shelter and thus stops being a house and becomes "a house in construction" (or in demolition).

So in this more canonical example of emergence, we have a system (the house) providing a function (shelter) that none of the building component can offer on its own. And it's all about how the roof interacts with the walls, within the "house system": the roof needs to be supported by walls, and in turns it will protect the walls e.g. from rain.

Thanks for this interesting and clear question.

I will start by noting that a heap is a most simple form of structure: elements are randomly stacked upon one another, without strong connections between them. The structure is, therefore, very weak. It can be disposed of by just kicking at it. A heap is not much of a structure, it lies at the low end of structures, as far as complexity is concerned.

Is a heap a case of emergence? Yes, but it's a very, very simple case, simplistic really. A heap is very limited in what it can do. It cannot fly or swim, for instance. Yet it can do a few things that individual grains of sand can't do, like block a vehicle, and in that sense I suppose one could say things like "a heap of sand emerged next to the kid as he started to dig into the beach".

One could consider a heap as emergent, though it seems a borderline case because barely structured at all.

Does this means that any and all forms of emergence can be seen as particular cases or instances of the Sorites paradox? Not really, for the already mentioned reason that a heap is barely a structure at all, as interactions between elements remain very weak.

A better metaphor would be some bona fide structure, like a house, a car or a boat. So the question would become: at what point does a house stops being a house, when you take out one piece of it at a time?

Unlike heaps, real structures have a function, which one can use to define them. Considering that a house primarily functions as a shelter, I would say that once you take out the roof, it ceases to offer shelter and thus stops being a house and becomes "a house in construction" (or in demolition).

So in this more canonical example of emergence, we have a system (the house) providing a function (shelter) that none of the building component can offer on its own. And it's all about how the roof interacts with the walls, within the "house system": the roof needs to be supported by walls, and in turns it will protect the walls e.g. from rain.

In conclusion, there's indeed no particular point at which a heap ceases to be a heap if you take out its composing sand grains one by one, because a heap is a mere addition of components, without a clear function. In case of a functional structure, it ceases to be it when rendered dysfunctional. If I sold you as "a car" a terminally broken wreck, you'd have good reasons to feel conned. A wreck is not a car anymore. So the Sorites paradox is more manageable when speaking of functional structures.