What exactly is a 'heap'?

[Kyng]

Here's an argument I came across recently:

Premise 1: A single grain of sand is not a heap.
Premise 2: Adding one grain of sand to a non-heap does not turn it into a heap.
Conclusion: There is no such thing as a heap of sand.

I don't think any of us would be willing to accept this conclusion. However, the structure of the argument is fine: it's the same form of argument that mathematicians use routinely (in proofs by induction). And both premises seem to be reasonable - at least, at face value. So, what gives?

Perhaps one of the premises is false. My first instinct when confronted with this argument was to deny the first premise: I would define 'heap' as 'a collection of object(s) arranged in no particular order'. This would mean that a single grain of sand would be a 'heap', albeit a very small one. However, this doesn't seem very satisfactory: if someone asked me for a heap of sand, and I gave them a single grain of sand, then they'd probably just throw the grain of sand back in my face . Even if I'm willing to accept that one grain of sand is a 'heap', most others won't.

So, perhaps the second premise is false? Perhaps there's some magic number of grains of sand (say, 10,000 grains) at which a non-heap becomes a heap. Unfortunately, I'm not happy with this solution either: it seems very arbitrary, because a pile of 9,999 grains of sand is not meaningfully different from a pile of 10,001 grains of sand. Heck, if I piled up 9,999 grains of sand, and then piled up 10,001 grains of sand, and asked which of these two piles had the most grains, I don't think any of us could give the correct answer with any confidence (I certainly couldn't!). Or, going back to my earlier attempt to solve this, I could just modify my definition of 'heap' to 'a collection of multiple objects arranged in no particular order'. At this point, one grain of sand is not a heap, but two grains of sand would be, provided that they weren't arranged in any particular order. This seems slightly more reasonable; however, if someone asked me for a heap of sand, and I gave them two grains of sand, then I suspect the outcome would be pretty much the same (the only difference is that I'd have twice as much sand thrown back in my face as before ).

So, I decided to look up a proper definition of 'heap'. The Oxford English Dictionary defines the 'heap' as "An untidy collection of objects placed haphazardly on top of each other". So, we'd have three separate conditions for a bunch of sand to be a 'heap':

1. There must be multiple grains of sand. One grain of sand is not a 'heap'.
   
2. The sand must be in an untidy arrangement. This block of sand is not a 'heap', because it's too orderly.
   
3. The grains of sand must be on top of one another. The grains of sand on the underside of sandpaper are not a 'heap', because they are not on top of one another (and also because sandpaper no longer has real sand on its underside )
   

So, if I have four grains of sand lying on the floor next to each other, they don't form a 'heap', because they aren't on top of one another. However, if I put a fifth grain of sand on top of those four, then I've created a 'heap' (thus refuting the second premise). The heap I've made is still very small; however, it has all of the required structural features to be a 'heap'.

So, are you happy with that solution - or, are you going to throw those five grains of sand back in my face ?

[JHG]

So, are you happy with that solution - or, are you going to throw those five grains of sand back in my face ? 

Third option-donate them to the sea.

[Lurkerish Allsorts]

are you going to throw those five grains of sand back in my face

[Kyng]

are you going to throw those five grains of sand back in my face

I see more than five grains of sand in that picture ...

(Though, since they're not piled on top of one another, I still maintain that they're not a 'heap' )

[Lurkerish Allsorts]

are you going to throw those five grains of sand back in my face

I see more than five grains of sand in that picture ...

(Though, since they're not piled on top of one another, I still maintain that they're not a 'heap' )

They were a heap in the pocket and in the hand prior to them becoming projectiles in an act of self defense.

[Moonshroom]

So, are you happy with that solution - or, are you going to throw those five grains of sand back in my face ?

You know the answer, my friend

[Kyng]

Well, I had another idea to defeat the inductive argument at the start of the opening post:

Premise 1: A single grain of sand is not a heap.
Premise 2: Adding one grain of sand to a non-heap does not turn it into a heap.
Conclusion: There is no such thing as a heap of sand.

The thought I had was: in order for it to be a 'heap', you mustn't be able to immediately tell how many objects there are in it (that is, you mustn't be able to 'subitize' the number of objects). After all, if the 'heap' consisted of five grains of sand, and we could tell just from looking at it that there were five grains... we wouldn't call it "a heap of sand": we would just call it "five grains of sand" .

This way, we defeat premise #2: adding one grain of sand to a non-heap can turn it into a heap. The exact point at which this happens varies from person to person, but it's usually somewhere around five. (Granted, this means our definition of 'heap' will also be subjective - but, do we really need an objective definition of such an informal word as 'heap' ?)

[Jarkko]

he thought I had was: in order for it to be a 'heap', you mustn't be able to immediately tell how many objects there are in it (that is, you mustn't be able to 'subitize' the number of objects). After all, if the 'heap' consisted of five grains of sand, and we could tell just from looking at it that there were five grains... we wouldn't call it "a heap of sand": we would just call it "five grains of sand" .

I would disagree here. If you're dealing with larger objects in the range of what is "heapable" then you can get away with being able to "subitize." Those heaps may not necessarily be prototypical (as in, what most speakers immediately think of when they hear the word "heap") but the native speaker of any given language that has a separate term for "heap" would still call it thus. The four general semantic constants of a heap, I find (as always, there are exceptions amongst individual speakers), is that it must have more than three or four members (of course this depends on the size of the object being heaped ), the members must have a certain maximum size, and the heap must have a certain overall size and a certain degree of disorder/discombobulation amongst its members. The real problem is, by most speakers' standards, six grains of sand is hardly a heap, even if it isn't subitizable per se due to such a unit's overly small size.

The problem here isn't so much one of logic as it is of the lexical semantics of the word "heap." For starters, people don't generally think in terms of removing individual grains of sand from "heaps" thereof, because for something to be a heap generally means that it is of decent size. I could see something like a heap of apples or a heap of potatoes or a heap of rocks containing as few as four individual items simply because it's much easier to get a disordered pile that way. But with sand? You need a lot more than five grains to make a heap of it. This is the problem with vaguer quantity nouns like "heap," "pile," or even "bunch." While there is a ballpark idea of what can define them numerically, the exact number is never definite. I mean come on. A hundred potatoes can definitely be a heap. But a hundred grains of sand, while sounding like a lot, can probably fit into a pile about the diameter of a Canadian two-dollar coin. Hardly heap-worthy IMO. On the flip side, there are things that are too big to be "heapable," but where the line is drawn is rather arbitrary.

This is where logic goes out the window and you end up having to rely on native-speaker intuition, and even that is never precise in numerical value of the constituent object but rather in terms of the overall size and relative disorder of the whole, provided it isn't fewer than three or four items!

[Kyng]

Thanks for the linguistic insights, @ Jarkko !

The thing about "the objects themselves having a certain maximum size" isn't something I'd even considered. However, I think it'd be a maximum size relative to the heap, rather than a maximum size in absolute terms. After all, we can have a 'heap' of scrap cars (or at least a heap of scrap metal which includes scrap cars), but the individual cars will be bigger than the average heap of sand .

Another thing I might add is, when I think of a 'heap', I think of a certain shape (which peaks in the middle, and then gradually slopes downwards as it moves further out). It's not really possible to have this with four or five objects: you'd need at least dozens, possibly hundreds. Of course, the exact minimum number will depend on the size of the objects being 'heaped' (and the individual's perception), but it exists .

But going back to the argument in the opening post, could we also apply that to the word 'pile'? I suppose in that case, there are two ways to defeat the argument: a) defeat premise 1 by allowing one object to count as a 'pile' (albeit in a meaningless technical sense), or b) defeat premise two by defining 'pile' as "two or more objects stacked on top of each other in a disorganised fashion" (in which case, adding one object can turn a non-pile into a pile )

[Jarkko]

The thing about "pile" though is that there can be (relatively) organized piles, such as a pile of clothing. It's as if "pile" is the hypernym and words like "heap" (disorganized) or "stack" (organized and generally made up of things with rigid, stable bases) are the hyponyms.