A single grain of sand is not a heap; that\'s obvious. A heap is a collection of things, you need several things to make it up. Two grains of sand isn\'t a heap either; a heap is a collection of several things.

The concept of a heap is fuzzy, though; there’s no precise number that marks the difference between heaps and non-heaps. Defining precisely how many things one needs in order to have a heap is therefore impossible. This is what gives rise to the paradox of the heap (also called the “Sorites paradox”, Sorites being the Greek word for heap).

Suppose that we have a collection of a million grains of sand. That is absolutely, definitely, undeniably a heap.

Because there is no precise number that separates heaps from non-heaps, removing a single grain of sand from a heap will never turn it into a non-heap. If you have a heap of sand, and you take away a single grain, then you still have a heap.

If you have a heap of a million grains of sand, though, and repeatedly take away a single grain of sand, doing so 999,999 times, then what have you got? Taking away a single grain of sand cannot turn a heap into a non-heap. What you have at the end of this process, then, is a heap.

What you have at end of this process, though, is a single grain of sand, and, as we said at the beginning, a single grain of sand is obviously not a heap.

Well... the way I would view this is that a \"heap\" is judged in proportion to other groups of objects. For example, if you take a hundred small pebbles, and place it in front of someone who has seen nothing bigger than grains of sand, then that is, to them, a heap. However, if you take the same hundred small pebbles and put it in front of a quarry worker who sees mounds comprised of millions of pebbles on a daily basis, that would not be considered a heap to him or her.

In short, it is all proportional and must be evaluated on an independent basis. There is a point, based on appearances, where a heap becomes a non-heap. Where this point lies is determined by your background on the subject, and so is different for every person; hence, though there is not an official point, there is a non-standardized one based on the subject\'s experiences.

I look at the starting point of a heap this way: a structure must have a foundation. Therefore, since even on a near-microscopic level, there is no way to balance two grains of sand, this is illogical. On a 2-D plane, 3 2ould work, but otherwise there is little chance of this. However, with four on a 3-D plane, 3 can provide a triangular base, and the 4th can be balanced atop the others. Therefore, although not a heap to us, this certainly would appear as a heap to extremely small lifeforms. therefore, the preception of a heap is largely dependent on how supportive the components of said heap are and the relative size of the subject observing it.

So really, unless on a two dimensional plane, isn\'t 4 the minimum number for a heap?

That doesn\'t make sense. How is one big stone a heap? I don\'t see any logic behind that. And how must a heap be three dimensional? It seems to me that if we see a filled triangle on a 2-D plane, there\'s no reason it can\'t be a heap. It doesn\'t have to be if there\'s no reason it should be, but there\'s no reaon it can\'t be. It might represent a pointed stone atop a few rectangular stones.

[Edited on 11-8-2005 by AimMan v2.5]

God, please don\'t disagree with someone just for the sake of argument. Aim has a pretty good point. Allow me to attempt a diagram:

2D: one stone
[]

-A single stone. Not a heap.

2D: two stones
[]
[]

-One stone on top of another stone. A 2D heap.

3D: three stones
--[]
-[][] [][]
(top) (side)

-Support stones all on the same level. Not a heap.

3D: four stones

--[]
-[][]
(top(fourh afixiated in center))

--[]
-[][]
(side)

-A 3D piramid is formed by the forth stone. A heap.

Just imagine the stones as being about one ton of granite each. Large enough for for to be considered a heap to any human.

This paradox didn\'t debunk heaps, it just pointed out that there isn\'t a deffinition of one.

Well... this one\'s simple!

\"There is no heap!\"



Yeah, I took the cheap way out. So sue me.

The paradox is not questioning the existance of heaps. It is questioning the deffinition of what it takes for a pile to become a heap.

I will now explain why I said that. It\'s actually quite simple to understand with the brains you people got.

What Mega X stated in his first post here is obviously true. So the reverse is as well.

If putting one grain sand with another one, it´s not a heap. So if I´d do that 999,999 times and get 1,000,000 grains of sand together, I still don´t have a heap.

My point now is, that either there are no heaps, or there are no non-heaps. Go figure.

So what you are saying is, you don\'t believe the term \"heap\" can be used to define any given amount of objects? I guess that\'s fine, but it would still help if you said why 1,000,000 objects piled up can\'t be a heap.

Like Mega X said, taking away one grain of sand from a pile of one million won\'t turn it into a non-heap. Doing so again won\'t as well. So if you do it 999,999 times you still got a heap. But what do you also have then? Just one grain of sand. Which cannot be heap. But because taking away one grain of sand cannot turn a heap into a non-heap, that one grain of sand must be a heap.

In reverse, putting one grain of sand with another won\'t turn it into a heap. Doing so again up to the time you have a pile of one million grains of sand, what do you have? You\'d say a heap, but because adding one grain of sand cannot turn a non-heap into a heap, the pile of one million grains of sand is still a non-heap.

Therefor, I state that there are no heaps OR non-heaps.

Your logic isn\'t making sense. In the first paragraph, you say that taking away 999,999 from 1,000,000 from a heap does not make it a non-heap. I see no reason why it wouldn\'t, seeing as it would now simply be one unit. 999,998 would have a similar effect, because there is no plane on which such a figure could exist. Taking one grain of sand from a heap of sand can make it a non-heap, depending on how many are currently in the pile, if you were listening to what I said earlier about it taking a minimum of 3 items to form a heap.

I\'m not quite sure what you\'re trying to say, and forgive me if I\'m assuming, but it seems like what you\'re saying is based purely on the information already given in the paradox which supposedly makes it a paradox. However, I believe it has been solved as not being such.

Aim, as Mega X stated, the line between heaps and non-heaps is very vague. There is no real line were heaps turn into non-heaps and the other way around. If you take that in mind, what I said, and what Mega X said, will probably make that much more sense.

I think that you always have a heap.

Chances are, as you removed grains of sand, you did something with them. And you moved them into another heap so that you could keep track of how many you removed.

However, yes it\'s mostly just perseption.
Heaps are a non discript means of measuring, not a weight or volume, which is why we no longer use them as means of selling grain.:lol:

However, I give the nod to Aimman for his observation. Well put.

ok heres the way it is,a heap is more than one object fumbled together
example:a heap of luandry, or a garbage heap:zzz:

ok heres the way it is,a heap is more than one object fumbled together
example:a heap of luandry, or a garbage heap:zzz:

Dude, don\'t double post.

I\'m being attacked in a post directed towards somebody else, and I can\'t even think of a counter. T_T

I\'ll just stay over here and be an indirect object. I can\'t think of any sort of response whatsoever. I guess I\'m beaten, but I\'d like to say that I understand the problem now.

It makes sense now, but it seems I was just thinking about it too hard. It\'s a pretty obvious paradox that could arise from the definition of any term for a nonspecific number of objects. I just seem to be trying to make sense of something that\'s already as plain as a word\'s definition. Like when I used to ask questions in math class.:lol:

... Lord! I can\'t think of a single smart thing to say. But I know there\'s something I\'m meaning to get at, if I could just clear my head.

A heap is a relative term, God. Since when does a \"heap\" not describe a tower?

A person who has never seen newspapers before to know they were common might say \"That\'s a heap of newspapers\" if there were, say, 20 newspapers. Regardless of whether they were carelessly thrown together, in a vertical stack folded perfectly, or lain tiling the floor.
\"Heap\" is a term that is entirely relative to the situation in it\'s meaning. If I was asked to describe a stack of newspapers folded perfectly and labeled it a \"heap\" I would understand being counted wrong, but we can\'t define a \"heap\" in all cases to be a stack in that particular sense.
Heap
1 : a collection of things thrown one on another : PILE
2 : a great number or large quantity : LOT
If we\'re referring to the first definition or the second, 2 technically fits the bill depending on perception.
By 2-D plane, Aim was referring to stones lined up or stacked without being placed behind another or in front of another, not to the stones beign 2-D.

..I missed out on all this pointless psychobabble? Why didn\'t anyone tell me?