The Paradox of Sand Heaps: Is It True or False?

[CPL.Luke]

so the paradox goes like this.

if n grains of sand is a heap, then n-1 grains of sand is a heap.

with the justification being that you can't destroy a heap by removing a single grain of sand.

is it just me or does this sound like bullcrap, as this would imply 0 grains is a heap and in turn -1 grains of sand is a heap doesn't this seem like we either have to accept that a heap is defined by that statement, or take it as a theorem of a heap hich can easily be shown to be false.

I spent part of a Q&A session with a lecturer trying to convince him that there is no paradox here, and he did not budge. Any thoughts?

 

[christianjb]

Yes, it's rubbish reasoning.

Similar thinking is used to deny evolution, because

If the offspring of species X belongs to species X

Then all descendants of X remain members of species X

Therefore evolution cannot produce a change in species.

 

[baywax]

Oxford Thesaurus

heap

noun
1 a heap of boxes pile, stack, mound, mountain, mass, quantity, load, lot, jumble; collection, accumulation, assemblage, store, hoard.

2 informal : we have heaps of room | a heap of troubles a lot of, a fair amount of, much, plenty of, a good deal of, a great deal of, an abundance of, a wealth of, a profusion of; (a great) many, a large number of, numerous, scores of; informal hundreds of, thousands of, millions of, a load of, loads of, a pile of, piles of, oodles of, stacks of, lots of, masses of, scads of, reams of, oceans of, miles of, tons of, zillions of. verb she heaped logs on the fire pile up, pile, stack up, stack, make a mound of; assemble, collect.

PHRASES heap on/upon they heaped praise on her shower on, lavish on, load on; bestow on, confer on, give, grant, vouchsafe, favor with.

Origin

Old English heap (noun), heapian (verb), of Germanic origin; related to Dutch hoop and German Haufen.

The paradox is that the word heap can almost mean what you want it to. These definitions seem to point to heap being a collection of things piled up and above a horizontal plane. I think this rules out -1 grain of sand or 0 grains of sand. At less than 2 grains of sand a heap ceases to exist. However, one could argue that with one grain of sand, there is a heap of molecules. This would hold until there was only one molecule. At this point it could be construed that there is a heap of atoms but, there would have to be a reference such as a horizontal plane to define the heap of atoms. In this case the atoms would hard to distinguish from the atoms of the horizontal plane and there would be no heap. If only laundry was as simple as that.

 

[Gelsamel Epsilon]

The paradox is demonstrating exactly that Luke. That there is a problem in drawing a line where a heap is no longer a heap. And also, the nature of language.

A heap is still a heap if you take one little sand from the heap of sand, let's keep taking one grain of sand, where do you draw the line where it suddenly stops being a heap?

 

[Wizardsblade]

Interesting, so someone would make that argument to say that someone else is imposing a personal opinion to the conclusion of a set of facts, even though the conclusion may not be as clear/exclusive as the "someone else" thinks?

So the paradox as an argument is supose to show "this agrument has an opinion implied in it"?

 

[Crosson]

The paradox is: We all agree that some things are heaps (not an opinion) and yet we do not agree on what a heap is. Same problem the astronomers faced with pluto.

 

[CPL.Luke]

but that isn;t a paradox, that just means we havn't created a proper definition yet. and clearly since we all aee that a heap is not 0 grais of sand or -1 grains of sand the proposed definition is not valid.

it should be noted however that you can replace heap with tallnes, or a person being rich.

however the entire thing seems an exercise in avoiding the invalidation of he theorem, I tried to show the philosopher that mathematical induction quickly deomnstrates that the sentance can't define a heap as the set of all n's which would define a heap is not closed under the opperation defined in the sentance.

however because the philosopher had never seen letters in his math before, I think he had trouble understanding what I was doing. The same thing happened when he was trying to talk about the solution to zeno's paradox.

 

[Rade]

One way to falsify the paradox is to "set a fixed boundary" to define what it means to say a heap (= a concept) of similar things (such as sand grains) exists. For example, would you say the following represents a heap ?

0
|
0
|
0
|
0
|
0​

I would say no because, while there is property of height, there is no property of extension (diameter).

How about this ?

0
|\
0 0

Now, perhaps not much of a heap, but I logically see no good reason why such a structure would not represent the smallest type of heap (as a concept) possible since we find both height and diameter and stability with 1 thing supported by 2. Therefore, when a heap has > 3 of some thing, one thing can be removed and the concept heap maintained, but, by definition, it not possible to have a heap of < 3 things. So the fixed boundary between heap and non-heap is mathematically between the numbers 2 & 3 and the sorites paradox solved.

 

[DyslexicHobo]

Not sure if this has been brought up, but the way I like to rebut those arguments is saying "well, if one cannot make a difference, then at what point DOES it make a difference?"

If they have an ambiguous answer to that, you can say "does two grains destroy a heap?", "does three grains destroy a heap?" etc.

 

[rocks]

You can't subtract a quantity from an adjective. Therefore the statement is not a paradox. It's just jibberish, no?

 

[Jimmy Snyder]

if n grains of sand is a heap, then n-1 grains of sand is a heap.

This statement is false. Therefore there is no paradox.

 

[LightbulbSun]

A heap can begin when you have at the very least two items stacked together. One item would not be defined as a heap. This argument seems to be more of a semantics argument rather than a paradox.

 

[out of whack]

if n grains of sand is a heap, then n-1 grains of sand is a heap.

with the justification being that you can't destroy a heap by removing a single grain of sand.

The problem is that the first claim is not proven true. The "justification" does not provide any proof of it, it only repeats it in different words: if n is a lot, then n-1 is a lot because you cannot destroy a lot by subtracting one. The claim is still baseless.

 

[Moonbear]

so the paradox goes like this.

if n grains of sand is a heap, then n-1 grains of sand is a heap.

with the justification being that you can't destroy a heap by removing a single grain of sand.

is it just me or does this sound like bullcrap, as this would imply 0 grains is a heap and in turn -1 grains of sand is a heap doesn't this seem like we either have to accept that a heap is defined by that statement, or take it as a theorem of a heap hich can easily be shown to be false.

I spent part of a Q&A session with a lecturer trying to convince him that there is no paradox here, and he did not budge. Any thoughts?

It's a paradox because it uses circular reasoning. It is attempting to define the term heap based on the definition of heap. If you define a heap as some number of particles n, then that only holds true if n-1 is also a heap. But, how would you know if n-1 is a heap? Only if (n-1)-1 is a heap. In the end, you are correct, you will get to some point where n-1-...-1-1 is no longer a heap, which would mean n-1-...-1 is not a heap, all the way back to (n-1)-1 and n-1 and n are not a heap, even though you started out satisfied that n and n-1 were a heap. That is the paradox, and an illustration of why it can be dangerous to attempt to quantify qualitative measures, as well of how ambiguous a qualitative measure is.

So, it's not about choosing any arbitrary n and thus any arbitrary n-1 and determining if those two define a heap, but looking at the logical progression of what happens if you continue with multiple iterations of that statement that shows the paradox.

 

[JonF]

Making a general claim about a non-well defined concept (heap) and then through slight of hand PMI getting a result contrary to a well defined version of the same concept?

Nothing illogical about that. Just not really useful.