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There is a glaring omission:
Premise 3: A heap is a large collection of grains of sand
Without this third premise, Premise 2 is only applicable once (is a large collection of grains of sand minus one grain still a large collection of grains?)
Of course, surely we can just do with one premise?
Premise 1: A large collection of grains of sand minus one grain is still a large collection of grains of sand.
A suggested definition for a heap: a combination of more than 2 objects that raise at least one of their members above the others. Thus four carefully arranged grains of sand of equal size could be heap (3 forming a tripod for a 4th - or two large grains supporting a third). Whereas a million grains of sand, with none on top of one another does not equal a heap.
~ender 2003-09-12 06:14:MST
The sorites isn't really about 'heaps' as such... but there is one philosopher -- I forget who just now -- who has half-seriously suggested that four is the least number of grains that can make a heap, just as ender described above.
This is part of the tradition of resolving the paradox by denying the 'tolerance' premise -- ie that there are no two elements a, b of a sorites series such that p(a) but not p(b). In other words, one resolution is to say that the definition of the predicate can be sharpened so that there is a definite cutover point. In this case, removing one grain from a heap of size 4 creates a non-heap, and so one of the steps in the paradoxical argument now fails. In other situations, however, this can be powerfully counterintuitive, IMHO. http://en.wikipedia.org/?title=Talk:Sorites_paradox&action=submit#
Ornette 16:58, 3 October 2005 (UTC)
I suggest that this article should instead be called The sorites paradox, and that Paradox of the heap should redirect here, rather than the other way around. In the philosophical literature in which this paradox is discussed, it is, I think, more commonly referred to as "the sorites paradox" than as "the paradox of the heap". Such a name change would be less prone to suggest that the philosophical problem which this article is about is a problem specific to heaps. Opinions? Matt 9 Nov. 2005
As long as whichever one redirects to the other, it doesn't matter, both names are used in the literature. Yesterdog 00:47, 16 May 2006 (UTC)
There are four articles in Wikipedia dealing with essentially one and the same philosophical topic:
Imprecise language,
Paradox of the heap,
Vagueness and
Continuum fallacy. (
Sorites paradox redirects to
Paradox of the heap.) I have done a little editing of the
Vagueness page, but really I think all four pages should be merged, or that at very least, they be rationalised to two pages, one a longer one on the philosophical problem of vagueness, and the other a quick summary of the sorites paradox with a link to the
vagueness page for a more in-depth discussion. What do people think? Matt 9 Nov. 2005
The section entitled "idiot's solution" did not make a lot of sense. I could not discern what the "solution" actually was that was being offered. Also note that Zeno's paradox of the frog jumping across the pond in jumps each 1/2 as big as the last etc. has very little to do with the paradox of the heap. They are different puzzles with different solutions. The new section seemed to suggest there was some sort of connection but left the reader with no idea what this connection actually consisted in. So I deleted the new section. User:Matt9090 14 Dec. 2005
This paradox reminded me of something I've heard often. It's often said that age x is essentially the same as age x±1 (ie. "What's the difference between 25 and 26?" as an argument for still doing something done at 25 at 26), which infers that the difference between x and x±1 (in terms of ages) is negligible. Obviously, this can be extended in both directions infinitely: by applying this principal recursively, the difference between any number and any other number can be considered negligible (in the case of ages, at least). Logically, of course, this doesn't make sense. I'm not sure if this is an appropriate concept to discuss in this article. -- Dvandersluis 20:41, 23 June 2006 (UTC)
Actually, I think this argument isn't very well-liked among philosophers. If you can find a lot who argue for it, feel free to put it up, but keep in mind one argument: 25 and 26 aren't essentially the same, they merely are so close that they're almost the same thing (much like how 1.1 rounds to 1.) However, 25 and 40 are more equivalent to 1 and 2.1. 204.95.23.122 20:50, 24 November 2006 (UTC)
The problem of describing resolutions to paradoxes is that one has to understand why something is a paradox to begin with and why a paradox is resolved by your solution.
From the page:
How is this not a resolution? We've resolved it by defining set boundaries.... it may not be a satisfactory resolution, but it would appear to give a simple though arbitrary answer. It is as much a resolution as the Setting a fixed boundary solution. It would seem there should be better phrasing or wording here as to what is meant. Maybe a sentence about how the the valued logic solution no better matches our intuition than the aforementioned solution, or something.
Root4(one) 04:16, 4 December 2006 (UTC)
This isn't a paradox at all. The so-called paradox is just an obscure look at the human mind's ability to establish abstract entities and associate them with words. A "heap" is a word used to describe one's relative perception of any arbitrary quantity of particles which resemble a familiar shape formed by gravity. You may as well ask how unattractive someone has to be in order to be considered ugly: obviously this is relative to the person perceiving.
The real problem comes from trying to define a heap by the number of particles in it, which has absolutely nothing to do with a heap. Who ever said anything about particle quantity? Why is that somehow implicit? The laptop I'm writing this on is not a laptop because of the number of buttons it has, it is considered a laptop because of its size, look, and feel. Trying to define something with irrelevant premises is obviously, always going to be impossible. It's like trying to measure the volume of a swimming pool with a yard stick.
With that said, it's not technically impossible to define a heap, it's just impractical. We don't know our brains well enough to examine the information they harbor. If we did, then we could determine a group of people's thresholds for what qualifies as a heap, which would consist of an abstract concept of its shape, volume, material, etc..., having varying degree of certainty between samples, since some people have obviously seen more heaps in their day and can better identify them. This could all be averaged to come up with a technical definition of the requirements that something must have in order to be classified as a heap. But what's important to realize is that those requirements would be massive with ranging sizes, shapes, colors, textures, and senses; not a simple range of particle quantities. If you must insist upon looking at it mathematically (numerically? whatever), then you can compare it to rounding a huge amount of data that we don't know how to interpret (mental abstraction) into a very simple integral form that we can interpret (particle quantity), and then trying to tell the difference between these integers after the conversion.-- RITZ 15:16, 27 January 2007 (UTC)
I think there is an error in the following quoted section of the article.
"On the face of it, there are three ways to avoid this conclusion. One may object to the first premise by denying that a large collection of grains makes a heap (or more generally, by denying that there are heaps). One may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may reject the conclusion by insisting that a heap of sand can be composed of just one grain."
The final method of rejecting the conclusion (by insisting that a heap of sand is composed of just one grain) is in fact an affirmation of the conclusion is it not? I've never read any philosophy at all, so I'm not editing this, but I assume this was just an oversight.
Some of the 'solutions' here are a bit dubious. The 'trivial solution' is perhaps misnamed. The 'Multi-valued logic' solution is badly written - it's not clear whether it's suggesting having three predicates for categorising putative heaps ('is a heap', 'is not a heap', and 'is neither a heap nor not a heap'), or that we should distinguish between three truth-values (true, false, neither true nor false). It also leaves out fuzzy logic, which has an infinite number of truth-values. And it uses an epistemic term ('unsure') for the middle category, suggesting it is an epistemic approach. The 'visual definition' and 'group consensus' solutions need citations; I'm not sure whether they represent original research, but they certainly aren't mainstream. It's hard to see how the visual definition solution is meant to work. The group consensus solution talk of 'probability' is very strange; (if 9 out of ten people thing a particular pile of sand is a heap, the chance of it's being a heap is 0.9?) - presumably it's just a variant on a fuzzy logic approach - once we get into the vague region, the truth value of the claim that the pile of sand is a heap is determined by group consensus. We also need a section on supervaluationism and the epistemic approach. If no one has any objections, I'll start making some changes in due course. 80.195.231.43 18:37, 5 February 2007 (UTC)
Consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:
* A large collection of grains of sand makes a heap. (Premise 1) * A large collection of grains of sand minus one grain makes a heap. (Premise 2)
Repeated applications of Premise 2 (each time starting with one less number of grains), eventually forces one to accept the conclusion that a heap may be composed by just one grain of sand.
And why is that exactly? How is one grain of sand "A large collection of grains"?
~dongray2
This is related, so I place it under here. It seems like something ought to be said about what "makes" means in the article I see possible definitions of make.
I'd add something, but I don't know what philosophers have said about this particular paradox, and I know that in this instance wording DOES matter. Root4( one) 17:23, 12 April 2007 (UTC)
I have removed the section on "visual definition". It didn't contain any citations and isn't part of standard philosophical discussion of this problem - not surprisingly, because it is specific to actual heaps of sand, instead of the abstract concept the paradox is really about. It was added by, and seems to be the original research of, a user whose only other editing activity consisted of adding a similarly questionable section to Monty Hall problem, getting into a flame war over it, and eventually being banned and restored. The content related to the actual paradox seems to be covered adequately in other sections. 129.97.79.144 15:13, 22 March 2007 (UTC)
I think "heap of sand" is a bad example, because "heap" is just a description of the shape. You could say a heap is a collection of objects, sufficiently numerous that they can't be counted in a single glance, and stacked together with more towards the centre and fewer towards the outside.
The number of items required for a heap is reasonably well defined, it must be about five or six, because you need to be able to stack them in a heap shape, and must not be able to count how many there are in a single glance. People can count up to about five or so items in a single glance and you need at least about three or four items to get the heap shape. You could easily have a heap of clothes or bowling balls with as few as five items. Even if you debate the second criterium I've offered you still can't drop below three and become a heap. No arrangement of two items has the appropriate shape necessary for a heap.
In fact the description of the shape of a heap seems like a better example of the paradox than the number of items in it. How flat does the heap have have to become before it's no longer a heap?
I'm not debating the paradox here, I'm just pointing out that the "heap of sand" example is not a particularly good one and we could probably find a better one. Cheers!
80.192.29.107 13:12, 25 August 2007 (UTC)
I just wanted to complement whoever's done the recent edits. A+ on clarity. Root4( one) 16:31, 7 January 2008 (UTC)
You know, it's not that hard of a problem. Things aren't defined by quantity, but instead by quality. A laptop is defined by its characteristics: two sections: a screen that moves for easier storage when put away, and the control and hardware section, containing the keyboard, mouse, and hardware. Similarly, a heap isn't defined by the number of things making up that heap but the relation of things to each other. A heap is a large pile of small things. Defining large and small is a bigger problem than this. -- Guugolpl0x ( talk) 19:29, 3 November 2008 (UTC)
It seems obvious that you cannot use something with an imprecise definition to precisely measure or categorise something else. It may be a problem, but it is certainly no paradox. -- 58.172.176.120 ( talk) 00:00, 19 June 2009 (UTC)
"1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2) Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand."
The last sentence stops one short. If 1 grain is still a heap, then according to Premise 2, that "heap (1 grain) of sand minus one grain is still a heap", so 1 grain minus 1 grain = 0 grains which is still a heap, by the above premises.
A "heap of sand" cannot logically be a heap of sand if it contains no sand, so, ipso facto, the wording of the premises must be flawed. 75.15.70.224 ( talk) 13:56, 18 September 2009 (UTC)
I somehat hastily removed the following comment from the article:
<!-- And, if you follow premise 2 again, composed of a negative number of grains of sand, possibly antimatter. -->
Dromioofephesus pointed out a logical consequence of accepting the conclusion here. Coming from a Wikipedian, it's WP:OR, of course, but this is the first time I've seen this particular argument. I'll be awarding a barnstar for anyone who can source it. Paradoctor ( talk) 11:08, 4 March 2010 (UTC)
I see the paradox as an attempt to interpret a situation using two languages, one is words the other is mathematics.
The pile of sand is described as a heap. Heap translated into mathematics might be 1,000,000 grains of sand. Repeatedly subtracting one grain a sand away from the heap until there is but one grain left and translating the result from mathematics to words, obviously there is a change in meaning.
Pauljalexander ( talk) 06:47, 23 April 2010 (UTC)
I don't have a reference for you but the basis for this idea came from many texts some include wikipedia pages on the [Philosophy_of_language], [Semiotics], [Linguistic_determinism] and [Theory_of_descriptions], writing of Alfred Korzybski and Bertrand Russell. Both of these authors describe mathematics as a language. Heap stands out as not being part of the mathematics language. Is the above argument not clear?
Pauljalexander ( talk) 04:20, 28 April 2010 (UTC)
In addition to the contribution above is the following. If half the heap is removed, a half heap would remain, removing half the heap again, would leave a quarter heap. Repeating this until there is but one grain would leave a fraction of the heap. This 'fraction of a heap' is a concept that needs little interpretation and no translation to or from mathematics is needed, because the concept of a heap is tied with the mathematical concept of division and fractions. A reference to quotation that Mathematics is a language, http://en.wikipedia.org/wiki/Josiah_Willard_Gibbs#Quotations
Pauljalexander ( talk) 06:24, 29 April 2010 (UTC)
It would seem the following would be true
1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2)
One grain of sand is a heap of sand. (Conclusion)
Hence Premise 2 can be rewritten:
A heap of sand minus a heap of sand is still a heap.
I think that clearly shows the fallacy of Premise 2. 4.253.119.18 ( talk) 14:46, 9 August 2010 (UTC)
Could someone knowledgeable flesh out the Supervaluationism resolution section? It only describes the "Pegasus likes licorice" statement, and not how it applies to the paradox. I don't understand the connection based on what's written. 67.171.199.11 ( talk) 08:40, 11 December 2010 (UTC).
What does "Pegasus fails to refer" mean? Seems like an incomplete sentence. That sort of shorthand is fine, perhaps, for the notational representations, but in the narrative format, it is confusing. What is it that Pegasus fails to refer to? A person? An actual person? What sort of failure? Thanks. 68.190.23.42 ( talk) 18:53, 13 November 2013 (UTC)
--
Why, clearly a heap of sand is only a heap if the heap-invariant somehow holds. Of course, in that case there must be some way we can order the grains. Ay, there's the rub, for what makes a grain of sand greater or lesser than another? Little which to me makes philosophical sense, but perhaps somebody else would have a better idea. Filburli ( talk) 01:50, 26 March 2013 (UTC)
This is not really my field, but there seems to be quite a bit of current philosophical discussion regarding sorites and Priest's inclosure schema, and whether or not there is a good case for sorites fitting within it cf. http://projecteuclid.org/euclid.ndjfl/1273002110 ( 20040302 ( talk) 10:11, 28 April 2014 (UTC))
It's not the purpose of an encyclopaedia to resolve Sorites (or any other paradox), but to describe it, it's history, explain why it is important, and possibly refer to different philosophers who have attempted to resolve it, (and their arguments).
But the talk page (and sometime the article) has been filled with attempts to resolve (or defuse, deflate, dissolve) Sorites. ( 20040302 ( talk) 10:19, 28 April 2014 (UTC))
User:Jochen Burghardt left me the following message regarding the [first] image:
Hyacinth ( talk) 22:38, 8 October 2014 (UTC)
I numbered the images to ease referring to them. In image 2, I still have the problem with "=". I think image 3 and its caption illustrates an important aspect of the paradox; but I'm not sure whether it is the main aspect (because I'm not really sure what the paradox' main aspect is...). I preferred your first version, without parantheses; in the actual version they are much bigger than the small heaps, which can be confusing. You could use instead e.g. red color for the dots ("...") to indicate that they are not heaps, but kind of meta symbols here. And the small heaps are hardly visible, perhaps you could decrease the image width by showing only 4 large and 4 small heaps, i.e. omitting the smallest 6 heaps. And you could clip almost all left and right white border. - Jochen Burghardt ( talk) 13:31, 9 October 2014 (UTC)
Picture 4 shows what I had in mind. However, it doesn't convince me either: contrary to what the caption claims, the small heaps still don't look more different in size than the large ones. - Jochen Burghardt ( talk) 06:57, 10 October 2014 (UTC)
New image, picture 5. Hyacinth ( talk) 23:41, 5 March 2015 (UTC)
I just want to say this article is fantastic. I learned a lot from it and it's very interesting, well written too. I would advise against making major changes with it. EggsInMyPockets ( talk) 17:55, 29 March 2017 (UTC)
i missed the meaning of the word from the article. an etymology would be useful to add, like here you can find: http://www.thefreedictionary.com/sorites
or at least the word sorites needs to turned int a link - there is a relevant article about what sorites is. 80.98.79.37 ( talk) 14:41, 27 May 2017 (UTC).
okay i put in the link though its not the first instance of the word, instead the text under the picture. 80.98.79.37 ( talk) 14:50, 27 May 2017 (UTC)
I know there have been various suggested resolutions before, but can't this be resolved by pointing out that a heap of sand is not actually a measure of quantity, and is basically another word for a pile of sand? or, indeed, mound of sand? it's not exactly something that needs a precise definition. (That, and it occurs to me that this paradox comes under the same category of paradox as Zeno's Paradox, and the one surrounding imprecision in coastline lengths. Namely that it is pretty much exclusively a purely philosophical paradox- if you try to argue the point in a non-philosophical setting, you will just be told to shut up.) Sstabeler ( talk) 14:17, 13 September 2017 (UTC)
@ Nikola Smolenski, Gregbard, Nbarth, Mesoderm, Charlotte Aryanne, Jeraphine Gryphon, Sandra lafave, and Raven530:
I suggest to "merge" the article " Continuum fallacy" (CF) into here. Actually, I believe this would amount almost to a deletion of "Continuum fallacy", as it doesn't provide much information that isn't already contained in Sorites paradox (SP). I suggest to keep the following parts.
The fallacy is the argument that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states.
Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomism, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.
For the purpose of the continuum fallacy, one assumes that there is in fact a continuum
They could be added to Sorites paradox#Variations where the continuum fallacy is already mentioned; some copyediting may be necessary.
None of the remaining (CF) examples (beard, sand) and references (Roberts, LaFave) deals with a continuum in the mathematical sense; the same applies to the Thouless reference already present here (SP). For this reason, I suggest to add a warning that in popular philosophy "continuum" is often abused for "too large to glance over". This would nicely fit as a footnote after "(or appears to be)" in the 2nd part. - Jochen Burghardt ( talk) 09:17, 18 April 2019 (UTC)
Of course, anybody who has grown one knows that some acquaintance eventually will ask, "Hey? Are you growing a beard?" If you answer, "Yes," Then that's a good candidate for the day when your non-beard became a beard. 74.111.96.172 ( talk) 21:04, 2 August 2021 (UTC)
The problem with naming it a 'paradox' is that it's only a paradox if it addressed by substance ontologies. It's not a paradox for many other ontologies - in fact, it is much more an argument used to demonstrate that the nature of a heap cannot be found in the substance of the heap. Cf., for instance, Tsongkhapa (or Candrakirti) using composition/aggregation as a means of establishing the lack of entityhood, essence, or (Buddhist 'self') in a heap (eg, the aggregated heaps of a person).
Therefore Sorites is used as an argument by non-essentialist thinkers, as well as thinkers who dispute objectivity.
It is an argument, not a paradox. 20040302 ( talk) 11:37, 15 August 2022 (UTC)
So, >removing a single grain does not cause a heap to become a non-heap But removing the second and so on grain would cause the heap to be a non-heap or able of becoming a non-heap at a certain boundary condition. 1, for example. Not 10000 (because why not 7776?), and no group consensus (i.e redefining the dictionary). 1 is because it is a line between singularity and plurality in natural math. It depends on whether the word 'heap' also preserves its original meaning or it is superceded in the given local scope! 46.96.149.182 ( talk) 10:00, 20 September 2022 (UTC)
I have encountered this paradox being called "Wang's paradox", but I'm not sure if that's just another term for it, a term for a variation of it, or perhaps even the name of an attempted resolution of it. I may report back later. Dingolover6969 ( talk) 14:24, 20 January 2023 (UTC)
Hi there, I came by this page again while discussing how Supervaluationism seemed valid for me, and when I noticed another answer I had thought after my first visit: the heap being either a pyramid of even three blocks of sand within a 2D arrangement, or perhaps a stereotypical pyramid arranged across three dimensions of space. The point is that a heap begins at the smallest scale of gravity-based and gravity-suspended stacking of sand grains. I think this is compatible with Supervaluationism, but still requires the discussion to happen. If it doesn't happen, folk are talking about theoretical sand grain, which are not spherical, 1-dimensional point coordinates. To know what I mean in literal exemplification, the international shortage of concrete sand is due to desert sand being too smooth. Ocean and river sand has more uniqueness and character that is stackable. - PhilienTaylor as FindingFilene (new account someday) Geometric solution is valid and needs persisting after citations of proof FindingFilene ( talk) 02:22, 29 April 2023 (UTC)
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The contents of the Continuum fallacy page were merged into Sorites paradox on 20 June 2020. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
There is a glaring omission:
Premise 3: A heap is a large collection of grains of sand
Without this third premise, Premise 2 is only applicable once (is a large collection of grains of sand minus one grain still a large collection of grains?)
Of course, surely we can just do with one premise?
Premise 1: A large collection of grains of sand minus one grain is still a large collection of grains of sand.
A suggested definition for a heap: a combination of more than 2 objects that raise at least one of their members above the others. Thus four carefully arranged grains of sand of equal size could be heap (3 forming a tripod for a 4th - or two large grains supporting a third). Whereas a million grains of sand, with none on top of one another does not equal a heap.
~ender 2003-09-12 06:14:MST
The sorites isn't really about 'heaps' as such... but there is one philosopher -- I forget who just now -- who has half-seriously suggested that four is the least number of grains that can make a heap, just as ender described above.
This is part of the tradition of resolving the paradox by denying the 'tolerance' premise -- ie that there are no two elements a, b of a sorites series such that p(a) but not p(b). In other words, one resolution is to say that the definition of the predicate can be sharpened so that there is a definite cutover point. In this case, removing one grain from a heap of size 4 creates a non-heap, and so one of the steps in the paradoxical argument now fails. In other situations, however, this can be powerfully counterintuitive, IMHO. http://en.wikipedia.org/?title=Talk:Sorites_paradox&action=submit#
Ornette 16:58, 3 October 2005 (UTC)
I suggest that this article should instead be called The sorites paradox, and that Paradox of the heap should redirect here, rather than the other way around. In the philosophical literature in which this paradox is discussed, it is, I think, more commonly referred to as "the sorites paradox" than as "the paradox of the heap". Such a name change would be less prone to suggest that the philosophical problem which this article is about is a problem specific to heaps. Opinions? Matt 9 Nov. 2005
As long as whichever one redirects to the other, it doesn't matter, both names are used in the literature. Yesterdog 00:47, 16 May 2006 (UTC)
There are four articles in Wikipedia dealing with essentially one and the same philosophical topic:
Imprecise language,
Paradox of the heap,
Vagueness and
Continuum fallacy. (
Sorites paradox redirects to
Paradox of the heap.) I have done a little editing of the
Vagueness page, but really I think all four pages should be merged, or that at very least, they be rationalised to two pages, one a longer one on the philosophical problem of vagueness, and the other a quick summary of the sorites paradox with a link to the
vagueness page for a more in-depth discussion. What do people think? Matt 9 Nov. 2005
The section entitled "idiot's solution" did not make a lot of sense. I could not discern what the "solution" actually was that was being offered. Also note that Zeno's paradox of the frog jumping across the pond in jumps each 1/2 as big as the last etc. has very little to do with the paradox of the heap. They are different puzzles with different solutions. The new section seemed to suggest there was some sort of connection but left the reader with no idea what this connection actually consisted in. So I deleted the new section. User:Matt9090 14 Dec. 2005
This paradox reminded me of something I've heard often. It's often said that age x is essentially the same as age x±1 (ie. "What's the difference between 25 and 26?" as an argument for still doing something done at 25 at 26), which infers that the difference between x and x±1 (in terms of ages) is negligible. Obviously, this can be extended in both directions infinitely: by applying this principal recursively, the difference between any number and any other number can be considered negligible (in the case of ages, at least). Logically, of course, this doesn't make sense. I'm not sure if this is an appropriate concept to discuss in this article. -- Dvandersluis 20:41, 23 June 2006 (UTC)
Actually, I think this argument isn't very well-liked among philosophers. If you can find a lot who argue for it, feel free to put it up, but keep in mind one argument: 25 and 26 aren't essentially the same, they merely are so close that they're almost the same thing (much like how 1.1 rounds to 1.) However, 25 and 40 are more equivalent to 1 and 2.1. 204.95.23.122 20:50, 24 November 2006 (UTC)
The problem of describing resolutions to paradoxes is that one has to understand why something is a paradox to begin with and why a paradox is resolved by your solution.
From the page:
How is this not a resolution? We've resolved it by defining set boundaries.... it may not be a satisfactory resolution, but it would appear to give a simple though arbitrary answer. It is as much a resolution as the Setting a fixed boundary solution. It would seem there should be better phrasing or wording here as to what is meant. Maybe a sentence about how the the valued logic solution no better matches our intuition than the aforementioned solution, or something.
Root4(one) 04:16, 4 December 2006 (UTC)
This isn't a paradox at all. The so-called paradox is just an obscure look at the human mind's ability to establish abstract entities and associate them with words. A "heap" is a word used to describe one's relative perception of any arbitrary quantity of particles which resemble a familiar shape formed by gravity. You may as well ask how unattractive someone has to be in order to be considered ugly: obviously this is relative to the person perceiving.
The real problem comes from trying to define a heap by the number of particles in it, which has absolutely nothing to do with a heap. Who ever said anything about particle quantity? Why is that somehow implicit? The laptop I'm writing this on is not a laptop because of the number of buttons it has, it is considered a laptop because of its size, look, and feel. Trying to define something with irrelevant premises is obviously, always going to be impossible. It's like trying to measure the volume of a swimming pool with a yard stick.
With that said, it's not technically impossible to define a heap, it's just impractical. We don't know our brains well enough to examine the information they harbor. If we did, then we could determine a group of people's thresholds for what qualifies as a heap, which would consist of an abstract concept of its shape, volume, material, etc..., having varying degree of certainty between samples, since some people have obviously seen more heaps in their day and can better identify them. This could all be averaged to come up with a technical definition of the requirements that something must have in order to be classified as a heap. But what's important to realize is that those requirements would be massive with ranging sizes, shapes, colors, textures, and senses; not a simple range of particle quantities. If you must insist upon looking at it mathematically (numerically? whatever), then you can compare it to rounding a huge amount of data that we don't know how to interpret (mental abstraction) into a very simple integral form that we can interpret (particle quantity), and then trying to tell the difference between these integers after the conversion.-- RITZ 15:16, 27 January 2007 (UTC)
I think there is an error in the following quoted section of the article.
"On the face of it, there are three ways to avoid this conclusion. One may object to the first premise by denying that a large collection of grains makes a heap (or more generally, by denying that there are heaps). One may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may reject the conclusion by insisting that a heap of sand can be composed of just one grain."
The final method of rejecting the conclusion (by insisting that a heap of sand is composed of just one grain) is in fact an affirmation of the conclusion is it not? I've never read any philosophy at all, so I'm not editing this, but I assume this was just an oversight.
Some of the 'solutions' here are a bit dubious. The 'trivial solution' is perhaps misnamed. The 'Multi-valued logic' solution is badly written - it's not clear whether it's suggesting having three predicates for categorising putative heaps ('is a heap', 'is not a heap', and 'is neither a heap nor not a heap'), or that we should distinguish between three truth-values (true, false, neither true nor false). It also leaves out fuzzy logic, which has an infinite number of truth-values. And it uses an epistemic term ('unsure') for the middle category, suggesting it is an epistemic approach. The 'visual definition' and 'group consensus' solutions need citations; I'm not sure whether they represent original research, but they certainly aren't mainstream. It's hard to see how the visual definition solution is meant to work. The group consensus solution talk of 'probability' is very strange; (if 9 out of ten people thing a particular pile of sand is a heap, the chance of it's being a heap is 0.9?) - presumably it's just a variant on a fuzzy logic approach - once we get into the vague region, the truth value of the claim that the pile of sand is a heap is determined by group consensus. We also need a section on supervaluationism and the epistemic approach. If no one has any objections, I'll start making some changes in due course. 80.195.231.43 18:37, 5 February 2007 (UTC)
Consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:
* A large collection of grains of sand makes a heap. (Premise 1) * A large collection of grains of sand minus one grain makes a heap. (Premise 2)
Repeated applications of Premise 2 (each time starting with one less number of grains), eventually forces one to accept the conclusion that a heap may be composed by just one grain of sand.
And why is that exactly? How is one grain of sand "A large collection of grains"?
~dongray2
This is related, so I place it under here. It seems like something ought to be said about what "makes" means in the article I see possible definitions of make.
I'd add something, but I don't know what philosophers have said about this particular paradox, and I know that in this instance wording DOES matter. Root4( one) 17:23, 12 April 2007 (UTC)
I have removed the section on "visual definition". It didn't contain any citations and isn't part of standard philosophical discussion of this problem - not surprisingly, because it is specific to actual heaps of sand, instead of the abstract concept the paradox is really about. It was added by, and seems to be the original research of, a user whose only other editing activity consisted of adding a similarly questionable section to Monty Hall problem, getting into a flame war over it, and eventually being banned and restored. The content related to the actual paradox seems to be covered adequately in other sections. 129.97.79.144 15:13, 22 March 2007 (UTC)
I think "heap of sand" is a bad example, because "heap" is just a description of the shape. You could say a heap is a collection of objects, sufficiently numerous that they can't be counted in a single glance, and stacked together with more towards the centre and fewer towards the outside.
The number of items required for a heap is reasonably well defined, it must be about five or six, because you need to be able to stack them in a heap shape, and must not be able to count how many there are in a single glance. People can count up to about five or so items in a single glance and you need at least about three or four items to get the heap shape. You could easily have a heap of clothes or bowling balls with as few as five items. Even if you debate the second criterium I've offered you still can't drop below three and become a heap. No arrangement of two items has the appropriate shape necessary for a heap.
In fact the description of the shape of a heap seems like a better example of the paradox than the number of items in it. How flat does the heap have have to become before it's no longer a heap?
I'm not debating the paradox here, I'm just pointing out that the "heap of sand" example is not a particularly good one and we could probably find a better one. Cheers!
80.192.29.107 13:12, 25 August 2007 (UTC)
I just wanted to complement whoever's done the recent edits. A+ on clarity. Root4( one) 16:31, 7 January 2008 (UTC)
You know, it's not that hard of a problem. Things aren't defined by quantity, but instead by quality. A laptop is defined by its characteristics: two sections: a screen that moves for easier storage when put away, and the control and hardware section, containing the keyboard, mouse, and hardware. Similarly, a heap isn't defined by the number of things making up that heap but the relation of things to each other. A heap is a large pile of small things. Defining large and small is a bigger problem than this. -- Guugolpl0x ( talk) 19:29, 3 November 2008 (UTC)
It seems obvious that you cannot use something with an imprecise definition to precisely measure or categorise something else. It may be a problem, but it is certainly no paradox. -- 58.172.176.120 ( talk) 00:00, 19 June 2009 (UTC)
"1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2) Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand."
The last sentence stops one short. If 1 grain is still a heap, then according to Premise 2, that "heap (1 grain) of sand minus one grain is still a heap", so 1 grain minus 1 grain = 0 grains which is still a heap, by the above premises.
A "heap of sand" cannot logically be a heap of sand if it contains no sand, so, ipso facto, the wording of the premises must be flawed. 75.15.70.224 ( talk) 13:56, 18 September 2009 (UTC)
I somehat hastily removed the following comment from the article:
<!-- And, if you follow premise 2 again, composed of a negative number of grains of sand, possibly antimatter. -->
Dromioofephesus pointed out a logical consequence of accepting the conclusion here. Coming from a Wikipedian, it's WP:OR, of course, but this is the first time I've seen this particular argument. I'll be awarding a barnstar for anyone who can source it. Paradoctor ( talk) 11:08, 4 March 2010 (UTC)
I see the paradox as an attempt to interpret a situation using two languages, one is words the other is mathematics.
The pile of sand is described as a heap. Heap translated into mathematics might be 1,000,000 grains of sand. Repeatedly subtracting one grain a sand away from the heap until there is but one grain left and translating the result from mathematics to words, obviously there is a change in meaning.
Pauljalexander ( talk) 06:47, 23 April 2010 (UTC)
I don't have a reference for you but the basis for this idea came from many texts some include wikipedia pages on the [Philosophy_of_language], [Semiotics], [Linguistic_determinism] and [Theory_of_descriptions], writing of Alfred Korzybski and Bertrand Russell. Both of these authors describe mathematics as a language. Heap stands out as not being part of the mathematics language. Is the above argument not clear?
Pauljalexander ( talk) 04:20, 28 April 2010 (UTC)
In addition to the contribution above is the following. If half the heap is removed, a half heap would remain, removing half the heap again, would leave a quarter heap. Repeating this until there is but one grain would leave a fraction of the heap. This 'fraction of a heap' is a concept that needs little interpretation and no translation to or from mathematics is needed, because the concept of a heap is tied with the mathematical concept of division and fractions. A reference to quotation that Mathematics is a language, http://en.wikipedia.org/wiki/Josiah_Willard_Gibbs#Quotations
Pauljalexander ( talk) 06:24, 29 April 2010 (UTC)
It would seem the following would be true
1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2)
One grain of sand is a heap of sand. (Conclusion)
Hence Premise 2 can be rewritten:
A heap of sand minus a heap of sand is still a heap.
I think that clearly shows the fallacy of Premise 2. 4.253.119.18 ( talk) 14:46, 9 August 2010 (UTC)
Could someone knowledgeable flesh out the Supervaluationism resolution section? It only describes the "Pegasus likes licorice" statement, and not how it applies to the paradox. I don't understand the connection based on what's written. 67.171.199.11 ( talk) 08:40, 11 December 2010 (UTC).
What does "Pegasus fails to refer" mean? Seems like an incomplete sentence. That sort of shorthand is fine, perhaps, for the notational representations, but in the narrative format, it is confusing. What is it that Pegasus fails to refer to? A person? An actual person? What sort of failure? Thanks. 68.190.23.42 ( talk) 18:53, 13 November 2013 (UTC)
--
Why, clearly a heap of sand is only a heap if the heap-invariant somehow holds. Of course, in that case there must be some way we can order the grains. Ay, there's the rub, for what makes a grain of sand greater or lesser than another? Little which to me makes philosophical sense, but perhaps somebody else would have a better idea. Filburli ( talk) 01:50, 26 March 2013 (UTC)
This is not really my field, but there seems to be quite a bit of current philosophical discussion regarding sorites and Priest's inclosure schema, and whether or not there is a good case for sorites fitting within it cf. http://projecteuclid.org/euclid.ndjfl/1273002110 ( 20040302 ( talk) 10:11, 28 April 2014 (UTC))
It's not the purpose of an encyclopaedia to resolve Sorites (or any other paradox), but to describe it, it's history, explain why it is important, and possibly refer to different philosophers who have attempted to resolve it, (and their arguments).
But the talk page (and sometime the article) has been filled with attempts to resolve (or defuse, deflate, dissolve) Sorites. ( 20040302 ( talk) 10:19, 28 April 2014 (UTC))
User:Jochen Burghardt left me the following message regarding the [first] image:
Hyacinth ( talk) 22:38, 8 October 2014 (UTC)
I numbered the images to ease referring to them. In image 2, I still have the problem with "=". I think image 3 and its caption illustrates an important aspect of the paradox; but I'm not sure whether it is the main aspect (because I'm not really sure what the paradox' main aspect is...). I preferred your first version, without parantheses; in the actual version they are much bigger than the small heaps, which can be confusing. You could use instead e.g. red color for the dots ("...") to indicate that they are not heaps, but kind of meta symbols here. And the small heaps are hardly visible, perhaps you could decrease the image width by showing only 4 large and 4 small heaps, i.e. omitting the smallest 6 heaps. And you could clip almost all left and right white border. - Jochen Burghardt ( talk) 13:31, 9 October 2014 (UTC)
Picture 4 shows what I had in mind. However, it doesn't convince me either: contrary to what the caption claims, the small heaps still don't look more different in size than the large ones. - Jochen Burghardt ( talk) 06:57, 10 October 2014 (UTC)
New image, picture 5. Hyacinth ( talk) 23:41, 5 March 2015 (UTC)
I just want to say this article is fantastic. I learned a lot from it and it's very interesting, well written too. I would advise against making major changes with it. EggsInMyPockets ( talk) 17:55, 29 March 2017 (UTC)
i missed the meaning of the word from the article. an etymology would be useful to add, like here you can find: http://www.thefreedictionary.com/sorites
or at least the word sorites needs to turned int a link - there is a relevant article about what sorites is. 80.98.79.37 ( talk) 14:41, 27 May 2017 (UTC).
okay i put in the link though its not the first instance of the word, instead the text under the picture. 80.98.79.37 ( talk) 14:50, 27 May 2017 (UTC)
I know there have been various suggested resolutions before, but can't this be resolved by pointing out that a heap of sand is not actually a measure of quantity, and is basically another word for a pile of sand? or, indeed, mound of sand? it's not exactly something that needs a precise definition. (That, and it occurs to me that this paradox comes under the same category of paradox as Zeno's Paradox, and the one surrounding imprecision in coastline lengths. Namely that it is pretty much exclusively a purely philosophical paradox- if you try to argue the point in a non-philosophical setting, you will just be told to shut up.) Sstabeler ( talk) 14:17, 13 September 2017 (UTC)
@ Nikola Smolenski, Gregbard, Nbarth, Mesoderm, Charlotte Aryanne, Jeraphine Gryphon, Sandra lafave, and Raven530:
I suggest to "merge" the article " Continuum fallacy" (CF) into here. Actually, I believe this would amount almost to a deletion of "Continuum fallacy", as it doesn't provide much information that isn't already contained in Sorites paradox (SP). I suggest to keep the following parts.
The fallacy is the argument that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states.
Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomism, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.
For the purpose of the continuum fallacy, one assumes that there is in fact a continuum
They could be added to Sorites paradox#Variations where the continuum fallacy is already mentioned; some copyediting may be necessary.
None of the remaining (CF) examples (beard, sand) and references (Roberts, LaFave) deals with a continuum in the mathematical sense; the same applies to the Thouless reference already present here (SP). For this reason, I suggest to add a warning that in popular philosophy "continuum" is often abused for "too large to glance over". This would nicely fit as a footnote after "(or appears to be)" in the 2nd part. - Jochen Burghardt ( talk) 09:17, 18 April 2019 (UTC)
Of course, anybody who has grown one knows that some acquaintance eventually will ask, "Hey? Are you growing a beard?" If you answer, "Yes," Then that's a good candidate for the day when your non-beard became a beard. 74.111.96.172 ( talk) 21:04, 2 August 2021 (UTC)
The problem with naming it a 'paradox' is that it's only a paradox if it addressed by substance ontologies. It's not a paradox for many other ontologies - in fact, it is much more an argument used to demonstrate that the nature of a heap cannot be found in the substance of the heap. Cf., for instance, Tsongkhapa (or Candrakirti) using composition/aggregation as a means of establishing the lack of entityhood, essence, or (Buddhist 'self') in a heap (eg, the aggregated heaps of a person).
Therefore Sorites is used as an argument by non-essentialist thinkers, as well as thinkers who dispute objectivity.
It is an argument, not a paradox. 20040302 ( talk) 11:37, 15 August 2022 (UTC)
So, >removing a single grain does not cause a heap to become a non-heap But removing the second and so on grain would cause the heap to be a non-heap or able of becoming a non-heap at a certain boundary condition. 1, for example. Not 10000 (because why not 7776?), and no group consensus (i.e redefining the dictionary). 1 is because it is a line between singularity and plurality in natural math. It depends on whether the word 'heap' also preserves its original meaning or it is superceded in the given local scope! 46.96.149.182 ( talk) 10:00, 20 September 2022 (UTC)
I have encountered this paradox being called "Wang's paradox", but I'm not sure if that's just another term for it, a term for a variation of it, or perhaps even the name of an attempted resolution of it. I may report back later. Dingolover6969 ( talk) 14:24, 20 January 2023 (UTC)
Hi there, I came by this page again while discussing how Supervaluationism seemed valid for me, and when I noticed another answer I had thought after my first visit: the heap being either a pyramid of even three blocks of sand within a 2D arrangement, or perhaps a stereotypical pyramid arranged across three dimensions of space. The point is that a heap begins at the smallest scale of gravity-based and gravity-suspended stacking of sand grains. I think this is compatible with Supervaluationism, but still requires the discussion to happen. If it doesn't happen, folk are talking about theoretical sand grain, which are not spherical, 1-dimensional point coordinates. To know what I mean in literal exemplification, the international shortage of concrete sand is due to desert sand being too smooth. Ocean and river sand has more uniqueness and character that is stackable. - PhilienTaylor as FindingFilene (new account someday) Geometric solution is valid and needs persisting after citations of proof FindingFilene ( talk) 02:22, 29 April 2023 (UTC)