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Suppose you have a heap of sand. You remove one grain. Is there still a heap? You remove another, until you get down to a heap with three grains, a heap with two grains, a heap with one grain, and finally a heap with no grains at all. But that’s ridiculous. There must be something wrong? Does removing one grain turn a heap into not-a-heap? But that's ridiculous too. How can one grain make so much difference?

Is there a solution to this problem?

Mauro ALLEGRANZA
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ActualCry
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    See Sorites Paradox. – Mauro ALLEGRANZA Feb 21 '23 at 09:53
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    Heap is a vague concept, so what is or is not a heap shifts with context. Intuitions about the effect of one grain are localized enough to be plausible without providing the context. The paradox then plays on subtle equivocation by assuming a fixed concept throughout the whole process. – Conifold Feb 21 '23 at 12:25
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    How is that any kind of 'paradox,' please?

    How is this really about anything other than the observer's ability to distinguish 'one' from 'some' or 'many'?

     – Robbie Goodwin Feb 21 '23 at 22:25
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    @RobbieGoodwin Roughly, a 'paradox' is a set of seemingly true statements which result in a contradiction. "If you have a heap of sand, and take only a single grain of sand from it, it's still a heap." and "A single grain of sand is not a heap." lead to a contradiction (because a finite heap can be reduced to a single grain by repeated removal of single grains). -- The naive resolution is to deny the first, however a counter example (where you have a heap, remove a grain, and then no longer have a heap) resists construction, which makes demonstrating the falsity difficult. – R.M. Feb 21 '23 at 22:55
  • @R.M. That I have never before seen 'seemingly' confuse the definition of 'paradox' matters not.

    What we have here is by no means a set of statements resulting in a contradiction, however many pins you hope your angels might dance on.

    This is purely about comprehension and language; the observer's ability first to distinguish 'one' from 'some' or 'many' and then to convey that difference to an audience, all in light of conflicting definitions of 'heap'.

    That it could be to do with philosophy is a significant reason so many people disdain philosophy.

    Oops!

     – Robbie Goodwin Feb 21 '23 at 23:28
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    @RobbieGoodwin There exists at least one phrasing of the issue which meets the definition of a paradox, and that is what one is invoking when one says "sorites paradox". The fact that there are other phrasings which don't matters not. Yes, questioning the definition of terms is indeed a possible resolution. And yes, one may skip the paradoxical phrasing to invoke that resolution. But just because one can travel to Canterbury by going around London rather than through it does not mean that London does not exist, or that it is not a valid path of travel. – R.M. Feb 22 '23 at 02:13
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    The question occurs in other situations. Given and list of numbers generated by some TRVE random number generator (radioactive decay for example), such a list may be used for random sampling (the Rand Table for example), one may change one number and still have a "random list." Etc. However, governmental laws may exhibit such behavior. One is considered a "chronic migraine sufferer" if one gets 15 or more migraines per month; apparently, those suffering from 14 or fewer are deemed not to suffer enough for "chronic migraine" treatment. – ttw Feb 22 '23 at 02:56
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    A solution is to just keep removing grains and call it a heap deficit. – candied_orange Feb 22 '23 at 20:31
  • @R.M. If the issue is as stated here, there exists no 'paradoxical' phrasing; only a Question badly worded.

    If there is a complicit wording, why hint at what you could clearly state?

    This is about logic, not term definition: the Question as Posted fails. It's not about 'skipping paradoxical phrasing…'; there would be none even if going to Canterbury round or through London had to do with sand, or heaps.

    What matters is the ability to distinguish between count and non-count nouns; between grains and heaps; not whether but when removing one grain turns a heap into not-a-heap.

     – Robbie Goodwin Feb 22 '23 at 23:01
  • Also, a heap is not just its number of grains. If I line up one million grains on a straight line, it is "less" a heap than a hundred grains put in a roughly conic shape. If I have a heap and I start to flatten it with a rack, when does it stop to be a heap? – Taladris Feb 23 '23 at 14:49
  • @RobbieGoodwin the reason that this paradox is not easily solved is that your proposed solution would have to be common knowledge. Any and all numeric definitions of the word 'heap' - like "a heap must contain at least 1253 grains of sand; removing the 1253d grain turns the 'heap of sand' into a 'teaspoon of sand' which must contain between 832 and 1252 grains of sand" would be met with "says who?" or "why" or "hahahahaha". It is a paradox precisely because the question assumes that such a definition should exist, when clearly it does not. – Jumboman Feb 23 '23 at 15:51
  • Sorry, RM… That's simply not so. 'paradox' is never about 'seeming'.

    Ether both statements are demonstrably true or if one fails that test, there is no paradox; merely a mistake. Oops!

    Who doubts, taking a single grain from a heap leaves a smaller heap, until a certain point. The actual problem here is a combination of what 'a heap' means, and when that 'certain point' is reached. How is that more philosophy than simple vocabulary?

    Of course a single grain is not a heap and any heap can be reduced to a single grain. How is that more philosophy than simple vocabulary?

     – Robbie Goodwin Feb 24 '23 at 20:36

9 Answers9

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The solution is to realise that the problem as posed is based on a false assumption that there is always a clear dividing line between two opposing classifications of degree. Take long and short, heavy and light, wide and narrow, too salty and not too salty, etc etc. It is impossible to define an unambiguous boundary between pairs of terms like that. The terms 'a long piece of string' and 'a short piece of string' are overlapping with blurred boundaries, so a piece of string can be both long and short depending on the context.

Marco Ocram
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    Without resorting to change of context, you can also say that boolean logic requires a hard cutoff somewhere, which our human way of thinking resists. Fuzzy logic is a better match for our intuitive sense of how strongly something is or isn't a heap. (I still remember writing an essay on the Sorites paradox in 1st year undergrad philosophy 20 years ago, and making this argument. :) – Peter Cordes Feb 22 '23 at 06:48
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    Reality contains many spectrums and continuous ranges, and we try to split those into concrete categories. That's the problem. – NotThatGuy Feb 22 '23 at 09:29
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    @PeterCordes: the problem is that the definition of a fuzzy set is also arbitrary, with a non-countable set of membership functions. – Taladris Feb 23 '23 at 14:37
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    This can be visualized with larger objects. If you have around 6+ tennis balls, it can be considered a heap. You can imagine going from 6 to 1 easily, and it is unlikely anyone will call 1 tennis ball a "heap". Some people wouldn't even call 6 a heap. If 6 doesn't work, use a bigger number until it does. – Nelson Feb 24 '23 at 03:30
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One solution would be to say that even 1 grain is a heap. That would be defining "heap" more precisely than its informal, intuitive meaning. What does "heap" precisely mean anyway? The whole "paradox" is built on the lack of precision in the informal understanding of "heap", so it doesn't seem to be such a formidable "paradox".

Frank
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    Some would even say that 0 grains is a valid heap, but those people are probably computer programmers and it's best to ignore them :) – Jeremy Friesner Feb 22 '23 at 18:30
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    To me, this definition doesn't match the concept of a heap, which is vague by nature. The paradox stems from the shared belief that some amounts are too few to be considered a heap. – Ashlin Harris Feb 22 '23 at 20:15
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    @JeremyFriesner Or mathematicians. In short, those people who know that zero is actually the first number. Everyone else needs to handle extra special cases. – cmaster - reinstate monica Feb 22 '23 at 22:03
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    @cmaster-reinstatemonica but one problem with 0, is that if can call a heap, then is a heap? meaning, if there is not at least one grain, there is really nothing to call a heap? – Frank Feb 22 '23 at 23:04
  • I forgot to mention that my comment is a bit bland without a grain of salt... Anyway, calling an empty heap a heap means treating a heap as a collection of things, or as some kind of container. Of course, the definition of "heap" can include further restrictions, like saying that a heap can only contain physical objects. And once again, programmers are unhappy because they routinely put data objects into a heap... – cmaster - reinstate monica Feb 23 '23 at 09:33
  • @Frank Not everything is a heap. Only heaps are a heap. So no grains of sand can form a heap so can 30 or 50000, but they form it, they aren't it. Heap is a "container" of sorts in that understanding. It's a way to think about/work with things, that makes something easier. Most of math is about that. Everything in math is just lot's of sets nested inside each other containing nothing other than sets. That's all math is in at least one fairly dominant approach. But that's inefficient so we call things 1 and 2 and heap. – DRF Feb 23 '23 at 11:58
  • @cmaster-reinstatemonica Maybe I'll restate my point. Right now, I'm looking at the empty table in front of me. It has nothing on it whatsoever. It has 0 grains on it. Therefore by the 0-definition of heap, there is a heap in front of me on that table. But that table is empty. Isn't that a contradiction? – Frank Feb 23 '23 at 14:47
  • No, not a contradiction. I would say that you can interpret the space above your table as an empty heap. Still, the table is empty. In this sense, your table is just another container that can contain some physical objects which are stored above it, and it just so happens that this empty container can also be seen as an empty heap. (This is known as "duck typing" among programmers: If something walks like a duck, swims like a duck, and quacks like a duck, I call it a duck. In this sense, your empty table is a heap of gold just the same as a heap of shit. Until you add something to it.) – cmaster - reinstate monica Feb 23 '23 at 15:08
  • @Frank To a mathematician, a depression would be a negative heap. – Barmar Feb 23 '23 at 15:32
  • @Barmar I don't think that's true, because it's not logically consistent. You can't have be . – Frank Feb 23 '23 at 15:52
  • @cmaster-reinstatemonica Contradiction, because now, can be interpreted as a "heap", a "tree" or whatever whatsoever. Or if it's not a contradiction, it's at least not useful at all. Better to define "heap" to start at 1 grain, that's more useful. The errand about mathematician and CS types indexing from 0 is not going anywhere here, IMHO. – Frank Feb 23 '23 at 15:54
  • @Frank It absolutely is not more useful. It's a lot less useful. If you want to stay out of proper definition territory, live with the uncertainty or learn some non binary logic. If you want a standard logic, get a proper definition and in that case having 0 grains form a heap is going to be much more useful in just about every situation. – DRF Feb 24 '23 at 08:55
  • @DRF how so? Can you explain? – Frank Feb 24 '23 at 14:37
  • @Frank Let us continue this discussion in chat. – DRF Feb 24 '23 at 14:40
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The answer is in the definition of a heap.

I offer the definition that a heap must have at least one layer stacked upon a base layer.

For grains of sand you need at least 3 grains in a base layer to support 1 grain of sand stacked upon it (thus forming a tetrahedral stack). Thus, properly arranged, a heap could be as little as 4 grains of sand; any fewer is no longer a heap.

Xavier
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    Some grains of sand retain the crystal structure of the mineral they're made of (others have been eroded and are more sphere-like). So in some cases, it might be possible to balance one grain of sand on top of another, and have a heap of 2 grains of sand for as long as you can avoid breathing on it. – Misha Lavrov Feb 22 '23 at 01:20
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    @Misha Lavrov OK, maybe theoretically possible, but very unlikely. Maybe we also need to define what a grain of sand is. – Xavier Feb 22 '23 at 01:30
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    +1 for considering arrangement as well as amount in the definition. άνθρωπος's answer suggests that a heap must also be haphazard, making the notion of a "properly arranged heap" an oxymoron. – Ashlin Harris Feb 22 '23 at 20:24
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So I would say that two possible answers to the paradox are

  1. Rigorously define a 'heap' to explicitly consider the number of grains of sand, or
  2. Keep the fuzziness around heaps, and observe that tiny changes in the amount of sand trigger correspondingly tiny changes in the probabilities that different people will consider it to be a heap.

The existing answers focus on solution number 1, but I don't believe anybody has discussed option 2 which I find is a more intuitive resolution.

Consider the chance of a given person at a given point in time considering a given amount of grains of sand as a 'heap' as some probability function depending on the number of grains, starting at 0 (or very close to it) and growing to nearly 1. Removing grains one by one does affect the probability something will be classified as a heap, just not by much.

enter image description here

Of course, other factors will be at play - have they been primed, whether they are a construction worker or a child (as per Mark Andrew's answer), whether they are in a heap-finding mood, or any number of other things.

Bug Catcher Nakata
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  • "the chance of a given person considering a given amount of grains of sand as a 'heap'" is not satisfactory as this probability is ever-changing (people die, or simply change their opinion, or don't have one), and is no less arbitrary than, for example, asking people to vote every 5 years to elect a Heapmaster who will decide what is the minimum number of grains necessary to make a heap. – Taladris Feb 23 '23 at 15:03
  • While others have also mentioned that heap is a fuzzy concept, this provides an interesting formalisation of that. – NotThatGuy Feb 23 '23 at 16:21
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    Note that in reality, the probability doesn't go to 1 for large numbers of sand grains. Firstly, there are some people who don't know the word 'heap' and would call it something else. Secondly, if the heap is too big most people will stop calling it a heap and start calling it a mountain. Therefore, there must be an ideal heap size that the largest number of people would call a heap. – DarthVlader Feb 23 '23 at 18:49
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    @DarthVlader Yes 100% agree – Bug Catcher Nakata Feb 23 '23 at 22:38
  • @Taladris the fact that the function varies in time doesn't seem particularly bothersome/unsatisfactory to me, I'll edit to specify a given person at a given point in time though. As for arbitrariness, its value is in aligning our day-to-day experience of heap-classification with the apparent slight-reduction-paradox. Electing a heapmaster is completely alien to day-to-day experience of heaps, so I don't see how it would add clarity. – Bug Catcher Nakata Feb 23 '23 at 22:38
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The solution, of course, is to learn to cook:

Image showing a heaped teaspoon and scraping a knife over it to make a level teaspoonful

Therefore, how many grains must be removed to "turn a heap into not-a-heap" depends on the spoon.

candied_orange
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There is no paradox - there is a fallacy

Let us compare with integers.

If I take an integer (Let's say 100) and subtract 1, I get 99, a positive integer; I subtract one again and get 98, another positive integer. From that Sorites would presumably argue that whenever I subtract 1 from any integer I still have a positive integer. However that is patently not true. When I reach 1, the subtraction will yield zero which does not qualify.

The above argument is no different in structure than that of the 'paradox'. All we need to do is define boundary conditions.

Example

Definition: A heap of sand is a quantity of sand such that a path can be traced from any single grain to all others without intervening gaps and that has at least one grain that is elevated from the underlying supporting surface by resting on one or more of the other grains.

Boundary condition: When I remove a grain of sand from a quantity of grains that constitute a heap, I will be left with another heap unless there are no elevated grains.

Note that the definitions are arbitrary and can be adjusted as required. All that is required in rigorous discourse is that terms should be agreed ahead of time. It is not sufficient to rely on folk terminology.

chasly - supports Monica
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    +1 for the introduction of definition and boundary condition to dissolve the paradox. But there is definitely a paradox: "a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true". That people are genuinely flummoxed before explanation (and many after) by naive forms of categorization is an empirically verifiable fact, which says more about human reason and intuition, than it does about anything else. – J D Feb 22 '23 at 17:27
  • I don't think any reasonable person would colloquially call 4 grains of sand, where one is elevated, a "heap". The point is that there isn't a boundary condition, because "heap" is a fuzzy concept. (Also, if you say "there is a fallacy", then you also need to explain what the fallacy is - being wrong about there being a paradox, or failing to define terms, would not automatically make it a fallacy). – NotThatGuy Feb 23 '23 at 15:20
  • @NotThatGuy - My point is that you can talk using human intuition or folk-reasoning or you can talk using strict definitions. The 'paradox' comes when you try to mix the two. If you don't like my definition of the minimum sized heap, you are free to devise your own definition. If you complain that you can't define what a heap is then, there is no point in making mathematical statements about one - you're just playing with words.. – chasly - supports Monica Feb 23 '23 at 18:10
  • @chasly-supportsMonica but that's exactly what a paradox often is: mixing two concepts that should not be combined. Zeno's paradox mixes the concept of infinity with the concept of distance. Galileo's paradox mixes infinity and simple counting maths. – Jumboman Feb 24 '23 at 12:24
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    @Jumboman - That's a good point. I do not have any formal education in philosophy. I may be using a definition of paradox that is not generally accepted in the field. – chasly - supports Monica Feb 24 '23 at 12:30
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The solution is to acknowledge that the common notion of a "heap" does not have a rigorous definition. Like all terms in everyday language it is defined by context.

Daron
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What’s the solution to the Sorites paradox?

Most responders have offered the same answer: the solution depends on the definition of "heap".

I agree, but I add that the definition itself depends on what you intend to do. To a road construction crew, a heap of sand might be several tons. To a child making a sand castle, a heap might be only three plastic buckets full. To a molecular scientist, three grains might be enough.

Mark Andrews
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Edited by Ashlin Harris:

A certain number of grains is not necessarily a heap. For instance, they could be arranged in a row.

A heap it is something without any artful structure; when a heap gets any order it is already not a heap. An ordered structure can have elements removed one by one and still be the same(*) structure. If you take an element from a heap, it might become something different(**).

*but incomplete, at that time a heap is always complete to a heap.

**by the structure, or you can make a new heap.

Old poor version:

i named it "the method of the ordering elements"

all sets* contained more then 2 grains are the heaps. But possible to remove all grains from the heap and create the string on numerated grains that not a heap. 2 grains is the heap and is the string at the same time.

So, all that you need something that not a heap, but consists of same grains.

easy.

*non-ordered.

**one grain is not a heap and not a string, it is the single element of a set or string.

άνθρωπος
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  • What do you mean by a "string" in this context? – chasly - supports Monica Feb 22 '23 at 18:55
  • @chasly-supportsMonica a "string" is a numerated set. It can be not only a string, but another kind of an order. Non-ordered "many grains" is a heap. I disagree that the heap needs any definition except that 1 and 2 grains are not a heap, or maybe 2 is my imagination, i still think about 2 needs or not. – άνθρωπος Feb 22 '23 at 19:38
  • So in other words, a certain number of grains is not necessarily a heap. For instance, they can be arranged in a row (or "string"). This answer also includes structure in the definition of heap. – Ashlin Harris Feb 22 '23 at 19:41
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    @AshlinHarris Yep, a Heap it is something without any artful structure, only one thought, that grains all over here. And when a heap gets any order it is not already a heap, or possible it is a heap of some kind of order. A "natural" heap have no any kinds, but same elements. Ordered heap can be named as a kind of the order: a string, a brick, something another. A heap is chaotic, ordered heap is not chaotic, if you take element of it will be incomplete "heap" by the order. There is a point about "Is the glass half empty or half full?" but it is another theme. – άνθρωπος Feb 22 '23 at 19:56
  • @AshlinHarris also a method of the ordering elements makes clear that part where you get and leave grains, because if you don't have the order, you will make another more heap. – άνθρωπος Feb 22 '23 at 19:59