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I'm not sure what the big deal here is. This has an obvious solution. "Heap" is a word. Words have meaning to people. Different people ascribe different meanin

I'm not sure what the big deal here is. This has an obvious solution. "Heap" is a word. Words have meaning to people. Different people ascribe different meanings to the same word. Each grain added to a pile of grain measurably changes the probability that an observer will classify the resulting collection as a heap. While it's not a linear function, it's generally a positive correlation, more grains means a higher probability of any individual calling it a heap. That is a collection of grain of size n+1 has a probability of being classified as a heap that is usually greater than the probability of a collection of grain of size n being classified as a heap. Where's the paradox?

n0us 10y ago
You are conflating probability with truth. A person also may incorrectly classify a collection of grain size n as a heap.
- dragonwriter 10y ago
  Perhaps he is not conflating them but asserting that, for named categories, there is no truth to membership of an item in them beyond the probability that an observer will ascribe an item to the named category.
hderms 10y ago
It's just interesting when you have discrete entities and you can't show inductively that there is any point where the classification changes from non-heap to heap, but clearly that point must exist. It's interesting from the same perspective as what makes the ship of Theseus interesting. the probability of something being viewed as a heap could only be measured statistically as the probability doesn't derive from some direct process. The issue can be viewed in both an axiomatic and a empirical way and both views lead to interesting conclusions.
- empath75 10y ago
  That point may not exist though, if it just gains more 'heapness' over time. Even if 99% of people agree that a collection of grains of sand is a 'heap' there might be some people who still insist that it is not.
  - coldtea 10y ago
    After a point, 100% will agree (anybody not agreeing that an obvious heap of say 100kg of sand is not a heap but just grains, can be discarded as a madman). But still the exact point can't be specified, we only know when it's over-surpassed. It's like in the famous trial, when they asked someone to specify is a film is "porn" or "erotic art", and he said: "I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description ["hard-core pornography"], and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that."
- Retra 10y ago
  That point doesn't need to exist. A "heap" is not a formally defined concept, and the Sorites paradox is basically what you get when you improperly assume it must be. Think about how you learned the meaning of the word "heap." People just used it. They used it in some varying contexts, and your brain picked up on a pattern. It's like if I said "This is my dog, Blue" and someone was all hung up on the paradox of "what color is the dog, really." Blue is the name of the dog. You know this because I used it as a name. You don't need a formal definition, all you need is to know how a name-context works, and you can tell it's the dogs name. "The probability of something being viewed as a heap" is not the right way of looking at it. What's important is the probability of it being correct to be called a heap, which is derived from the correlations in usage patterns of the word "heap" around various things.
- mcphage 10y ago
  > but clearly that point must exist Rather, it's very clear no such point exists, and the assumption that it does is both nonsensical and leads to foolish conclusions.
JadeNB 10y ago
I think that the problem is that the sorites paradox continues to manifest even if it is applied only to a single person. Any reasonable person will agree that one grain of sand is not a heap, and that some sufficiently large pile of sand will a heap; but I think that few reasonable people would be able to pinpoint, either a priori or a posteriori, any particular number of grains as constituting the minimal possible heap. If you apply this probabilistic interpretation to a single person, then your conclusion is that even individuals (rather than communities) attach words to concepts only probabilistically, which is not a paradox but is, I think, rather disturbing.
coldtea 10y ago
The idea that a concrete thing (a number of grains on top of another) can not be said to be in a specific state, but rather its pile-ness is a non-linear function of the number of items it contains, is already the paradox. Common logic would expect concrete things to be either X or not X (unless X is something subjective like "is it pretty?", "is it art?", "is it ethical?", etc). Instead it turns out (paradoxically) that even a quality of a concrete thing based on its physical characteristics, like "is it a pile", can be subjective.
SilasX 10y ago
IMHO, that's the lesson -- your mental model of the world shouldn't have sharp categories. [1] If you use concepts such that things are either a heap or not-a-heap, "you're gonna have a bad time". Instead, you should recognize a smooth boundary between categories -- things are more or less heaplike, so you should be able to continuously reduce its rating as grains are added or removed. [1] https://news.ycombinator.com/item?id=11687391 https://news.ycombinator.com/item?id=11687391

n0us 10y ago

You are conflating probability with truth. A person also may incorrectly classify a collection of grain size n as a heap.

dragonwriter 10y ago
Perhaps he is not conflating them but asserting that, for named categories, there is no truth to membership of an item in them beyond the probability that an observer will ascribe an item to the named category.

hderms 10y ago

It's just interesting when you have discrete entities and you can't show inductively that there is any point where the classification changes from non-heap to heap, but clearly that point must exist. It's interesting from the same perspective as what makes the ship of Theseus interesting. the probability of something being viewed as a heap could only be measured statistically as the probability doesn't derive from some direct process. The issue can be viewed in both an axiomatic and a empirical way and both views lead to interesting conclusions.

empath75 10y ago
That point may not exist though, if it just gains more 'heapness' over time. Even if 99% of people agree that a collection of grains of sand is a 'heap' there might be some people who still insist that it is not.
- coldtea 10y ago
  After a point, 100% will agree (anybody not agreeing that an obvious heap of say 100kg of sand is not a heap but just grains, can be discarded as a madman). But still the exact point can't be specified, we only know when it's over-surpassed. It's like in the famous trial, when they asked someone to specify is a film is "porn" or "erotic art", and he said: "I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description ["hard-core pornography"], and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that."
Retra 10y ago
That point doesn't need to exist. A "heap" is not a formally defined concept, and the Sorites paradox is basically what you get when you improperly assume it must be. Think about how you learned the meaning of the word "heap." People just used it. They used it in some varying contexts, and your brain picked up on a pattern. It's like if I said "This is my dog, Blue" and someone was all hung up on the paradox of "what color is the dog, really." Blue is the name of the dog. You know this because I used it as a name. You don't need a formal definition, all you need is to know how a name-context works, and you can tell it's the dogs name. "The probability of something being viewed as a heap" is not the right way of looking at it. What's important is the probability of it being correct to be called a heap, which is derived from the correlations in usage patterns of the word "heap" around various things.
mcphage 10y ago
> but clearly that point must exist Rather, it's very clear no such point exists, and the assumption that it does is both nonsensical and leads to foolish conclusions.

JadeNB 10y ago

I think that the problem is that the sorites paradox continues to manifest even if it is applied only to a single person. Any reasonable person will agree that one grain of sand is not a heap, and that some sufficiently large pile of sand will a heap; but I think that few reasonable people would be able to pinpoint, either a priori or a posteriori, any particular number of grains as constituting the minimal possible heap. If you apply this probabilistic interpretation to a single person, then your conclusion is that even individuals (rather than communities) attach words to concepts only probabilistically, which is not a paradox but is, I think, rather disturbing.

coldtea 10y ago

The idea that a concrete thing (a number of grains on top of another) can not be said to be in a specific state, but rather its pile-ness is a non-linear function of the number of items it contains, is already the paradox. Common logic would expect concrete things to be either X or not X (unless X is something subjective like "is it pretty?", "is it art?", "is it ethical?", etc). Instead it turns out (paradoxically) that even a quality of a concrete thing based on its physical characteristics, like "is it a pile", can be subjective.

SilasX 10y ago

IMHO, that's the lesson -- your mental model of the world shouldn't have sharp categories. [1] If you use concepts such that things are either a heap or not-a-heap, "you're gonna have a bad time". Instead, you should recognize a smooth boundary between categories -- things are more or less heaplike, so you should be able to continuously reduce its rating as grains are added or removed. [1] https://news.ycombinator.com/item?id=11687391 https://news.ycombinator.com/item?id=11687391