Varies from individual to individual. Its subjective, unlike other matters.

So, 1?

if that looks like a heap to you, it is 1, for you.

Sure. Heap is a spectrum. No puzzle.

A single grain is a minimal heap. A completely bald man is minimally hairy. Black is minimally white.

The puzzle requires an intuition to the contrary.

a single grain is 1, a completely bald man is by definition someone with 0 hairs, not 1. so... he is hairless, hairy on the other hand is 66.66666666%+1

So you've pumped the required intuition, and a single grain is no longer merely the smallest heap?

when I think about it more the smallest heap is 3, but only if arranged with 1 grain sitting atop two other grains (one part of it resting on one grain and another part resting on another grain).

Ahem, we of the sorites appreciation society are not amused :meh:

Try bald vs. hairy, black vs. white etc. — bongo fury

i guess we've reached an impasse.

A complete impasse, or a minimal impasse? :grin:

like imagining a heap of sand that never changes after a grain is removed or added.
— sime

... leading to the conclusion (incompatible with a premise, or there's no puzzle) that a single grain is a heap. Does that happen also with your "infinite" element, so that it can evaluate to 1? — bongo fury

No, that isn't the case. To summarise, a heap of sand can be defined as the list:

Heap := [ Heap, grain]

The list remains constant, regardless of how many grains are added to it or subtracted from it. Any finite number of grains of sand does not have this property. That is precisely what it means to say that a heap of sand has no inductive definition.

Where i differ with the OP, is his belief that it is an approach unconnected to ideas of fuzziness or ambiguity. This isn't the case, because the practical usage of infinity, such as the use of infinite loops in computer programs, is to defer the termination of the program to the environment. Or in the case of heaps of sand, the semantics which concern the precise moment when an actual heap of sand is considered to be mere grains of sand, isn't linguistically specified a priori but is decided by speakers on a case specific basis.

Any finite number of grains of sand does not have this property. — sime

So... isn't a heap?

the semantics which concern the precise moment when an actual heap of sand is considered to be mere grains of sand, isn't linguistically specified a priori but is decided by speakers on a case specific basis. — sime

Agreed. But what is the smallest number of grains that would need considering by speakers as a particular case? Is it 1?

Any finite number of grains of sand does not have this property.
— sime

So... isn't a heap? — bongo fury

There is no a priori linguistic definition of "heap" in terms of any specific number of grains of sand, which is why "heap" must be logically represented as referring to a potentially infinite number of grains of sand.

The role of potential infinity in a logical specification is to act as a placeholder for a number that is to be later decided by external actors or the environment, rather than the logician or programmer.

Agreed. But what is the smallest number of grains that would need considering by speakers as a particular case? Is it 1? — bongo fury

It is you and only you who gets to decide the answer to that question whenever you are next confronted by a growing or diminishing collection of sand grains.

I would be surprised if there isn't a population study that has attempted to quantify the mean number of grains of sand at which speaker of English judge a collection of sand to be a heap. A few hundred grains?? A few thousand?

There is no a priori linguistic definition of "heap" in terms of any specific number of grains of sand, — sime

Yes, that is the problem.

which is why "heap" must be logically represented as referring to a potentially infinite number of grains of sand. — sime

So... the answer to this question...

Any finite number of grains of sand does not have this property.
— sime

So... isn't a heap?
— bongo fury — sime

... would be? 5 million grains, say... isn't a heap, in your logical representation?

But what is the smallest number of grains that would need considering by speakers as a particular case? Is it 1?
— bongo fury

It is you and only you who gets to decide the answer to that question — sime

Not if I'm a semantically competent speaker of English, it isn't. I know full well that a single grain is so far from being a heap in this language as to make it an obvious case of a non-heap. So the smallest number of grains that would need considering as a particular case would seem to be much larger than one, no? Or are you ok with,

Sure. Heap is a spectrum. No puzzle.

A single grain is a minimal heap. A completely bald man is minimally hairy. Black is minimally white. — bongo fury

a heap isn't defined only by the geometric amount of grains, but by the artihmetic amount to, in simpler words, it isn't defined only by how many grains there are but by how they are placed in relation to each other. if you have a fine layer 1 grain thick of 10000000 grains you don't have a heap, you have a film of grains, if you have 3 grains arranged in a pyramid you have yoruself a heap.

This paradox is fun to think about. Remember though that thinking of perception (like the threshold of hearing a noise) differs for people. SO defining what is out there in discrete terms will not result in the same answer for everyone. I think you are approaching this from a subjective angle for or less, which is how I see it

(like the threshold of hearing a noise) differs for people — Gregory

Sure, but does the distribution of personal thresholds of heap-recognition, and hence usage of "heap", extend all the way back to a single grain? If so, no puzzle.

A single grain is a minimal heap. A completely bald man is minimally hairy. Black is minimally white.

The puzzle requires an intuition to the contrary. — bongo fury

I think you are approaching this from a subjective angle for or less, which is how I see it — Gregory

Bits of what you say make sense. So I doubt if your zero attention to syntax is forgiveable. I don't know your situation so probably shouldn't judge. But jeez.

I personally have no problem with someone calling a single grain of sand a heap, even if that isn't my cup of tea. What harm could it do, and who am I to decree otherwise?

There is no a priori linguistic definition of "heap" in terms of any specific number of grains of sand,
— sime

Yes, that is the problem. — bongo fury

Why is linguistic imprecision a problem? "Heap" trades referential precision for flexibility, whilst retaining the necessary semantics for useful, albeit less precise communication.

Why is linguistic imprecision a problem? — sime

Well obviously it's a puzzle if we accept also the premise that calling a single grain a heap is absurd. If calling it a heap is tolerable then, as I keep saying, no puzzle.

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] Well, certainly, it's the very smallest size of heap.

Game over. People often finish up claiming 2 had been their position all along. Perhaps it should have been, and the puzzle is a fraud.

I think it reveals aspects of the behaviour of antonyms that are fundamental to both syntax and semantics.

This paradox is fun to think about. Remember though that thinking of perception (like the threshold of hearing a noise) differs for people. SO defining what is out there in discrete terms will not result in the same answer for everyone. I think you are approaching this from a subjective angle for or less, which is how I see it — Gregory

Yes, to me it is fun to think about. I believe it's a good example of how our brain works in dealing with specifics (one grain of sand), and generalities (a pile of sand).

Try this approach: Start by imagining a single grain of sand. Now, add another grain of sand. We can easily imagine two grains of sand that are close together (not far apart). Add another grain - it's also easy to imagine three grains of sand that are close together. Now - when we try to add another grain - such that we would have four grains of sand - it gets harder to imagine. Do you visualize all four grains at the same time, or do you visualize two groups of two? The brain automatically tries to regroup numbers greater than three into new "visual" groups - hence; two groups of two. Adding more grains changes the image again, A group of five, or more, grains causes the brain to sub-divide the grains again into new distinct groups with a maximum of three grains each until one gets to three groups of three - or nine grains total. However, the brain simply can't visualize nine grains of sand in a group - only three groups of three. Try it yourself.

As a result of this simple "thought experiment" one could conclude that the maximum number of grains of sand (where one can visualize the individual grains) is nine. Any number of grains greater than nine results in an "image" of a pile - not individual grains. We have knowledge (math) that we can add more grains to the pile - or take grains away - but it's the image that will not change in our minds, not the actual number.

Ancient philosophers didn't have the knowledge of brain mechanics that we do today so they didn't think in terms of how the brain actually counts. However, they did understand the mechanics (math) of adding, or subtracting, grains of sand to a pile. They were just not able to "visualize" what was happening by adding or subtracting mentally. I believe the Sorites Paradox is a mental paradox - not a physical one.

Yes, to me it is fun to think about. I believe it's a good example of how our brain works in dealing with specifics (one grain of sand), and generalities (a pile of sand).

Try this approach: Start by imagining a single grain of sand. Now, add another grain of sand. We can easily imagine two grains of sand that are close together (not far apart). Add another grain - it's also easy to imagine three grains of sand that are close together. Now - when we try to add another grain - such that we would have four grains of sand - it gets harder to imagine. Do you visualize all four grains at the same time, or do you visualize two groups of two? The brain automatically tries to regroup numbers greater than three into new "visual" groups - hence; two groups of two. Adding more grains changes the image again, A group of five, or more, grains causes the brain to sub-divide the grains again into new distinct groups with a maximum of three grains each until one gets to three groups of three - or nine grains total. However, the brain simply can't visualize nine grains of sand in a group - only three groups of three. Try it yourself.

As a result of this simple "thought experiment" one could conclude that the maximum number of grains of sand (where one can visualize the individual grains) is nine. Any number of grains greater than nine results in an "image" of a pile - not individual grains. We have knowledge (math) that we can add more grains to the pile - or take grains away - but it's the image that will not change in our minds, not the actual number.

Ancient philosophers didn't have the knowledge of brain mechanics that we do today so they didn't think in terms of how the brain actually counts. However, they did understand the mechanics (math) of adding, or subtracting, grains of sand to a pile. They were just not able to "visualize" what was happening by adding or subtracting mentally. I believe the Sorites Paradox is a mental paradox - not a physical one. — Don Wade

Great post! I like working on questions relating to traditional ontology and modern psychology. Kant fits into the picture nicely as well I've found. His concept of schemata and such are very interesting