In science, many such paradices are resolved by what they call in scientific writing "working definition".

Where a useful, real-life definition is impossible to find, the scientist creates a working definition, WD. In the case of the heap, the WD may say: "We shall consider any haphazardly thrown together comparatively identical objects a HEAP if hit has 100 or more elements, and a NON-HEAP if it has fewer than 100 elements."

Similar WDs exist all over the place. You want to, say, gather a bunch of people and test their behaviour on hunger. How do you define a hungry person? A person who feels hungry? A lean person? A scientists will have to come up with a WD, since common language and thought is fuzzy or unclear on what constitutes a hungry person. So the WD may say "a person is hungry if he hasn't eaten in four hours or mre; and a person is not hungry if he has eaten within the past hour hours." This definition may not even cover the ultimate truth, that is, picking people who feel hungry; but it is a working definition inasmuch it is qualtitative enough to select people without hesitation or doubt.

The term ''heap'' in common usage doesn't actually mean a ''certain'' number of grains. — TheMadFool

Agreed.

More accurately a ''heap'' includes in its definition the size of the components, the shape of the collection, in addition to the number of objects in the collection. — TheMadFool

(Assuming that by "its definition" you mean any reasonable characterisation of its usage...) Yes, there is a range of related aspects that help us decide.

Therefore, to isolate one variable, the number of objects in the collection, may be a mistake. — TheMadFool

Happy, myself, to make that mistake.

doesn't detract from the problem of vagueness, the central message of the paradox. — TheMadFool

Yes, and we can always play the bald man or other versions, for variety.

If you are against either of these reductions, then hooray. If your talk of "incommensurability" isn't, after all, about trying to separate usage of heap from the naturals, then even better. — bongo fury

Let's first give a definition to incommensurability:

two or more quantities having no common measure.

So your correlation 2 goes totally against the definition and I'm not exactly sure what you mean by correlation 1.

What the important point here is that in mathematics there indeed is this incommensurability: that you simple cannot measure everything to everything else with some common measure. To assume it would be so is simply incorrect. And this is totally logical.

Therefore, to isolate one variable, the number of objects in the collection, may be a mistake. Nonetheless this is an issue for the heap paradox specifically and doesn't detract from the problem of vagueness, the central message of the paradox. — TheMadFool

Yes. As I said, It's basically because "heap" as a measurement system isn't measurable the same way like the number system. You just get smaller and larger comparisons to "heap". With let's say "a mountain of sand" compared to a heap is bigger and a perhaps smaller comparison to a heap is let's say "few grains of sand". If we want to go into measuring specific grains of sand, the scale system "few", "a heap", "a mountain" simply doesn't cut it. It's incommensurable.

When we look at paradoxes, the usual reason for them is that we have the premises wrong. Just as here, where we make the totally incorrect and false presumption that everything could be reduced to being measured by a common measure, notably with natural numbers.

In the case of the heap, the WD may say: "We shall consider any haphazardly thrown together comparatively identical objects a HEAP if hit has 100 or more elements, and a NON-HEAP if it has fewer than 100 elements." — god must be atheist

Yes, the arbitrary threshold solution: correlation 2, above.

I'm glad you chose so as to give some breathing space to non-heap, this time! Going for 4-grains-or-more seemed a bit extreme, but fair enough, you wanted to think outside the box of entrenched usage? Anyway, this way (100) you get more of a sense of "absolutely not!" at a single grain.

We lose, of course, any sense of give, or tolerance. You make a good case for not caring: the scientific context may be about nothing to do with usage of "heap"... nor of "hunger", etc.

If we can interest you in trying to characterise either of these usages (or of some more emotive predicate... poverty? ... where your threshold is liable to dispute), then you may want to examine a possible distribution of thresholds, to accommodate some fuzziness. And it begins again...

But what about the big picture, a poll of judgements, or of individual thresholds? What if the tail end of such a distribution (of thresholds) reaches back to a single grain? From your observations about means, we guess that it will.

Then, for some enthusiasts at least, this play of the game is over. From their point of view, you won't play. You decline to agree that a single grain is absolutely not a heap. You admit that this grain is, in the current idiom, "on the spectrum" of (usage of) heap. Albeit at one far end of that spectrum. You've lost one of the two required (and puzzlingly opposed) intuitions that we are trying to reconcile. — bongo fury

When we look at paradoxes, the usual reason for them is that we have the premises wrong. Just as here, where we make the totally incorrect and false presumption that everything could be reduced to being measured by a common measure, notably with natural numbers. — ssu

What other method would you choose to describe how a heap of sand stops being a heap? Isn't it the number of sand grains in a collection that determines the heap-ness?

What other method would you choose to describe how a heap of sand stops being a heap? Isn't it the number of sand grains in a collection that determines the heap-ness? — TheMadFool

When does a beautiful girl stop being beautiful and become 'OK looking' or 'ordinary' or even be outright 'ugly'? If you cannot draw a specific line, then is the notion of being beautiful in peril?

Even if you would very rudely give numbers for attractiveness, it still begs the question of what is the measure. Being beautiful is obviously fundamentally something that you simply CANNOT put into a measurement system of exact numbers, points or decimals. The whole point of being 'beautiful' or there being a 'heap' of sand is that you cannot measure it exactly on an arithmetic scale. It's based quite on a personal judgement and the only comparison that we can agree on is that a heap of sand is smaller than a mountain of sand.

Isn't it the number of sand grains in a collection that determines the heap-ness? — TheMadFool

No.

That's how you just get to the paradox: you are insisting that an exact number of sand grains determines what a heap of sand is. You simply dismiss there being the notion of incommensurability, but falsely think that everything can be measured by exact numbers. (Perhaps because of reductionist thinking that every amount of sand has an exact number of sand grains, which is true but meaningless here.)

That's how you just get to the paradox: you are insisting that an exact number of sand grains determines what a heap of sand is. — ssu

Well, to insist we must. Because this follows from the first fundamental law of Calculus (sorta). If there are two points on a function, f(A) and f(B), such that f(A) < f(B), and f is a continuous function between A and B, then there must be such a point x where x is between A and B, and f(x) is between f(A) and f(B) and f(x) = (f(A) + f(b))/2.

Therefore if we call f(A) a heap, and f(B) a non-heap, both a function of number of grains of sand, then there must be a x which falls between A and B, and f(x) is between f(A) and f(B).

Well, to insist we must. - . If there are two points on a function... — god must be atheist

You aren't getting the point. The measurement system of heap of x < mountain of x isn't straight forward calculus as you cannot answer exactly how much bigger is a mountain of sand compared to a heap of sand. Hence you cannot add them up together and divide them into two, because you are using the number system. In order to talk about mathematical functions, you do need the number system and arithmetic to calculate functions. With heaps it isn't so!

That's the whole point: 'heaps' or beauty do not have a common measure with an arithmetical system like the natural numbers. That's the whole point of incommensurability.

In order to talk about mathematical functions, you do need the number system and arithmetic to calculate functions. With heaps it isn't so! — ssu

That's the beauty in Calculus. You can describe incredibly complex relationships without using numbers, or many numbers.

In fact, the first fundamental law of Calculus only uses 2 to find the midpoint between two values. But it does not go beyond that in any more ways of using numbers.

And it applies to ALL continuous functions! You know how many of those there are between two points? Quite a few. An infinite number of them, actually.

So to interpret it to the common man's world: Between a heap and a non-heap there are many gradations, and at one point a gradation will be a step between a heap and a non-heap.

If you insist it's a perceptual (by human perception by the observer person) decision, then the heapness is regulated by perception, yielding possibly many different results, when you take a large number of humans each to say "WAIT! Taking away this grain of sand made the heap into a non-heap."

IN this case, the fundamental law of Calculus applies, but it yields a (possibly) different value of number of sand from human to human.

So your correlation 2 goes totally against the definition — ssu

Not sure I understand. I'm correlating two systems (both apparently in working order) of grain-collection labels: one is the system of two labels, heap and non-heap; the other system is the naturals, with respect to grain-collections in particular... a single grain, a pair of grains, 3 grains, etc.

The apparent behavior of the first does seem to correlate in some way with the second. The game is to try and discuss the correlation without paradox. But we can map certain of one system onto certain of the other, and get the feeling that some "commensuration" or correlation is there to be described adequately, somehow.

Correlation 1 is the common policy of settling for a different grade of heap for each natural, which is tantamount to giving up on heap vs non-heap. (So game over in that kind of play.)

Does this help?

And very often (any slippery slope ethical dilemma, any artistic play with discrete perceptual categories, e.g. musical pitches), you want to work with the usage as it is, not precisified — bongo fury

Sure, depending on your aims. All I was saying is that the dividing line simply tells you about how people formulate their concepts, how they use words, and I was noting that going by any consensus doesn't give us a correct answer, just a common or conventional answer.

There is no correct answer for stuff like this, by the way.

That's the beauty in Calculus. You can describe incredibly complex relationships without using numbers, or many numbers. — god must be atheist

And for this you need arithmetic to apply and there needs to be a number system.

IN this case, the fundamental law of Calculus applies, but it yields a (possibly) different value of number of sand from human to human. — god must be atheist

Actually, it really doesn't genuinely apply.

It's as wrong as to try to put infinity, as a number, or an infinitesimal, as a number, on the number line. You simply cannot do it. And thus people don't regard either as numbers. Yet both are extremely useful in mathematics, so there isn't anything wrong with them.

In fact, the first fundamental law of Calculus only uses 2 to find the midpoint between two values. But it does not go beyond that in any more ways of using numbers. — god must be atheist

This again is a fallacy here, because you simply deny the existence of incommensurability. Think about it: if you have a heap of sand and a mountain of sand, what then is the middle, really? It would be something like "an amount more than a heap and less than a mountain". Is that useful? Likely not, and still you don't have any idea when a heap turns into 'more than a heap and less than a mountain'. The laws you refer to don't really solve the issue at all.

Again, you simply do not get from one system to another, even if you argue that all amounts of sand are made of individual grains of sand.

Not sure I understand. — bongo fury

Ok, you said:

correlation 2: an arbitrary individual threshold... a policy with some good PR (e.g. "you have to draw the line somewhere, and that's that"), but which will inevitably deprive the usage of its useful fuzziness / tolerance — bongo fury

"You have to draw the line somewhere" is itself the problem. When you don't have a common measure, just how are you going to draw the line somewhere? You simply need that common measure to draw the line somewhere. This is similar to where just assumes Calculus, but forgets what it means not having a common measure.

In fact, the first correlation too goes also against incommensurability:

) Hooray if, for example, you want to resist this correlation because you have a sense of clarity or absolutism about certain cases of heap and of non-heap, and a sense that the same clarity will transmit from these cases to certain others. — bongo fury

The issue won't transmit so easily, because notice the definition of incommensurability: two or more quantities having no common measure.

If they have no common measure, how do you think the clarity will transmit from one to the other? Think about it this way: try to do the following algorithm in arithmetic:

1) Start from the natural number 57.
2) Add a somewhat large natural number to it.
3) Substract from the sum a small natural number.
4) So in which natural number are you in the end...exactly?

This is what looks like when you mix two incommensurable systems together. The problem is simply to assume that you can do it, and that you get an exact answer. Yet "somewhat large" or "a small number" is quite practical sometimes, assuming there is an universal agreement just what the range is. Not so if you just take as here.

One should really stop are really think about what it means when in mathematics two quantities have no common measure. That in math we have quantities that are unmeasurable. The answer isn't that if it's a quantity, it has to have a measure. The issue is that it isn't measurable to others.

If you [ssu] are against either (I meant both) of these reductions, then hooray. If your talk of "incommensurability" isn't, after all, about trying to separate usage of heap from the naturals, then even better. — bongo fury

No second hooray, then: you think usage of heap should be kept separate from the naturals. But still hooray!

"You have to draw the line somewhere" is itself the problem. — ssu

Yes, I said that too. I'm saying it's an inappropriately crude reform of actual usage, and that radical reform may be unnecessary anyway. We both recommend following actual usage instead. But I'm happy to follow usage with reference to numerical comparisons, despite the challenges of the heap puzzle. I'm against unnecessary reform, but enamoured of careful correlation, or "commensuration" with number talk. Partly because ordinary usage of heap seems pretty deeply soaked into ordinary usage of number words. You are so appalled by inappropriate reductions of systems to arithmetic that you won't hear of any such intermingling.

Hooray if, for example, you want to resist this correlation because you have a sense of clarity or absolutism about certain cases of heap and of non-heap, and a sense that the same clarity will transmit from these cases to certain others.
— bongo fury

The issue won't transmit so easily, because notice the definition of incommensurability: two or more quantities having no common measure. — ssu

Yes! I do notice that, and I'm grateful you followed my reasoning, poorly expressed. Yes, I was talking about a sense of likely transmission from case to case, a sense fueled by having started the game well away from the fuzzy border in question. I agree that we have to be careful how we formulate this intuition. We have to deny the transitivity of the transmission in some acceptable way. No need to assume this is not feasible, though.

4) So in which natural number are you in the end...exactly? — ssu

Who said exactly? We are interested in tracing vague discourse. Such as,

Yet "somewhat large" or "a small number" is quite practical sometimes — ssu

... Well, quite! And here you aren't going to deny that our usage of "small" and of "large" map in some interesting way onto the naturals, are you?? No, apparently not...

assuming there is an universal agreement just what the range is. — ssu

The challenge of the heap game is to describe the fuzzy/tolerant bounds of this tacitly agreed range. Although I'm not sure I read that last quote right.

I think we're approaching some kind of agreement.

you think usage of heap should be kept separate from the naturals. — bongo fury

If people get puzzled with the Sorites paradox, then yes.

I think my point is that the so-called "problem" isn't vagueness of language, but incommensurability, which is a mathematical feature itself. When talking about math, better use logic.

You are so appalled by inappropriate reductions of systems to arithmetic that you won't hear of any such intermingling. — bongo fury

Oh it's not me, it's the logic in mathematics. You see a crude counting system, like "nothing, 1,2,3, many" is logical in it's own way, if one hasn't the need to count things more than up to three. For some animal it can be a splendid counting system: why would they need to count to several thousands? And so is with "heap of x" versus "mountain of x" as a simple scale system.

The challenge of the heap game is to describe the fuzzy/tolerant bounds of this tacitly agreed range. — bongo fury

Tacit agreement is the word.

We typically have some idea of the range. If I say "There were many mosquitoes near the swamp" or say "There were many aircraft carriers in the harbour", you hopefully have some idea of how many mosquitoes can be in a swamp and how many aircraft carriers can be in one place at a time. Hence you can easily understand that "many mosquitoes" can be a number in the hundreds if not in thousands and "many aircraft carriers" is number likely more than three, but very likely less than ten, as there simply aren't many in the World (less than 20 are in service around the World and there surely isn't a get-together-party for aircraft carriers organized in some harbour).

This again is a fallacy here, because you simply deny the existence of incommensurability. — ssu

This is true. I deny the existence of any word that I can't pronounce. And "incommensurability" is one of them.

This again is a fallacy here, because you simply deny the existence of incommensurability. Think about it: if you have a heap of sand and a mountain of sand, what then is the middle, really? It would be something like "an amount more than a heap and less than a mountain". Is that useful? Likely not, and still you don't have any idea when a heap turns into 'more than a heap and less than a mountain'. The laws you refer to don't really solve the issue at all. — ssu

This is not the actual problem. You re-worded them to suit your model. That is not fair.

It's as wrong as to try to put infinity, as a number, or an infinitesimal, as a number, on the number line. You simply cannot do it. And thus people don't regard either as numbers. Yet both are extremely useful in mathematics, so there isn't anything wrong with them. — ssu

If it's wrong, it's wrong. It's as wrong as any other wrong. You are giving me another example of wrong, which you think my solution was, without your touching my solution and saying what's wrong with it.

You said two things about my solution:
1. I don't understand incomm... whatever. That is true. Maybe I understand the concept, but not the word you used to name it.
2. My solution is as wrong as something else that is also wrong.

How do these two claims prove that my solution was wrong? They do not. At all.

The issue won't transmit so easily, because notice the definition of incommensurability: two or more quantities having no common measure. — ssu

This actually does not apply to this problem. Having no common measure does not apply. Because both the heap and the non-heap comprise grains of sand(, or grains of grain, whatever).

Grains are the common measure to both. One grain, two grains,... n grains, n+1 grains, etc.

Both the heap and the non heap have an n, lets call them nn for no-heap n and hn, for heap-n.

"You have to draw the line somewhere" is itself the problem. When you don't have a common measure, just how are you going to draw the line somewhere? You simply need that common measure to draw the line somewhere. This is similar to ↪god must be atheist where just assumes Calculus, but forgets what it means not having a common measure — ssu

Actually, the fallacy is created by you, SSU, when you declare there is no common measure. The grains are the common measure. You just very conveniently lean on this concept, and keep hammering it in, but it is not applicable to this problem.

Let me introduce a new concept to the problem: it is not only the grains of sand in the heap that makes or breaks a heap, but the shape of the mass of grains. You can have 100,000 grains, all flat, on a flat surface, and they don't form a heap. And you can have 5000 grains in a cone shape, and they will form a heap.

So numbers alone is not the deciding factor in heapness or non-heapness of a quantity of grains; their shape is, too.

Therefore I suggest that we focus our attention on the duality of the relationship between grains of sand a heapness: on one hand, it's the shape, on the other hand, it's the number of grains.

And therefore I say unto you, my dear fellow travellers, that the minimum-four-grain heap opinion I offered early in this discussion is the right one: it is not any number (greater than three) that is the only plug-in variable, but the shape as well.

I think we're approaching some kind of agreement. — ssu

So near, and yet so far.

With your admirable aversion to correlation 2 (reforming a perfectly good vague predicate according to an arbitrary bi-partition of the naturals), and your interesting insights about natural development of more complex vague systems (your few-grains... heap... mountain, etc., and by the way I suggest smidgen for the first), you really should be an enthusiast for the heap puzzle, not one of its detractors!

But you repeatedly misapprehend my meaning and TheMadFool's too, if they don't mind me speaking for them.

Isn't it the number of sand grains in a collection that determines the heap-ness?
— TheMadFool
No. That's how you just get to the paradox: you are insisting that an exact number of sand grains determines what a heap of sand is. — ssu

The problem is simply to assume that you can do it, and that you get an exact answer. — ssu

No! You think we are committing correlation 2, but all 3 of us are opposed to that, I assure you. Others are more or less cool with it, their doubts (if any) assuaged by gestures towards allowing a distribution of different sharp thresholds. Which process, as I was saying previously, hopefully draws them back into the game at some point.

What we (I, and TheMadFool if I'm not wrong) are admitting is that usage of many vague labels like heap relates to numbers without difficulty in some cases, e.g. a single grain, and the problem is to describe the fuzzy border further along. Your passion against settling for an arbitrary sharp border, which we applaud, stops you from admitting this, and from appreciating that a vague category usually correlates in this puzzling way with some or other more fine-grained (often continuous) series.

Rude and unwelcome, as you say, to contrive such a background series in some cases. But consider musical scales. Twelve equal ("well tempered") divisions of the octave is some people's sole map of the terrain. Diatonic keys/scales all defined, for them, as subsets of these divisions. But plenty of alternatives persist, some with more wiggle room at each step, some with less, all of them dividing the octave differently (but nearly all into fewer than 12 steps). Potentially, all of them are independently viable, so that correlation or commensuration is unnecessary - and to some tastes undesirable. And the practitioner of any one system has no need to refer her judgements to any physical measure of the pitch continuum. But in practice plenty of mixing happens (blues scale theories anyone?) and it would seem a bizarre privation not to refer any good description of the result (how it divides the octave) to the scales that are being mixed and to the background continuum (or a discrete approximation of it). No?

No reason to fear the interrelating.

What we (I, and TheMadFool if I'm not wrong) are admitting is that usage of many vague labels like heap relates to numbers without difficulty in some cases, e.g. a single grain, and the problem is to describe the fuzzy border further along. — bongo fury

So basically your argument is the vagueness of the language.

Your passion against settling for an arbitrary sharp border, which we applaud, stops you from admitting this, and from appreciating that a vague category usually correlates in this puzzling way with some or other more fine-grained (often continuous) series. — bongo fury

Ok, it's seems you didn't get my point, because I don't find anything close to my reasoning in this.

Ok, it's seems you didn't get my point — ssu

Well I was definitely talking about usage, and aware that you weren't, directly.

Nonetheless I thought our views might be commensurable :wink:

Maybe not.