Sorites paradox and an aspect of objectivity
I hope everyone is familiar with the sorities/heap paradox. If not here's a Link:The Sorites paradox
I'll offer a slightly different, actually the paradox in reverse, version to say what's on my mind.
One grain of sand doesn't a heap make. Adding another will still not be a heap but carry this on for some time and we arrive at a heap of sand. The paradox is basically about how one grain of sand doesn't count and yet continue this for an adequate length of time and we have a heap of sand. Mathematically I think it can be stated as how 0+0+0+0+0+0+0+0 > 0? Each step doesn't count and yet after a certain time we have something that matters. I think it's about vagueness primarily because a heap is vague term. Any way what you should keep in mind about the paradox is simply that many nothings add up to something.
Coming to objectivity we can consider it a method for arriving at truth. There may be many definitions of objectivity but what I want to stress on is the requirment that there be an adequate number of observations. A single person's testimony amounts to very little these days. Each claim , whatever it may be, needs corroboration if it's to fly in any epistemological setup. What is notable is just like one grain of sand, a single person or observation fails to be objective. Yet, just like many grains of sand in a heap, multiple people or observations make them objective.
An important area of difference between the sorites paradox and objectivity is the former is physical and sand grains have volume and becoming a heap isn't that difficult to imagine. However, in objectivity, specifically the requirement that there be mutliple observations, things are different. Each single observation carries a not-true truth value. However mutliple observations add up to true. In logical terms how can a conjuction of non-truths (subjective) leave us with a truth (objective)?
Comments.
I'll offer a slightly different, actually the paradox in reverse, version to say what's on my mind.
One grain of sand doesn't a heap make. Adding another will still not be a heap but carry this on for some time and we arrive at a heap of sand. The paradox is basically about how one grain of sand doesn't count and yet continue this for an adequate length of time and we have a heap of sand. Mathematically I think it can be stated as how 0+0+0+0+0+0+0+0 > 0? Each step doesn't count and yet after a certain time we have something that matters. I think it's about vagueness primarily because a heap is vague term. Any way what you should keep in mind about the paradox is simply that many nothings add up to something.
Coming to objectivity we can consider it a method for arriving at truth. There may be many definitions of objectivity but what I want to stress on is the requirment that there be an adequate number of observations. A single person's testimony amounts to very little these days. Each claim , whatever it may be, needs corroboration if it's to fly in any epistemological setup. What is notable is just like one grain of sand, a single person or observation fails to be objective. Yet, just like many grains of sand in a heap, multiple people or observations make them objective.
An important area of difference between the sorites paradox and objectivity is the former is physical and sand grains have volume and becoming a heap isn't that difficult to imagine. However, in objectivity, specifically the requirement that there be mutliple observations, things are different. Each single observation carries a not-true truth value. However mutliple observations add up to true. In logical terms how can a conjuction of non-truths (subjective) leave us with a truth (objective)?
Comments.
Comments (54)
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The rest follows automatically.
To wit, my criteria for a heap is for it to look like a heap. To have the shape of a heap. That can be achieved by no fewer than 4 sand elements (if they form a tetrahedron) but they don't necessarily form it, so minimum 4 grains of sand can or can not form a heap. Similarly, any number of sand more than four in number can form a heap or a non-heap. Three or fewer can only form a non-heap.
So, the necessity of multiple observations for objective is in the domain of fuzzy logic?
Can you explain further? My problem is how can a bunch of lies (multiple corroborative observations each by itself not-true/false) add up to the truth (objectivity)?
I guess if we look at individual observations as carrying an "uncertain" truth value (multi-valued logic) until all observations agree we could accept this requirement for objectivity. Actually even this doesn't solve the problem because how can many "uncertains" yield a "certainty" of truth value?
That makes sense but the definition of "heap" in this case would be private and others will probably disagree with you.
All lies about facts contain an element of truth.
If a mind is capable of distilling the common elements in the lies, then chances are that the mind is touching on the truth.
This is the process what I tried to expound on in the only discussion I started so far on the forums, to show that the cave images in Plato's/Socrates' "Republic" can be assembled into Forms. My idea encountered dismal reception by the populus on this forum.
This is also how science works. A hundred people take measurements of a length of rope. They each come up with slightly different values. Therefore they don't just draw a consensus of how long the rope is, by computing the average length measured; they state a mean, and a deviance. For instance, they may say that the length of the rope is 5 feet, plus or minus three inches. That means that out of 100 measurements, approx. 66 will be between 4'9" and 5'3".
Exact numbers only exist for mathematicians. To a physicist, every measurement is expressed as it falls in a range.
You're absolutely right. That's why I made no bones about it, and did not declare that my proposition is the ultimate answer. I came out straight away and said under what circumstances my opinion holds.
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.
That I don't know.
Can you expand on that?
Quoting bongo fury
Kindly rephrase this. I couldn't understand.
Also can you comment on the multiple observations requirement for objectivity vis-a-vis the sorites paradox.
By "threshold" I hoped to refer to what you were calling a "private definition of heap".
E.g., GMBA's 4-grains-or-more. From their observations about taking mean averages I guessed they were interested in a larger pattern or distribution of different individual thresholds, in order to reconcile their own sharp threshold with their own intuition of fuzziness. And I wondered whether this was likely to end up compromising their intuition of clarity, in the case of a single grain (your starting point).
Ok so far?
Sorry for not relating my comments directly to the OP. I was interested in that particular exchange between you, as an instance of the heap game, considered as a challenge to reconcile clarity with fuzziness. For which I am an enthusiast.
Surprise...surprise !...I don't see any hands !...Maybe that's because the social dynamics we call context always renders such 'paradoxes' superficial.
Any "slippery slope" issue.
... is where, at any rate, 'a heap of sand' has seemed a pertinent analogy.
Give me an example you have come across.
A grain isn't nothing, by the way. It's just not a heap. Adding things to get something else isn't that unusual. We do it with things like houses, musical compositions, journeys by foot, etc.--all sorts of things.
Quoting TheMadFool
You're endorsing argumentum ad populums. There's no way around argumentum ad populums being a fallacy. What lots of people say only tells you what lots of people say.
Yes, and the word "heap" is intentionally vague. It's a messy word for a messy pile of stuff. If we wanted to be more specific, we could tighten it up. How about this:
Heap (rev 1) - An untidy collection of things piled up haphazardly such that the minimum slope of the sides intersect the ground surface at an angle of 7 degrees or more.
Which, when you want to know about usage, is what you want to know.
And, otherwise of course, not. But I am interested in usage, so I am.
But otherwise, we can use it as it is. With certain embarrassing difficulties on slippery slopes, admittedly.
Well, or if you want to know what people like (their preferences), or what their opinions about something are, etc., sure.
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. So you need both intuitions, clarity and fuzziness. The heap game, and other natural incursions by logical thought, can make you doubt this is possible, so you abandon one or the other. Then you come to, and realise you are in a mess without both. E.g. IMO fuzzy logic: analog-digital interfacing in wolf's clothing. Or e.g. Brexit.
Yes. I think the vagueness of the word matches the vagueness of what it describes.
Not sure about that. If it at least means we agree that vague words are useful in all their vagueness, then cool.
But does your conception of vagueness allow you to deny absolutely that a single grain is a heap?
EDIT: It seems it all depends on your starting point. Given a collection of sand grains that's definitely a heap and you carry out the procedure it becomes impossible to deny that a single sand grain isn't a heap.
However, begin by denying that a single grain of sand is a heap you eventually must deny even a truckload of sand is a heap.
What this illustrates is that some concepts are simply vague and didn't require precise definitions because despite their vagueness conversation/discourse wasn't hampered.
This is a good point. Clarity or precision may ease the pain of argumentation but it may be impossible or even undesirable to remove vagueness from discussions. I can't think of a situation where vagueness is a crucial aspect. Do you have an example?
Except that you do want your conversation/discourse to withstand the pressure of logical clarification. The puzzle suggests that any clarification renders the clarity at one point (e.g. a single grain) incompatible with vagueness/tolerance further along. E.g.,
Quoting TheMadFool
I agree, but try setting a limit (higher than zero grains) on your projection of larger and larger samples of usage. E.g. on your statistical "support" for projected possible usage. It doesn't look very scientific to say no one could ever call a single grain a heap, after all. The intuition of clarity is lost rather easily. The heap game usually pumps it, though, which is fun, and gratifying if (like me) you think the intuition of clarity at some point is important.
Quoting TheMadFool
I think one of the reasons why the Sorites Paradox is important and comes up so frequently isn't not only that we have a problem with vagueness. It's also that mathematics, as we understand it today, is built upon or founded on the practical need for counting. Hence we start with counting natural numbers. Now math has developed from this practical need, but it's logical foundations might not be good to be chained to counting. Now a heap confuses this thinking that "Let's start with counting" and we tend to just think of it as problem of mixing math with definitions from a spoken language.
I think it isn't just a problem of vagueness as it isn't so even in Mathematics. The problems what we have with infinity and what many mathematicians and philosophers had with what is now termed limits just shows that everything doesn't start from natural numbers and counting. The obvious reasoning what has puzzled people for long is that there cannot be a largest number and there cannot be a smallest number. Yet infinity and the infinitesimal, or limits are very useful. Both they have a different logic to them. And so does a heap.
Basically you have incommensurability between a heap and an exact number of grains. The paradox rises when we don't take into account the incommensurability between the two.
So that's what's wrong. Simply that we think every logical system can be reducted to a simple system of arithmetic. Why the paradox is so persistent is that we don't understand that incommensurability is part of the foundations of mathematics, which is extremely important for the whole system to be logical.
Here are some definitions of "heap" I got from the web:
These are all consistent with my understanding of the meaning of the word. Based on that, I'm willing to state that a single grain is not a heap. Absolutely? Nothing in language, or anything else, is absolute.
Which will be all the time, then, because our usage of heap quite clearly appeals to numerical comparisons to decide cases, and withholds the term from, well, small numbers of grains. It relates in an obvious though not necessarily exact way to usage of large number and, like that sibling concept (which provides its own popular variant on the game), it (heap) inevitably involves some sort of commensuration with the series of natural numbers. Some kind of imperfect correlation, deserving of adequate formulation, why not.
Quoting ssu
(Leaving aside the politics of reducing or not reducing formal systems to arithmetic).
Hooray if that means you want to accord respect to usage of heap, in some way that resists correlating it perfectly with the naturals, in either of the two common but unsatisfactory perfect correlations:
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.
If "incommensurability" means settling for both reductions, to be used according to context, then boo, and the game isn't pumping intuitions as it ought, or at any rate sometimes can.
I just don't see a paradox. I think this has been pointed out already to a degree. However what hasn't been mentioned is the fact that our human brain is capable of tracking a certain amount of objects accuratley without counting (I think it's around 7-13 but am not sure). This obviously varies between humans. If the amount of elements is to big to track without counting we call it a heap or other simular concepts, that rather seem to be based on size and not ammount. It's worth mentioning that there are primate species (at least one) that seemingly can track more elements without counting. Aswell as a bunch of experiments where rats had to press a button a certain ammount of times to get a food reward. With increasing number the succes rates decrease to blind guesses. So we seem to have species related differences.
I also want to agree with the objection calling this view out for using the ad populum fallacie.
I don't see much legitimacy in using a brain mechanism used for perceptions without specific ammounts be it heaps(visual), noises, smells, touche(s?), tastes to try to establish some kind of quantity for objectivity. Because the quantity of humans beliefing a fact does not change the fact and the paradox exactly arises because one tries to applying quantity (based on countimg) to an area where it does not apply.
No, but I think when people are inspired to declare that a single grain is absolutely not a heap, they mean it is a safe distance outside the range of potential application of the label "heap".
This sense of distance is destroyed by one of the usual suggested reforms of usage (correlation 1, above), which is to have a grade of heap for each natural number.
Are you comfortable with that reform, for some or all vague predicates?
Wasn't there some point or utility in dividing the domain into relatively few zones? (A slightly different point, I know, but connected with the sense of distance.)
An interesting aspect of the heap paradox is that, as all you have pointed out, is incommensurability is an issue. The term ''heap'' in common usage doesn't actually mean a ''certain'' number of grains. 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.
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.
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.
Agreed.
Quoting 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.
Quoting TheMadFool
Happy, myself, to make that mistake.
Quoting TheMadFool
Yes, and we can always play the bald man or other versions, for variety.
Let's first give a definition to incommensurability:
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.
Quoting 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.
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...
Quoting bongo fury
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?
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.
Quoting 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.)
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).
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.
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.
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?
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.
And for this you need arithmetic to apply and there needs to be a number system.
Quoting 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.
Quoting 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.
Ok, you said:
Quoting 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:
Quoting 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.
No second hooray, then: you think usage of heap should be kept separate from the naturals. But still hooray!
Quoting 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.
Quoting 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.
Quoting ssu
Who said exactly? We are interested in tracing vague discourse. Such as,
Quoting 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...
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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 is true. I deny the existence of any word that I can't pronounce. And "incommensurability" is one of them.
This is not the actual problem. You re-worded them to suit your model. That is not fair.
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.
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.
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.
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.
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.
Quoting ssu
Quoting 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.
So basically your argument is the vagueness of the language.
Quoting bongo fury
Ok, it's seems you didn't get my point, because I don't find anything close to my reasoning in this.
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.