Your brain might be persuaded, grain by grain, to the position of pointing a heap-word or a heap-picture at a single grain. That doesn't solve the puzzle.

The puzzle is how to avoid arriving at that position, without denying the validity of any one step along the way.

You've lost one of the two required (and puzzlingly opposed) intuitions that we are trying to reconcile. — bongo fury

I like your solution. The "image" of a pile of sand is different from the "image" of a grain of sand. So then the reason why we refuse to accept that anything has happened to the pile of sand when we remove a single grain of sand is that to our brains, these two are different "images" or concepts.

It's kinda like how a human being isn't just the cells that constitute faer. We can't talk of altering the human by taking or adding cells - cells and humans, though the latter is made up of the former, are entirely different, in your words, "images".

That's as far as I could get.

Thanks for the response. It appears you are on the right track. There are many similar examples of the sorites paradox. They are easily missed though because our minds are occupied by the "single" groupings - as your example shows.

It's easy to understand the pile can be change by adding, or subtracting, grains of sand - but it's the "images" in our brain that doesn't change simply by adding or subtracting a single grain. We just don't have that many images of piles of sand such that we can visualize a "different" pile to account for every single grain. Sometimes the paradox can appear complex, but the principle of this solution is simple.

I suppose it boils down to having multiple images for a "pile" of sand and removing grains of sand one by one simply switches between one of these images of a pile and another image of a pile, the result being the pile remains a pile despite grains of sand being taken away.

However, an intriguing point to note is that though we have multiple images for a pile of sand and even if all these images are piles, I feel that if asked whether all these images we label pile are identical?, the reply will be "no". It's like all of us are human but we're not identical to each other.

Too, it just dawned on me that a pile is about a certain type of shape, a shape that approximates a hill or mound. A pile is not about the number of grains of sand. Although physics might demonstrate there's a correlation between the number of grains of sand and the shape of the entire collection of sand grains, it's obvious that a very large number of sand grains need to be removed before a pile of sand loses its hill-like shape. In my humble opinion we make a similar mistake when we believe that the circular shape of a circle of people depends on the number of people in it. No, we can have a circle of 3 people, 4 people, 5 people, 6 people,.., n people.

We just don't have that many images of piles of sand such that we can visualize a "different" pile to account for every single grain. — Don Wade

The puzzle doesn't require us to guess, nor to fail to guess, the numerical size of a heap. It tells us the size, at each step. Whether we can reliably point a numerically distinct heap-picture, at each step, is no more relevant than whether we can reliably point a specific number-word, at each step. The puzzle does that part for us.

Then we are asked if we think that that particular numerical size of grain collection deserves to be pointed at by a word ("heap") which is a good deal less specific than the number-word. Even though we are in no doubt as to the perfectly specific number.

You're assuming that we should always be as specific as possible. The puzzle is about the behaviour of words that are deliberately non-specific.

I suppose it boils down to having multiple images for a "pile" of sand and removing grains of sand one by one simply switches between one of these images of a pile and another image of a pile, the result being the pile remains a pile despite grains of sand being taken away. — TheMadFool

The problem, as stated in the sorites paradox, is not being able to determine at what point a group of grains of sand becomes a pile. Whatever number one may choose that constitutes an image of when a group of grains becomes a pile - then one grain added, or subtracted, should "change" that image. But, it doesn't. The "image" of the pile remains the same whether a grain is added, or subtracted. It's not the math - it's the way our brain works in creating the image.

How many grains of sand does it take to make a pile?
How many images of grains of sand does it take to make an image of a pile of sand?

The first question gets the sorites paradox going. The second question is nonsensical.

The problem, as stated in the sorites paradox, is not being able to determine at what point a group of grains of sand becomes a pile. Whatever number one may choose that constitutes an image of when a group of grains becomes a pile - then one grain added, or subtracted, should "change" that image. But, it doesn't. The "image" of the pile remains the same whether a grain is added, or subtracted. It's not the math - it's the way our brain works in creating the image — Don Wade

I see. So, the image doesn't change despite the removal/addition of sand grains from the pile. Do you see any possibility of superimposing my take on the issue onto your image theory of the Sorites paradox?

As I mentioned, a pile even though composed of sand grains isn't about numbers for if it were then taking away/adding sand grains should make a difference to the pile. Ergo, a pile isn't about the number of sand grains and one way to look at it is your image theory. My own opinion is that a pile is a shape - a hill-like one resembling a bell curve - and this particular shape is retained in spite of changes in the number of sand grains in a pile. You could say that the image is a shape, both don't change even with a considerable increase/decrease in the number of sand grains.

Is the "shape" a paradox?

The first question gets the sorites paradox going. The second question is nonsensical. — jkg20

The paradox is in the image - not in the actual number of grains of sand. There is no paradox in the actual number of grains of sand in the pile. The paradox happens when we try to visualize the "change" in the pile, in our mind, when the acual number of grains is changed physically. There are many things we do not see even though they may happen in plain sight. A classic example is" the "gorilla in the room". http://theinvisiblegorilla.com/gorilla_experiment.html .

Even though the pile-size may change, the visual image of the pile remains the same. — Don Wade

It is a feature rather than a bug that language is vague at base. So the paradox doesn’t need “solving”. Being able to speak in generalities is the point. We can gloss over the multi fold differences that don’t make a difference ... to us, for some reason, at that moment.

How are you actually imagining your pile? Is there at least one grain stacked upon another? Is there more than a single layer of grains?

That generalised image must mean the smallest pile is 4 grains - three as a triangular base and one perched on top. Take that away an only have a clump or group of grains? Move the grains gradually apart and at some point they are not even a group?

Vagueness always exists when we form some verbal or imagistic generalisation. That is just everyday epistemic vagueness.

Where the metaphysics gets interesting is when the vagueness is ontic - a fact also of the world itself.

That generalised image must mean the smallest pile is 4 grains - three as a triangular base and one perched on top. Take that away an only have a clump or group of grains? Move the grains gradually apart and at some point they are not even a group? — apokrisis

There was a group of Germans that studied "visual grouping" back during the early 1900's. The study was called "Gestalt". https://en.wikipedia.org/wiki/Gestalt_psychology . How the mind groups information including visual information is very interesting.

I never understood what the problem was. The answer is 4. 1 is obviously not a heap; 2 is either a short row or a carefully balanced column; 3 is a row, a column, a plane or a wall; it takes 4 grains to have a layer of three and one on top, heaped up.

Slightly more formally: treating a grain as a centre of gravity with a spatial boundary, four points are required to define a three dimensional space, so four grains are required to define a three dimensional shape that constitutes a heap.

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

Try bald vs. hairy, black vs. white etc.

Chair vs. settee.

Try bald vs. hairy — bongo fury

Obviously it depends whether it's my pate, or my wife's chin. Let's assume that I am not bald and my wife's chin is not hairy, and don't fucking argue if you know what's good for you.

Slightly more formally: treating a grain as a centre of gravity with a spatial boundary, four points are required to define a three dimensional space, so four grains are required to define a three dimensional shape that constitutes a heap. — unenlightened

One can always add formal precision to a definition or constraint. And yet vagueness also remains. It is inherent in the world itself.

In ordinary language, “heap” has connotations of careless formation. A pile without a formal design, just randomly built up. So a tetrahedral volume is the opposite of a careless pile. Can it thus really be called a heap if we precisify our definition and emphasise the connotation of random formation?

Applying chaos theory, we could indeed apply a formal definition of “sufficiently random”. The question now is what is the least number of grains that could be piled in such a fashion. Sphere packing maths does just this. It contrasts regular stacking and random settling. https://en.wikipedia.org/wiki/Random_close_pack

That would argue more than four grains would be needed to arrive at a stable yet random heap.

So my point is that formal definitions of natural situations can indeed be tightened up as much as desired. That is not the issue.

What is of interest is whether formality can ever exhaust the availability of tiny differences or further distinctions. Is reality in fact atomistic as logicism likes to presume. (spoiler: no).

And then going with that acceptance of ontic vagueness goes the recognition that formality in fact doesn’t function because it is precise but because it can apply epistemic generality. Logic is semiotic. It is an exact way of ignoring the underlying vagueness of the world by making choices about what differences don’t count as making any damn difference, from some agent’s point of view.

↪jkg20
The paradox is in the image - not in the actual number of grains of sand.

The only images that might be considered paradoxical are of the Rescher variety. Where does the paradox lie in the image of a grain of sand? Where does the paradox lie in the image of a pile of sand? If your suggestion is that the paradox lies in the fact that we are somehow trying to force the image of a grain of sand to "align" with an image of a heap of sand, I refer you to my original quotation.
The paradox is in the concept of a heap. If one were an antirealist, that might lead one to thinking that paradox is also a feature of the world. On the other hand, one might just blame it all on vagueness and walk away.

Where does the paradox lie in the image of a pile of sand? — jkg20

https://www.bing.com/search?q=paradox%20definition&pc=cosp&ptag=G6C24A11441EEDE3&form=CONMHP&conlogo=CT3210127 . The paradox is not the first thing you see. However, it is there.

This solution is not predicated on vagueness or fuzzy logic - it is simple recognizing the limits of how our brain creates images of objects. — Don Wade

Yes. I'm not qualified to follow the complex logic & arcane terminology of your link : Supervaluationism ; Hysteresis ; Resolutions in utility theory ; etc. But a simple philosophical change of perspective can allow you to see the Whole instead its Parts. No abstruse math required --- not even addition (summation). Just re-focus the eye of your mind. :smile:

Holism : Philosophy
the theory that parts of a whole are in intimate interconnection, such that they cannot exist independently of the whole, or cannot be understood without reference to the whole, which is thus regarded as greater than the sum of its parts. Holism is often applied to mental states, language, and ecology.

Holism as a philosophical perspective :
https://www.newworldencyclopedia.org/entry/holism#Holism_as_a_philosophical_perspective

This paradox correlates with Mr. Zeno's paradox ( "The paradox of the large and the small"). Aristoteleans say quantum mechanics is weird because it approaches prime matter. But is a leaf closer to prime matter than a tree? Even if matter is pure extension, if you start on the east side of a building and try to get to the smallest limit, you can only go west. So the large IS the small

As many Pyrhonnians in history have concluded, the infinite is contained in the finite so there really isn't a distinction between the two

the infinite is contained in the finite so there really isn't a distinction between the two — Gregory

So, the un-bound is restricted by the bound, or the un-limited is confined within limits. Sounds like, not a paradoxical koan puzzle, but a simple contradiction in terms. If anything, I would expect the opposite relationship to be true : our finite space-time world exists within the context of Eternity & Infinity. Is there a rational interpretation of that koan? :smile:

If finite things (infinitesimals) make up finite objects, then the finite stands outside an infinite

I'm going to have a busy day, but we can talk more latter. I'm going to finish reading two books im the middle of and try to get a thread up on infinity inside of finite

"Infinity, the affirmation as the negation of the negation", Hegel

Your proposed solution sounds to be in a logical sense the same as mine, which is to consider the heap to be a co-inductive type.

First consider the recursive definition for the set of inductively constructed lists.

Inductive List := { [ ] AND x : Inductive List }

Evaluating this equation by substituting the left-hand side into the right-hand side is analogous to building every possible collection of sand-grains by starting from nothing and adding a grain at a time. Yet we don't call a collection of sand-grains constructed by this process a heap, because "heaps" aren't defined by induction.

In contrast, the set of co-inductively constructed lists is defined by switching the AND for an OR:

Co-inductive List := { [ ] OR x : Co-inductive List }

The set of co-inductive lists contains the inductive set of lists, but because it isn't obligated to build lists by starting from empty, it also includes an additional list of "infinite" length. By "infinite" we are merely referring to the fact this set contains the definition of list that might never terminate upon iterated evaluation.

The former type of list corresponds to the natural numbers, whereas the latter type corresponds to the conatural numbers, otherwise known as the extended natural numbers that includes a numeral for infinity, which in computing can be used to denote programs will never halt to produce an output.

I think that the grammar corresponding to "a heap of sand" is analogous to the "infinite" element of a coinductive list, which as you say is like imagining a heap of sand that never changes after a grain is removed or added.

Sorry for butting in without reading the many answers it has already revised but, what you are saying isn't just a variant of emergentism? I mean the critical point determines when a heap/pile becomes such.

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?

I mean the critical point determines when a heap/pile becomes such. — DeGregePorcus

Sure. The critical point could be 53. But it could be 530. So, could it be 1? If so, no puzzle. If not, what's the lowest number it could be?

I guess the critical point is one of proportion and perspective and that is why the brain is such a relevant issue to this seemingly emergentist solution... To a man-sized being a heap has more than a "few" grains arranged one upon another, to a giant that's invisible and you need more grains or larger grains, to a bug you need finer grains... and so forth.

To a man-sized being a heap has more than a "few" grains arranged one upon another, — DeGregePorcus

Yes, but how many?