A proposed solution to the Sorites Paradox
The sorites paradox ,(paradox of the heap - and similar puzzles), has been debated by many for thousands of years: https://en.wikipedia.org/wiki/Sorites_paradox . Some consider the paradox to be based on assumptions of vagueness or fuzzy logic. I propose a different reasoning and solution.
The brain visualizes a "grain of sand", or a "pile of sand", using different "property-grouping" inputs for each item. The brain can only focus on one property grouping at any specific time. The brain will also not allow changes to take place between property groupings - once the groupings has been established. Therefore a "pile of sand" is not changed - in the mind - by adding a "grain of sand", or taking away a grain of sand. In the mind the pile still remains the same. The (property grouping) for the pile remains the same. Even though the pile-size may change, the visual image of the pile remains the same.
This solution is not predicated on vagueness or fuzzy logic - it is simple recognizing the limits of how our brain creates images of objects. This solution can be used in many of the puzzles similar to the sorites paradox. I also believe this is the first time this solution has been published.
The brain visualizes a "grain of sand", or a "pile of sand", using different "property-grouping" inputs for each item. The brain can only focus on one property grouping at any specific time. The brain will also not allow changes to take place between property groupings - once the groupings has been established. Therefore a "pile of sand" is not changed - in the mind - by adding a "grain of sand", or taking away a grain of sand. In the mind the pile still remains the same. The (property grouping) for the pile remains the same. Even though the pile-size may change, the visual image of the pile remains the same.
This solution is not predicated on vagueness or fuzzy logic - it is simple recognizing the limits of how our brain creates images of objects. This solution can be used in many of the puzzles similar to the sorites paradox. I also believe this is the first time this solution has been published.
Comments (51)
The puzzle is how to avoid arriving at that position, without denying the validity of any one step along the way.
Quoting bongo fury
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.
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.
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.
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.
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 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.
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.
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 .
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.
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.
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.
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.
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.
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.
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
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
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.
... 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?
Quoting 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?
Yes, but how many?
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.
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.
So... isn't a heap?
Quoting sime
Agreed. But what is the smallest number of grains that would need considering by speakers as a particular case? Is it 1?
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.
Quoting 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?
Yes, that is the problem.
Quoting sime
So... the answer to this question...
Quoting sime
... would be? 5 million grains, say... isn't a heap, in your logical representation?
Quoting 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,
Quoting bongo fury
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
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.
Quoting bongo fury
Quoting 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.
Quoting 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.
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.
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.
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.
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