But MN isn’t about looking at and labeling things, it’s just about nit picking over whether or not the arm of a chair is truly a part of the chair or even whether it’s really a chair in the first place because of some vague insecurities over our senses. At least, that’s the impression I get.

Anything that involves debates over the definition or ‘meaning’ of “part” is semantics. And pointlessly silly.

Feels more like semantics to me. The idea of whether ‘parts’ really exist depends entirely upon what one’s definition of ‘part’ is and how obsessively one wants to nit pick things.

What interests me is what is there when we are not looking, labeling, etc.

I don’t think any of this is semantics.

Stephen

The Problem Of The Heap certainly is.

And the concept that small parts not being able to combine into bigger objects which have enhanced utility is demonstratably false. A car is certainly more useful than it’s components, as is a brain more useful than it’s cells. The whole thing sounds like a misunderstanding of the Heisenberg Uncertainty Principle to me, that because an individual particle isn’t well-defined, that groups of particles also cannot be - which is obviously untrue. Molecules do not act like quarks, and macro-scale objects consisting of billions of molecules do not act like molecules. There is obviously order added at the different levels of vast scale.

Also, I find the concept that things may not exist if we don’t observe them to first violates Occam’s Razor, and second, is also highly egocentric.

Or: you’re just changing the definition of “heap” as you go on.

I agree with DM. Sounds like nothing more than cheap word games.

Definition of heap: A pile of substance or of a number of objects. (Compact Oxford English Dictionary)

From this definition, there is no inconsistency to say that 999,999 grains of sand is still a heap of sand. If one defines 1,000,000 grains as a heap of sand and therefore 999,999 grains is not a heap of sand, that is an arbitrary definition. Why not 500,000 or any other number?

No word games or semantic indeterminacy here. It is just that the boundaries of many, if not all worldly objects, are inherently vague. Hence, the paradox of the heap is difficult to refute. MN resolves the paradox by denying the existence of the heap as well as everything, except for fundamental quantum objects. It is a drastic solution.

But MN isn’t about looking at and labeling things, it’s just about nit picking over whether or not the arm of a chair is truly a part of the chair or even whether it’s really a chair in the first place because of some vague insecurities over our senses. At least, that’s the impression I get.

Anything that involves debates over the definition or ‘meaning’ of “part” is semantics. And pointlessly silly.

Not really. MN is about finding out what is ultimate reality, irrespective of how we experience or interpret it.

Part can be defined as:

1. a piece or segment which is combined with others to make up a whole

Or: you’re just changing the definition of “heap” as you go on.

I agree with DM. Sounds like nothing more than cheap word games.

Definition of heap: A pile of substance or of a number of objects. (Compact Oxford English Dictionary)

From this definition, there is no inconsistency to say that 999,999 grains of sand is still a heap of sand. If one defines 1,000,000 grains as a heap of sand and therefore 999,999 grains is not a heap of sand, that is an arbitrary definition. Why not 500,000 or any other number?

No word games or semantic indeterminacy here. It is just that the boundaries of many, if not all worldly objects, are inherently vague. Hence, the paradox of the heap is difficult to refute. MN resolves the paradox by denying the existence of the heap as well as everything, except for fundamental quantum objects. It is a drastic solution.

Here’s a different solution:

I’ll refer to the “heap” as a set of sand grains, where Ax is the xth sand grain:

[A1, A2, A3, . . . A999999, A1000000]

Take away A1000000, and you get [A1, A2, A3, . . . A999998, A999999]

And take away A2 through A1000000, you get [A1].

No problem.

Take away the last A1, and you get the null set, []. Still no problem.

Therefore, this heap problem still looks to me like a poor definition problem, not any kind of legitimate logical or mathematical problem. By specifically defining the “heap” as a set in this manner, this definition problem goes away.

I’ll refer to the “heap” as a set of sand grains, where Ax is the xth sand grain:

[A1, A2, A3, . . . A999999, A1000000]

Take away A1000000, and you get [A1, A2, A3, . . . A999998, A999999]

And take away A2 through A1000000, you get [A1].

No problem.

Take away the last A1, and you get the null set, []. Still no problem.

Therefore, this heap problem still looks to me like a poor definition problem, not any kind of legitimate logical or mathematical problem. By specifically defining the “heap” as a set in this manner, this definition problem goes away.

Which set is the “heap”?

Is it [A1, A2, A3, . . . A999999, A1000000], [A1, A2, A3, . . . A999998, A999999], [A1], the null set, [] or are they all “heaps”?

And how would you define a set with a negative number of grains of sand?

Defining the “heap” as a set of grains of sand is no solution at all. The paradox remains unresolved.

Which set is the “heap”?
Is it [A1, A2, A3, . . . A999999, A1000000], [A1, A2, A3, . . . A999998, A999999], [A1], the null set, [] or are they all “heaps”?

Yes.

And how would you define a set with a negative number of grains of sand?

Not relevant. You can’t have a heap with a negative number of grains of sand, either.

Defining the “heap” as a set of grains of sand is no solution at all. The paradox remains unresolved.

Why not? I’m using that example to show how the problem is in the undefined nature of what is a “heap.” Just because one chooses a word that is poorly defined does not mean that the grains of sand do not exist, and demonstrating the grains of sand as a set which can be added to or subtracted from is merely a way of showing that approaching the same problem with a word that DOES make sense makes the apparent paradox go away.

Which set is the “heap”?
Is it [A1, A2, A3, . . . A999999, A1000000], [A1, A2, A3, . . . A999998, A999999], [A1], the null set, [] or are they all “heaps”?

Yes.

So,

[A1, A2, A3, . . . A999999, A1000000] is a heap of sand
[A1, A2, A3, . . . A999998, A999999] is a heap of sand
.
.
.

I see no paradox. [A1] and [] are both valid sets.

They are, but the cardinalities of [A1, A2, A3, . . . A999999, A1000000] etc. are different, therefore they are different sets which cannot be known by the same name, “heap”.

Furthermore, sets do not have the properties of their elements. For instance, the set of natural numbers N, is an object in it’s own right, but it is not a number. The set of men is not a man. So, the set of sand grains is not sand grain. Hence, to define [A1, A2, A3, . . . A999999, A1000000] etc. as “heaps of sand” is inappropriate and misleading.

The paradox is about grains of sand, not sets of grains of sand. To replace it with sets and declare that there is no paradox is no solution.

I would object to this theory, but I see that it follows from it that I don’t exist. A fortiori, I can’t object. I console myself with the thought that proponents of the theory do not exist either, and therefore cannot propound the theory.

I see no paradox. [A1] and [] are both valid sets.

They are, but the cardinalities of [A1, A2, A3, . . . A999999, A1000000] etc. are different, therefore they are different sets which cannot be known by the same name, “heap”.

Furthermore, sets do not have the properties of their elements. For instance, the set of natural numbers N, is an object in it’s own right, but it is not a number. The set of men is not a man. So, the set of sand grains is not sand grain. Hence, to define [A1, A2, A3, . . . A999999, A1000000] etc. as “heaps of sand” is inappropriate and misleading.

The paradox is about grains of sand, not sets of grains of sand. To replace it with sets and declare that there is no paradox is no solution.

There is nothing about either the set or the heap which says that it must have the same properties as it’s elements.

And no, it’s not a solution. It’s an attempt at a demonstration of the problem of the definition of the word “heap.” Until you come up with a more precise definition of “heap” than “pile,” this “paradox” will always be only derived from the ambiguity in the language which doesn’t define how small a heap can be. Until you make the language clear, you can’t have any kind of logical paradox, because such needs clear definitions. By delivering one form of a clear definition, in the form of a set and subsets, this apparent paradox goes away. I challenge you to deliver another clear definition of a “heap” which still retains the paradox you’re talking about.

There is nothing about either the set or the heap which says that it must have the same properties as it’s elements.

And no, it’s not a solution. It’s an attempt at a demonstration of the problem of the definition of the word “heap.” Until you come up with a more precise definition of “heap” than “pile,” this “paradox” will always be only derived from the ambiguity in the language which doesn’t define how small a heap can be. Until you make the language clear, you can’t have any kind of logical paradox, because such needs clear definitions. By delivering one form of a clear definition, in the form of a set and subsets, this apparent paradox goes away. I challenge you to deliver another clear definition of a “heap” which still retains the paradox you’re talking about.

No, but upon reflection, it is clear that a collection of things is not the thing. It is a “collection of things”, a different object in it’s own right.

This paradox can be reconstructed for a variety of predicates, for example, with “tall”, “rich”, “old”, “blue” and so on. Bertrand Russell argued that all of natural language, even logical connectives, are vague; most views do not go that far, but it is an open question.

Proposed solutions:

1. Setting a fixed boundary:

A common first response to the paradox is to call any set of grains that has more than a certain number of grains in it a heap. If one were to set the “fixed boundary” at, say, 10,000 grains then one would claim that for fewer than 10,000, it’s not a heap; for 10,000 or more, then it is a heap.

However, such solutions are unsatisfactory as there seems little significance to the difference between 9,999 grains and 10,000 grains. The boundary, wherever it may be set, remains as arbitrary and so its precision is misleading. It is objectionable on both philosophical and linguistic grounds: the former on account of its arbitrariness, and the latter on the ground that it is simply not how we use natural language.

2. Unknowable boundary or epistemic uncertainty:

Timothy Williamson and Roy Sorensen hold an approach that there are fixed boundaries but that they are necessarily unknowable.

There are more exotic “solutions” like supervaluationism, truth gaps, gluts, and many-valued logics, hysteresis and group consensus which are found in the same wiki.

There is one more thing to consider. The world is by nature, vague. In other word, all objects are vague and it is not possible to define them precisely because they are vague. This is ontic vagueness, as distinct from semantic vagueness.

I would object to this theory, but I see that it follows from it that I don’t exist. A fortiori, I can’t object. I console myself with the thought that proponents of the theory do not exist either, and therefore cannot propound the theory.

We, who do not exist, depend on the fortuitous grouping of vague fundamental quantum entities which do exist. “In vagueness we trust” is the mantra to repeat to “create transformation” from existence to non existence.

The word “sorites” derives from the Greek word for heap. The paradox is so named because of its original characterization, attributed to Eubulides of Miletus.[1] The paradox goes as follows: consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:

1,000,000 grains of sand is a heap of sand (Premise 1)
A heap of sand minus one grain is still a heap. (Premise 2)

Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand (and consequently, if one grain of sand is still a heap, then removing that one grain of sand to leave no grains at all still leaves a heap of sand, and indeed a negative number of grains also form a heap[2]).

On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. Peter Unger defends this solution. Alternatively, one may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may accept the conclusion by insisting that a heap of sand can be composed of just one grain.

I still think that this shows that someone who is trying to show a paradox here is changing their definition of “heap” midway through the process. If, literally, a “heap” is defined ONLY by those two premises, then there is no reason why a “heap” cannot have 0 or 1 grains of sand in it. There is no reason why one would need to avoid this conclusion as that last paragraph implies there must be. It looks to me very much like an example of a semantic “bait and switch,” where the reasoner only starts with Premises 1&2, but then without specifically stating so, adds the common usage definition of “heap” to the mix - without justification. Those two solutions that you list would still be adding another premise at the outset, essentially coming to a different definition of “heap,” with different conclusions to be drawn.