Heap or No Heap: A Possible Solution

Philosopher Timothy Williamson begins his essay on vagueness with the following thought-experiment:

"Imagine a heap of sand. You carefully remove one grain. Is there still a heap? The obvious answer is: yes. Removing one grain doesn’t turn a heap into no heap. That principle can be applied again as you remove another grain, and then another… After each removal, there’s still a heap, according to the principle. But there were only finitely many grains to start with, so eventually you get down to a heap with just three grains, then a heap with just two grains, a heap with just one grain, and finally a heap with no grains at all. But that’s ridiculous. There must be something wrong with the principle. Sometimes, removing one grain does turn a heap into no heap. But that seems ridiculous too. How can one grain make so much difference? That ancient puzzle is called the sorites paradox, from the Greek word for ‘heap’."

- Timothy Williamson, Aeon, "On vagueness, or when is a heap of sand not a heap of sand?" *

Williamson is right: there is something wrong with the principle "removing one grain does not turn a heap into no heap." One solution proposed by some contemporary philosophers is to replace traditional logic by something called 'fuzzy logic.' Curious readers can consult Google for more information about fuzzy logic, if they wish, but TW thinks that approach doesn't work, and besides, there is nothing broken here that needs fixing. He insists that traditional logic works just fine. The problem confronting us in the sorites paradox, he writes, is vagueness and ...

"Vagueness isn’t a problem about logic; it’s a problem about knowledge. A statement can be true without your knowing that it is true. There really is a stage when you have a heap, you remove one grain, and you no longer have a heap. The trouble is that you have no way of recognising that stage when it arrives, so you don’t know at which point this happens."

I agree with TW that the common use of the word 'heap' is loose and vague, but he throws in the towel too soon. Isn't it part of the job of philosophers to try to clarify troublesome terms so that they might better serve us in answering tough questions? TW declines to do this and so leaves us with the puzzle unsolved on account of terminal vagueness. I will argue here that there is a way of recognizing the stage when no-heap arrives and that the solution requires no exotic logical moves, just one additional concept. Here we go.

As in many other discussions of a philosophical problem, it is legitimate here to ask for a definition of the term 'heap.' At the outset it is vague indeed, and leaving it vague practically guarantees that no logical solution will be found. But why should we accept the initial vagueness?

Before offering a definition of 'heap,' I want to contrast it with another, not-so-vague term: holon. A holon (the term was first coined by Arthur Koestler) is any entity that is a whole and also a part of a larger whole.

Holons have a number of characteristics, but the crucial one for this discussion is that every holon has internal structural integrity or agency, that is, an internal force or principle that enables the holon to resist dissolution into its component parts. When a holon becomes part of a higher holon, the principle of functional organization of the lower is conferred or imposed by the higher holon. For example, a protein molecule is a holon with its own agency, and it is also part of a higher holon, a cell, which organizes the activity of the protein for its own benefit. If the cell moves around, the protein molecules and all the rest of the cell's component parts move with it; none of them can stay behind or go off on their own. For another example, an atom is a holon within which the strong and weak forces keep the component parts - protons, neutrons, and electrons - together as a functioning unit. The higher-level holon imposes its own agency on the lower components. Thus, when the atom changes its location in space, the whole atom moves. None of the components are lost or left behind. (Nuclear fission is an exception.)

Now we can define 'heap' as any collection of holons whose agency is not superseded by the higher agency of the collection. Each member is a holon, but the collection is not a higher holon; it has no agency of its own and hence no power to organize the activities of its members; no power of self-preservation. So if a member holon disappears or moves to another location, the rest of the collection does not disappear or move with it. Any such collection we can call a 'heap.'

This definition provides the basis for a solution to our puzzle. Recall Williamson's principle: "Removing one grain doesn’t turn a heap into no heap." By defining 'heap,' we have removed the vagueness from the principle. Since the principle is no longer vague, there's a greater chance that it can be used to solve our puzzle. Let's take a look.

We begin with our pile of sand. It is obviously a heap, but now we understand precisely what that means: it is a collection of grains, each of which is a holon whose agency is not subordinated to the agency of a higher holon. Next we apply the principle: removing a single grain of sand doesn't turn the heap into no heap. The remainder is still a heap - a slightly smaller heap, but still a heap. Remove another one and the heap is still a heap, not a no-heap. Continue removing one grain at a time. The heap becomes smaller and smaller. At what point does it cease to be a heap? In the original puzzle, there seemed to be no clear answer to the question; hence the paradox. With our new definition of 'heap,' there is a precise answer: as grain after grain of sand is removed, we eventually arrive at just two grains. Is it still a heap? Yes! It is still a collection of holons whose agency is not taken over by a higher holon. It's a very small heap, but still a heap.

Now remove one more grain. Do we still have a heap? No, because there is no longer a collection. No collection, no heap. The remaining grain of sand is a holon, standing alone with its magnificent self-contained agency, no longer a member of a collection with which it was only loosely connected in the first place. So the answer I propose to our puzzle is that a heap ceases to be a heap when the number of member holons has been reduced to one. In other words, the boundary between heap and no-heap is the boundary between the last grain of sand and the second-to-last.

This solution seems sound, but it requires us to amend our guiding principle to read "removing one grain does not turn a heap into no heap unless there are only two grains in the heap.

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