The Paradox of the Heap
A single grain of sand is not a heap; that’s obvious. A heap is a collection of things, you need several things to make it up. Two grains of sand isn’t a heap either; a heap is a collection of several things, more than just a couple.
The concept of a heap is fuzzy, though; there’s no precise number that marks the difference between heaps and non-heaps. It’s not as though 37 grains of sand aren’t a heap but 38 are. Defining precisely how many things one needs in order to have a heap is impossible.
This is what gives rise to the paradox of the heap (also called the “Sorites paradox”, sorites being the Greek word for heap).
Suppose that we have a collection of a million grains of sand. That is absolutely, definitely, undeniably a heap.
Because there is no precise number that separates heaps from non-heaps, removing a single grain of sand from a heap will never turn it into a non-heap. If you have a heap of sand, and you take away a single grain, then you still have a heap.
If you have a heap of a million grains of sand, though, and repeatedly take away a single grain of sand, doing so 999,999 times, then what do we have at the end of the process? Is it a heap or not?
Taking away a single grain of sand cannot turn a heap into a non-heap. We had a heap of sand at that beginning of the process. All we did was take away single grains of sand. Therefore what we have at the end of this process can only be a heap.
What we have at end of the process, though, is a single grain of sand, and, as we said at the beginning, a single grain of sand is obviously not a heap. The single grain, then, both is and is not a heap.