The (Problem of the) Heap of Sand is the classical representation of vagueness, where you create an inference: if you take away a grain from the heap then it remains a heap. This is a good inference for a long time, but eventually breaks down due to the vagueness of what a heap is (eventually you will have taken so many grains away that it wont resemble a heap anymore). And you can’t even tell when the inference will break down, exactly. So, as you can see, vagueness has a very precise and non-vague formulation, something similar to this strange property of vagueness (that it is a clear and distinct concept, even though it is a concept about indistinctness) is common for mathematical concepts. That is, mathematical concepts are chosen and defined so that they contain the terms necessary to get round their own difficulties—their faults or cracks, as all concepts are imperfect. Example: “Completeness”.

The difference between vagueness and completeness, is:

1) that the term completeness is in direct opposition to the claim that every concept has a “fault” or “crack”, in this it is a “perfect” term. The definition of completeness, however, must be vague, because the terms used to define completeness all have cracks.

2) The definition of vagueness, by contrast, is perfect, but the term refers to an essential fault of all terms, and is always perceived. (perception is always vague) Vagueness is defined as how the if, then fails— The “heart of logic” (the paragon of precision) and the most reliable way to preserve truth. It is clear and distinct in its description of how the if, then fails. Since each of the terms in the definition of vagueness have cracks, the definition is precise.

The term vagueness refers perfectly to its referent because it is well defined as an indistinctness, the definition of completeness is vague because the term refers to something that can’t be perceived, a perfect whole.
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