If you could call 10,000 grains a heap of sand, then so would be 9,999 and so on. So one grain of sand is also a heap…what a paradox.
Unless you develop some more specific meaning for a heap which while not defining a specific numeric requirement of grains of sand (such as 5271 grains = heap) you can still have a functional definition for heap.
I would imagine it to be something along the lines of: any amount of sand comprising of at least 2 specific layers of sand with a bottom layer being of greater length and width than the higher layer(s).
Right: if we’re strict about our definitions, we don’t have to define a ‘heap’ circularly in this way.
The paradox plays on the ambiguity of the group-word: how many wolves are a ‘pack’? How many zeppelins are ‘a lot’? How many bananas are a ‘bunch’?
Interestingly enough, this turns out to be a kind of a ‘discrete’ version of Zeno’s paradox of continuity. If the definition of a ‘set’ (of anything) is continuous through one element, then one of anything is a group (of one) of anything.
In fact, the distinction between groups and elements here is quite fundamental for set theory, the axiomatic foundations of algebra and calculus. You might enjoy Alain Badiou, who does a really interesting reading of set theory and its relation to ontology, as well as politics more generally–questions about who belongs versus who is included, and so forth. He’s very outspoken about the treatment of immigrants (in France, they’re called the ‘sans papiere’ or ‘without papers.’) It seems strange at first to connect these, but after reflection it does kinda make sense for a militant mathematician to care about how people get counted!
Then lets define a heap of sand starting from 10000 grains of sand, and whatever goes below that, even 9999 is not a heap!!
If no one does it, I will. No more paradox.
A heap of sand is something that acts in a certain way. You could go numerically and ask “How many grains must a heap have for it to be a heap?”, but that’s not the definition of heap. The heap of sand is defined by its use.
That’s not a paradox, that’s a generalization.