My question : I don’t understand this. Given an accurate definition of a heap, it should be trivial to say how many grains of wheat are required before you can call the collection a heap. I would say that heaps require either no grains of wheat or somewhere between 3 and 5. (i.e. enough to balance a peak on a base, in however many dimensions the problem is specified).

A heap is a property of a collection, not of its’ members, so it seems like sophistry to suggest than analysis of its members could lead to any determination of whether it is forming a heap or not.

A heap is a collection. What defines a heap depends on the circumstances. The circumstances generally arise when we want to refer to a group of particulars WITHOUT needing to count them. The fact that you want to count the particulars of a heap shows that you are no longer referring to a heap.

It’s simple. alll of these so-called paradoxes can be answered.

Think about it, if you know that a pile of indeterminate amount is a heap, and you know that 1 grain in there is not, then at some point there is a change. If you keep adding one grain, it will never change, and so you enver have a heap.

So, obviously, if we can have a heap, some part of that reasoning that says we can’t must be flawed. In this case, the originators assumed that a pile’s “heapness” is binary, it either is or it isn’t. Instead, describe it as a scale of heapness (from 0 to 1), where 1 grain of wheat is a 0, and pile of 10,000 grains is a 1. Each grain added therefore makes the pile 1/10000th more of a heap than before.

The limitation is the word used to describe it. When someone calls a pile of wheat a heap, they mean, “that pile is at least a 0.x on the heapness scale, which to me is a heap.”

Your post was stupid. Your post was stupid on two counts. The first point on which your post was stupid was that it merely repeated common misconceptions arising from the misplaced application of counting to the concept of a heap. The second count on which your post was stupid was that it ignored a solution, presented above your own post, to the initial problem and to these aforesaid misconceptions. Please re-read the post above yours carefully.

Of course, this is yet another example of JJ missing the point. My post was mostly tongue-in-cheek. Obviously my model for "heapness’ is silly, taking an overly analytical route to demonstrate the paradox’s assumptions are flawed.

See, what my little “model” implies, in between sloppily written lines, is that the pile grows imperceptibly larger with each grain, and at some point, our brains decide that the pile is big enough to qualify as a heap. That point is purely subjective.

All you are saying is that last part…which is perfectly true. Outr concept of heap is (very roughly) proportional to the size of the pile of wheat. I’m simply relating grains to pile size, which we do judge.

My whole point was WE DO NOT JUDGE grains to pile size. We just say ‘that heap…over there’. A heap can have one or more, it does not matter, because a heap is NOT COUNTED.
JJ