Well yes, we can clearly see that the second premise is false after some inductive reasoning.
But there’s also another route, the non-inductive route: can you give me a single example of a heap of sand that becomes a non-heap when you remove a grain?
The point is not that heaps are magic or induction is broken or anything like that. The point is that humans are awful at finding the boundaries of their categories. And as Wei Dai would note, we can’t just get around this by playing taboo when the thing we’re supposed to be finding the boundary of enters directly into our utility function.
If you have four grains of sand arranged in a tetrahedron, you could conceivably call it a (very small) heap. When you take away one of the grains, you will no longer have a heap, just three grains of sand.
This is assuming that your definition of “heap” includes some of it being on top of the rest of it, which I’m fairly sure is standard.
To avoid this paradox you can make the following rule:
(“The heap of sand minus one grain is still a heap ” is true) if and only if (The heap of sand minus one grain still constitutes a heap) in the style suggested in this lesswrong post
But there’s also another route, the non-inductive route: can you give me a single example of a heap of sand that becomes a non-heap when you remove a grain?
Yes that’s pretty easy, since it’s only a question of what you call a heap. The paradox is basically feeding you a bait by asking you to think of a million grains of sand, which you obviously can’t quantitatively visualize and that could result into abandoning trying to find definitive criteria.
As I’m not a native speaker of english I’m not sure if my idea of a “heap” corresponds to what it’s generally used to refer to, but I’d draw the mininum boundary at 1 grain always not being a heap of grains. In my opinion a heap refers to a count of objects above 1. In addition I think it also refers to a geometric structure where objects are arranged in such a fashion that some objects are supporting other objects on top of them. You could also try and make a distinction between stacks and heaps.. Anyway I think you should just drop suggested the million grains and start from “3 grains” and ask yourself “if I remove one grain, is what’s remaining a heap?”
Well yes, we can clearly see that the second premise is false after some inductive reasoning.
But there’s also another route, the non-inductive route: can you give me a single example of a heap of sand that becomes a non-heap when you remove a grain?
The point is not that heaps are magic or induction is broken or anything like that. The point is that humans are awful at finding the boundaries of their categories. And as Wei Dai would note, we can’t just get around this by playing taboo when the thing we’re supposed to be finding the boundary of enters directly into our utility function.
If you have four grains of sand arranged in a tetrahedron, you could conceivably call it a (very small) heap. When you take away one of the grains, you will no longer have a heap, just three grains of sand.
This is assuming that your definition of “heap” includes some of it being on top of the rest of it, which I’m fairly sure is standard.
That’s what I would have said.
To avoid this paradox you can make the following rule:
(“The heap of sand minus one grain is still a heap ” is true) if and only if (The heap of sand minus one grain still constitutes a heap) in the style suggested in this lesswrong post
Yes that’s pretty easy, since it’s only a question of what you call a heap. The paradox is basically feeding you a bait by asking you to think of a million grains of sand, which you obviously can’t quantitatively visualize and that could result into abandoning trying to find definitive criteria.
As I’m not a native speaker of english I’m not sure if my idea of a “heap” corresponds to what it’s generally used to refer to, but I’d draw the mininum boundary at 1 grain always not being a heap of grains. In my opinion a heap refers to a count of objects above 1. In addition I think it also refers to a geometric structure where objects are arranged in such a fashion that some objects are supporting other objects on top of them. You could also try and make a distinction between stacks and heaps.. Anyway I think you should just drop suggested the million grains and start from “3 grains” and ask yourself “if I remove one grain, is what’s remaining a heap?”
This gets quickly to “This plucked chicken has two legs and no feathers—therefore, by definition, it is a human!” that is it’s hard to find a really solid definitive criterion and so you should instead just try and imagine a situation where you would no longer call the the remaining grains of sand a heap
http://answers.yahoo.com/question/index?qid=20100805132521AAcGBqs
:)