Obviously the problem is that the premise isn’t true for any arbitrary N
This is a tangent, but...
This is certainly not obvious (I submit that if it were, the Greeks would have noticed) and I don’t think it’s true either.
Treating heapness as a boolean, the Sorites paradox indeed implies that there are no heaps, or that any number of grains is a heap. If N grains of sand together are not a heap, then N+1 grains of sand also are not; the piles are visually the same, and so anyone who knows what ‘heap’ means would look at them and classify them as both “heap” or “not heap”. I submit that if you ran that experiment for different values of N (showing each respondent only one instance of N and N+1), you would find no N for which most respondents classified one as a heap and the other as not a heap.
Since this is a paradox, we know there’s a problem with one of our assumptions. These days, the obvious one is that heapness is not a boolean. There’s a cluster in thingspace we call “heap”, and some things are clearly part of the cluster, like a pile of 2000 grains of sand, and some things are clearly not part of the cluster, like a single grain of sand.
Then to do our mathematical induction, we must agree that for any arbitrary N, if N is clearly part of the cluster then N+1 is also clearly part of the cluster. But there is some point at which you become uncertain. I submit that there is some N such that N grains of sand is clearly not a heap, and N+1 grains of sand may or may not be a heap; also, there is some N such that N grains of sand may or may not be a heap, and N+1 grains of sand is a heap.
Though really if we wanted to formalize this, we’d probably set different thresholds depending on whether we’re adding or removing grains of sand; the N at which something stops being clearly a heap is lower than the N at which something starts being clearly a heap, and we might even cut out the “uncertain” category from this model. And this would probably match people’s intuitions if they watched people adding or removing grains of sand and were told to classify.
Or, you can still treat “heapness” as a boolean and still completely clobber this paradox just by being specific about what it actually means to have us call something a heap.
This is a tangent, but...
This is certainly not obvious (I submit that if it were, the Greeks would have noticed) and I don’t think it’s true either.
Treating heapness as a boolean, the Sorites paradox indeed implies that there are no heaps, or that any number of grains is a heap. If N grains of sand together are not a heap, then N+1 grains of sand also are not; the piles are visually the same, and so anyone who knows what ‘heap’ means would look at them and classify them as both “heap” or “not heap”. I submit that if you ran that experiment for different values of N (showing each respondent only one instance of N and N+1), you would find no N for which most respondents classified one as a heap and the other as not a heap.
Since this is a paradox, we know there’s a problem with one of our assumptions. These days, the obvious one is that heapness is not a boolean. There’s a cluster in thingspace we call “heap”, and some things are clearly part of the cluster, like a pile of 2000 grains of sand, and some things are clearly not part of the cluster, like a single grain of sand.
Then to do our mathematical induction, we must agree that for any arbitrary N, if N is clearly part of the cluster then N+1 is also clearly part of the cluster. But there is some point at which you become uncertain. I submit that there is some N such that N grains of sand is clearly not a heap, and N+1 grains of sand may or may not be a heap; also, there is some N such that N grains of sand may or may not be a heap, and N+1 grains of sand is a heap.
Though really if we wanted to formalize this, we’d probably set different thresholds depending on whether we’re adding or removing grains of sand; the N at which something stops being clearly a heap is lower than the N at which something starts being clearly a heap, and we might even cut out the “uncertain” category from this model. And this would probably match people’s intuitions if they watched people adding or removing grains of sand and were told to classify.
Or, you can still treat “heapness” as a boolean and still completely clobber this paradox just by being specific about what it actually means to have us call something a heap.