The ancient paradox of the heap, the Sorites, goes as follows. If you have a heap of sand and you remove a single grain, what remains is still a heap of sand; that’s obvious. But according to what we’ve just deemed obvious, you can remove a single grain from this remaining heap too and you’d still have a heap. And so on, never mind how many times you repeat the process: it follows from what we’ve deemed obvious that you shall always have a heap. However, eventually you will be left with a single grain, which equally obviously is no heap. We have a contradiction. My solution is simple: what is true of a single step need not be true of many steps taken together, if we’re dealing with concepts like ‘heap’ for which there are vague boundaries between what is a heap and what is not. So although every step in the argument above is valid, validity does not ‘accumulate’: the long argument, which ends with the single grain, isn’t valid. Validity, in the case of vague concepts, is like the reality they describe: no single day turns a child into someone who’s no longer a child, but many together do; no single step in the Sorites argument turns a valid argument into an invalid one, but many together do. I think this approach is Wittgensteinian because it maintains that ‘a picture held us captive’, a picture of how arguments must work. Once we dismiss the picture, there’s no more work to be done: we realise that all is in order and philosophy ‘leaves everything as it is’. This picture is derived, like much else in philosophy, from mathematics. And precisely because mathematical concepts do not have the kind of vagueness that ‘heap’ does, arguments applied to vague concepts behave differently than arguments in mathematics. - www.3ammagazine.com