In the Perceptrons introduction by Minksy and Papert, they give a proof that the predicate of whether set of points is connected is not conjunctively local of any order k. They do this by stating that, because of the limited number of points k, there must be some middle square that would not contain one of these points. Therefore, the connected and non-connected figures would return the same result.While I agree with the assertion that if there is a middle square which does not contain any of the k points, then we could not distinguish between the connected and non-connected figures, I don't understand how we get to that premise.My question is: How can we say that there must be a middle square which does not contain any of the k points? A point is not a square, so couldn't we have many more points than squares? In fact, I don't really understand how we can even call these k+1-wide figures, given that k refers to a number of points, not a distance measurement. How can we say something is k points wide, when points can be infinitely subdivided and have no actual length? 
Understanding the proof in Minsky's Perceptrons that the connected predicate is not local of any order k
neural networks;math