Q&A: What Is a Point
What Is a Point
Question
Hello Rabbi,
In mathematics people deal a great deal with the concept of a point. From high-school math classes in analytic geometry, to set theory, which says that the cardinality of the points between 0 and 1 on the line has cardinality “aleph,” as distinct from cardinality “aleph-null.”
I would be happy for some clarification as to what that point is. After all, if we divide a segment into a finite number of parts, each part will have some size greater than 0. If we divide it “infinitely” many times, we get “infinitely” many points, each point having size 0. But in reality there is no such thing as dividing infinitely many times, because infinity is not a number. Rather, it only means that the more parts we divide it into, the smaller each part becomes.
If so, unlike a segment, which I see and experience and which does exist, there really is no such thing as a “point.” For any length by which I divide the line, it has a size greater than 0. So what are those “points” that mathematicians discuss? Do they “exist”?
Answer
This is a complicated issue in mathematics, and it is hard to teach it here. In general, you cannot arrive at the concept of a point by dividing a segment. But the problem is not that the number of divisions is infinite, since that at most only means that you will never actually complete the process of division, but it does not mean that no such division exists. By the same token, you could argue that there is no such thing as a segment either, since in order to get it you would have to join infinitely many points to one another.
You may perhaps understand the problem better if you notice that a point is not a segment of length 0. A point is a different kind of entity, since its dimension is 0 (whereas the dimension of a segment is 1). Therefore, you cannot view a continuous line as a collection of densely packed points side by side.