Response from Daniel J. Velleman on October 11, 2005

It seems to me that you are reading things into the original question, and my answer to it, that were not there. I do not see, either in the original question or in my answer, any reference to "points at infinity". The orignal question talks about a line going on forever, and my answer talks about the line extending infinitely far in either direction from some point P on the line. But this just means that for every number x, there are points on the line more than x units away from P in either direction, not that there are points that are infinitely far away from P. I claimed that the parts of the line on either side of P are congruent, and you can see this by observing that if you rotate the line 180 degrees around P, each side gets moved so that it coincides with the other side.

My previous answer was based on a particular definition of "center". There is another, slightly different definition of "center" that could lead to the sorts of worries that you raise. Suppose we define the center point to be the point that is equidistant from the endpoints. This works fine for a finite line segment, and leads to exactly the same center as the definition I originally proposed. But for an infinite line, if you tried to apply this definition then you would, indeed, find yourself looking for endpoints of the line--points at infinity--and you would find yourself trying to compute the distances from those points at infinity to other points. So this definition of "center" would lead to the sorts of worries that you raise, but it is not the definition I was using in my previous post.

Response from Daniel J. Velleman on October 8, 2005

As with so many questions in mathematics, the answer will depend on exactly how you define your terms. In this case, we will have to decide how to define the word "center". Now, you hint at a possible definition in your question, when you speak of the parts of the line on either side of a point as being congruent. Let's make this definition explicit. Suppose we define a center point of a line or a line segment to be a point with the property that the parts of the line or line segment on either side of that point are congruent. Then, for example, in a line segment of length 1 inch, the point that is 1/2 inch from each end will be the unique center point of the segment; the parts of the segment on either side of that point both have length 1/2, and are therefore congruent. But if we apply this definition to a line that extends infinitely far in both directions, then we find that every point is a center point, because, as you observe, the parts of the line on either side of any point extend infinitely far, and are therefore congruent to each other.

There is no contradiction or paradox here. With this definition of "center", an infinite line does not have a unique center point, it has many center points. I think what made this situation seem puzzling to you is that you were using a definition of "center" according to which a line has more than one center, but you also used the word "the", in the phrase "the center", which only makes sense if there is a unique center.

Response from Daniel J. Velleman on October 5, 2005

It's not so easy to know if you've avoided talk that could lead to a contradiction. Is it OK to talk about a set that contains all sets but one? All sets but two? No--it turns out those sets lead to contradictions too. What if you don't explicitly refer to a set that contains all sets, but such a set is used implicitly in some piece of reasoning? Where do you draw the line? How do you know if you've crossed the line? Rewriting the axioms is a way of drawing the line.

One way to see why avoiding contradictions is so important is to think about proof by contradiction. To prove a statement P, mathematicians sometimes assume that P is false and then try to deduce a contradiction. This method of proof is based on the idea that if you can deduce a contradiction, then the assumption that P was false must be incorrect, so P must be true. But if contradictions can arise even if you haven't made a false assumption, then you'll be able to use proof by contradiction to prove false statements. (In fact, this is the basis for the fact of logic that from contradictory premises you can prove anything.)

Also: Even if, as a practical matter, it weren't so important to avoid contradictions, isn't it more intellectually satisfying to try to track down the source of the contradiction, rather than just avoiding certain kinds of sets without really understanding why you have to avoid them?

By the way, division by 0 is not just something that mathematicians avoid. In the case of division, mathematicians also have clear lines that say what is allowed and what isn't allowed, and they have reasons for drawing those lines. For example, suppose we define "c divided by d" to be the unique solution for x in the equation dx = c. Then it is a theorem, provable using basic properties of multiplication of real numbers, that for any numbers c and d, if d is not equal to 0 then the equation dx = c has a unique solution, and if d = 0 then the equation dx = c has either no solution or infinitely many solutions. So you can prove that division by any number other than 0 is defined, and division by 0 is undefined.

Response from Richard Heck on October 6, 2005