Response from Daniel J. Velleman on October 19, 2005

Yes, lines in mathematics go on forever, and such things most likely don't exist in our universe. So why do mathematicians study them?

There are most likely only a finite number of elementary particles in the universe. Should mathematicians say that the positive integers end at some finite number? The study of the positive integers would actually be a lot more difficult (and a lot less attractive) if we placed some bound on the numbers to be studied, based on the number of particles in the universe. It is easier (and more interesting) to study the infinite collection of all positive integers, even if most of those numbers will never be used for counting objects in the physical universe.

In general, mathematicians don't think of themselves as studying things that exist in the physical universe. Rather, they study abstractions, like infinite lines or the positive integers. These abstractions may be motivated by things in the physical universe, such as the lines we draw on paper or the process of counting physical objects. But mathematicians leave these motivating examples behind and study the abstractions themselves. It is an interesting fact, which many have found surprising, that the study of these abstractions turns out to be very useful in the study of the physical universe.