Many years ago, when I was still at school, I remember one maths lesson in which a pupil had a heated argument with the teacher over the use of infinity in the calculus. The pupil refused to accept that this was valid because infinite sets "did not exist", and the teacher didn't seem to to know how to answer this and instead got rather angry. Well, after reading this post by Alexandre Borovik at A Dialogue on Infinity, I now know what the teacher should have said: that we use the infinite as an approximation to the very large. Hence it does not matter whether infinite sets exist or not. As Borovik explains, calculating with infinities is usually much easier than calculating with the very large.