When I was at university, one of our lecturers (who was described as a ‘philosophical skinhead’ by one reviewer) held a seminar on Xeno's paradox of infinity. The discussion went something like this:
Student (doing the presentation): A ball bounces an infinite number of times, and so never stops.
Rocky (for ‘twas his name): No, a ball bounces an infinite number of times, and then stops.
Student (becoming agitated): That’s not right; the concept of infinity involves not stopping. That’s why it’s infinite.
Rocky: No, it doesn't.
This went on for two hours. As the bell went, Rocky asked the student: ‘What do you mean by the term “infinite”?’ The student explained. And Rocky replied: ‘That's not what I mean’, and left.
We, poor astounded students, swore violently and took ourselves off to the pub to drown our sorrows in Guinness, cursing his name.
Two years later, I realised what he was getting at. The word ‘infinite’ is a negation, and we commonly assume that it is the negation of the word ‘finite’. But it is also the negation of the word ‘definite’. Rocky was using the second sense, and was saying that a ball bounces and ‘in-definite’ number of times, then stops. No paradox; no contradiction. If someone had actually challenged him on the definition of the term, we could all have retired to the pub a lot earlier and sung his praises instead. Needless to say, it’s one of the few things I remember about my philosophy degree.