On Dec 30, 2012, at 1:06 PM, Joe Niederberger <niederberger@comcast.net> wrote:

People who think that dispelling confusion in this area is just a matter of following definitions carefully are simply unaware of the subtleties, as meekly accepting prevailing convention will generally help one steer clear of trouble. In other words, the people who see no room for confusion from haven't understood the distinctions
that great thinkers of the past have wrestled with.

Are you saying that we don't understand Kirby?

 

The distinctions are certainly relevant to anything regarding limits (I'll again mention David Tall as someone who has looked at the .9999... issue in great detail) and again our current mathematical language shows a bit of inconsistency regarding these concepts. People still regularly speak of "limit as N goes to infinity". 
That tends to call up an image of a *ongoing process* that never quite completes. Given that, its not surprising that some people should be confused when the teacher turns rights around and begins to speak of "the limit" (say, 1 in the case of .999...) as something static that is simply there like any other mathematical object.

I was surprised that my son understood that statements involving infinity involved an ongoing process that never completes. I don't mean that he is formal or pedantic and says things like "increases without bound", and he still says "infinity" like it is a number even though he knows it isn't, but I think that is just language. But put him in a context, like when we are looking at the sequence 1/n, he knows that we can keep doing that a finite number of times no matter how large and never reach 0 but at the same time 0 is still the ultimate destination.

Of course, what he doesn't yet know is how that all connects and plays out in larger and more sophisticated arguments. That is where the structure, formality and precision gain importance.

Bob Hansen