Robert Hansen posted Dec 31, 2012 2:46 AM (GSC's remarks interspersed): > > On Dec 30, 2012, at 1:06 PM, Joe Niederberger > <email@example.com> 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? > Yes. (I have not understood him fully myself as I've not made models of his quite complex arguments - but I have understood that you are indeed simply "unaware of the subtleties").
Just as you have not understood:
"model"; "graphical representation"; many other things, including "OPMS" (regarding the last of which - possibly because of your lack of understanding - you have here consistently and continuingly been putting up false arguments over the past several years). > > > > > > 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. > I would guess that your son has actually understood infinity in a deeper way than any 'formal definition' (or 'formal thought') about it than you could give him at this stage of his life could teach him.
The power that children have to learn things is quite wonderful - it's something to be thankful about - despite all the nonsense that we adults might do to foist our twisted thinking upon them. See, for instance, "How a Child Learns", attached herewith.
GSC ("Still Shoveling Away!") > 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 >