Date: Dec 30, 2012 1:06 PM
Author: Joe Niederberger
Subject: Re: A Point on Understanding
>"Contradiction" is an English word that has survived the centuries without being co-opted by any sub-sect or religious body for purely its own purposes, although of course they're welcome to piggy-back, as is their wont.
>I hope you're not so dismissive of student difficulties when they're trying to get their minds around concepts with inherent conundrums.
It appears to me the "contradictions" as such surrounding these topics (be it .9999... or triangulations of a sphere) all are related to a very old debate regrading so-called "completed" or "actual" versus "potential" infinities. Great mathematicians of the past have sometimes viewed "completed" infinities with great suspicion, though today those are in a distinct minority.
"I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction."
- -- C.F. Gauss [in a letter to Schumacher, 12 July 1831]
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.
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'll also add that Devlin with his view of "multiplication" as a *process* appears to be a bit confused between two opposing views.
Cheers and Happy new years,