Date: May 7, 2013 6:57 PM
Author: Joe Niederberger
Subject: Re: Based on the quadrilateral tree

Its not even *just* a matter a definitions, this one seems pretty much a matter of mere labels with little further impact. 

On the other hand, we see via:
Timothy Gowers argues that even the meaning of .999... is at bottom a matter "of convention" -- "However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."

I happen to agree with Gowers, but also see a pretty big gulf between the two cases, as well as with the "unit square" convention I mentioned earlier.

To me, the upshot, for education, is that these sorts of matters (definitions, mere labels, definitions depending on other definitions, etc.) ought to be understood by those teaching, but apparently, that's not universally the case.

Joe N