I have to say I am with Tony and Anne on this point. Contrary to what Jonathan writes:
"Tony, it does help to discuss labels of circles and other geometric objects, especially if there several circles, several triangles, several angles, etc. in the same discussion or diagram so that it is clear which object we are referring to. However, whether a geometric figure is a circle or not does not depend on labels (or even if it is labeled). Students should know this fact. Labels are not parts of definitions of geometric objects because the property of a geometric object does not depend on its name for it; rather, we use labels for convenience or sometimes out of necessity to make it clear which object is being mentioned."
I think it is very important in a number of situations to have labels. One simple example follows. One can think of many more, as Tony has indicated. My daughter was, many years ago, in year 7 at a private girls school of some repute in Melbourne, Australia. She came home having to study geometry over Easter. The prescribed text's "definition" of a polygon was "A polygon is made of straight lines." I began by throwing some pick-up-sticks on the floor. "No, not like that!" my daughter exclaimed, and showed me a drawing of what I would think of as a polygon. I drew some lines, going from point to point, but not closing up the figure. She objected and indicated the figure should close up. I drew 4 points, labelled 1,2,3,4 (as if cyclically arranged at the corners of a square, yet to be drawn) and proceeded to join them 1 to 3 to 4 to 2 to 1. My daughter objected that the two "diagonal' lines crossed. We talked back and forth (definitions and refutations?) until she and I agreed - after looking at several other texts - that, for us, pre-requisite data to define a polygon is a set of points labelled by positive integers, joined from first to last, in order, and with a couple of other conditions obtaining (no three points in a line, no lines crossing except at specified points). She was happy, she could do the set problems without confusion and after Easter she taught her year 7 teacher something about polygons (related to the example above with 4 points).
Can one define polygons in other ways. Yes one can. Is the labelling an intrinsic part of the definition as she and I framed it: yes it is. Do the properties of the resulting figures depend on the labelling: yes they do.
Tony's examples are even more compelling. And I haven't even mentioned clarity of thought and perception as did Tony and Anne.
As Tony points out there is an essential difference between an arbitrary after-the-fact naming of parts, and labelling that is intrinsic and essential to the definition and to a deeper understanding.
TONY GARDINER wrote: >> whether a geometric figure is a circle or not does not depend on >> labels (or even if it is labeled). Students should know this fact. >> Labels are not parts of definitions of geometric objects >> > > No wonder students have difficulties if those of us who teach them > are so confused, so happy to parade our confusion in public, and so > reluctant to rethink when it is clear that we have got something > wrong! > > There is a very basic confusion here. > > If God were to exist, then he would probably not owe his existence to > whatever "name" we might use. However, as the OT use of "Jahweh" - > or "JHWH" - illustrates, we mortals may need a name in order to grasp > the idea and to communicate with one another about it. > > For all I know it may be true that the *idea* of a "function" does > not need the use of letters as names for entities/variables/elements > of the domain/etc. - such as "f" and "x". But I defy anyone to make > the idea of function *clear* and useable without invoking such > labels. > > In exactly this sense, the definition of a circle OBLIGES anyone to > give a name to the centre and to a radius. > > And the definition of an isosceles triangle includes the base (and > apex) and so requires us to label the vertices - say ABC (at which > point one can say clearly: > "If in triangle ABC we have AB = AC, > we say it is *isosceles with apex A and base BC*". > > Those who prefer to try circumlocutions (for circle, or for isosceles > triangle) will fail, and will at the same time increase the level of > totally unnecessary confusion in the world. > > The things that mathematics forces on us are quite different from > "life-style choices". Noone cares whether I am for or "against" > geometry books giving correct definitions. But anyone who insists > that > * given a labelled triangle ABC > * ABC and BCA are the same triangle > should never be let near a geometry class. > > I tried to say this once - and I clearly failed. > I apologise for having now had to try to say it more clearly in case > it was genuinely not understood. > > Tony > >