[Maths-Education] concentric and congruent circles

[Hugh.Burkhardt at nottingham.ac.uk]

This is a good opportunity to make the point, to students as well as 
to ourselves, that:
	definitions in mathematics are human constructs **, involving choices
	there is usually a cost benefit analysis involved in standard 
definitions, for example:

Why do we say that a square is a rectangle (when, in informal 
language, it is obviously not an oblong)?  Because excluding it would 
put a hole in a continuum, as geometry software shows (There may be 
other better reasons; that it fits the definition is not enough, see 
next example)  Saying concentric circles of the same radius aren't 
concentric circles has the same problem.

Why is 1 not a prime?  (It has no factors except one and itself.)  We 
choose explicitly to exclude it because letting it in would ruin the 
unique prime factorisation theorem  (You can multiply by as many 1s 
as you choose)

Pure mathematicians choose (for reasons that seem to me artificial) 
to demand that "functions" are unique mappings, thus excluding square 
roots, arcsin, etc and much of complex  numbers.  Other 
mathematicians don't want to lose these, distinguishing "single 
valued functions" etc.

It is rare that students are taught about this vital aspect of choice 
in definitions. Once you see it, it clears up a lot of things. 
(Units are similarly matters of convenience -- but that's another 
topic)

Hugh Burkhardt


**  As Kronecker (I think) said:
	"Die ganzen Zahlen hat Gott gemacht; alles andere is Menschenwerk"