Why is dictionary.keys() a list and not a set?

[Alex Martelli]

Christoph Zwerschke <cito at online.de> wrote:

> mathematics, everything is a set and set theory is the "theory of 
> everything". When I grew up pedagogues here in Germany even believed it
> would be best if kids learn set theory and draw venn diagrams before 

An alternative theory, of course, is "God made the natural numbers; all
else is the work of man" -- and that one is by a German, too (Kronecker,
if I recall correctly).  The hope to found all of mathematics on set
theory was primarily a _British_ effort, as I see it (Russell and
Whitehead), and failed a long time ago... I'm not sure what, if
anything, a mathematician of today would propose as the foundational
theory... perhaps modal logic, but I'm really just guessing, being,
myself, an engineer rather than a mathematician (perhaps category
theory? it's hard to say...).

What, if anything, is the theoretically proper foundation, is of course
a separate issue from where is it best to start _teaching_ maths... with
geometry as the Greeks would have it, with arithmetic as in traditional
schooling, or with sets as the modern pedagogues would have it.  Me, I'm
partial to arithmetic, but sets are just fine with me if they turn out
to work better (I wonder if proper controlled studies have ever been
done before revolutionizing the teaching of elementary maths,
though...!).  Again you see my engineer's bias: "whatever works"!-)

But OO really requires a different mindset, particularly when operating
under a regime of "mutable" objects.  "A circle IS-AN ellipse" in
Euclidean geometry... but inheriting Circle from Ellipse doesn't work in
OO if the objects are changeable, since you can, e.g., change
eccentricity in an Ellipse but not in a Circle...


Alex