[FOM] on andrej bauer on gs on |x| (II)
Andrej.Bauer at andrej.com
Mon Apr 3 09:02:27 EDT 2006
I did indeed misunderstand what you wrote as a claim that your definition
1) if x <= 0 then |x| = -x
2) if x >= 0 then |x| = x
is a valid definition of |x| for _all_ real x. I think it was not only my
You have further commented that you used to prefer the definition |x| =
max(-x,x), but now you prefer to define the absolute value via rational
intervals. I would like to comment on that.
It is in the constructive mindset (pun intended, I always intend all the puns)
to preoccupy oneself with constructions. Thus, for example, you have a
particular construction of the reals set in your mind: a real is a certain
set of rational intervals. To this I would say that it is methodologically
better to keep things as _algebraic_ and _abstract_ as possible, without
going overboard on the "abstract" part. Too many sets and sequences of things
can be hard on one's mind.
Thus, for example, instead of working with a particular construction of reals,
I would say it is better to work axiomatically from the axioms of an ordered
field with suitable additional properties (Archimedean axiom + completeness).
>From this point of view, it is more reasonable to define |x| via a pasting
lemma which uses only the axiomatized structure of the ordered field. This
often leads to slick proofs.
Let me give an example. In defining |x| I would prefer to avoid any talk of
the absolute value of an interval (unless my interest was in computation with
intervals for the sake of intervals), and I would do it as follows. First,
given two functions
f : (-infinity, 0] --> R
g : [0, infinity) --> R
we may paste them together into h : R --> R by the formula
h(x) = f(min(x,0)) + g(max(0,x)) - (f(0) + g(0))/2
If g and f are continuous then so is h, obviously since h is just a
composition of continuous functions (even uniformly continuous if f and g are
such). Furthermore, if f(0)= g(0) then h is a continuous extension of both f
The idea behind the definition of h is of course that (-infinity,0] and
[0,infinity) are both retracts of R.
Now define |x| as the pasting of the functions f(x) = -x and g(x) = x. The
formula for h in this case simplifies to |x| = min(x,0) + max(0,x).
I find the above construction cleaner and easier to understand (once you draw
a picture of what f(min(x,0)) looks like) than any definition that goes back
to a particular construction of R. It also teaches us about the importance of
If you like "reals as interval", for thought provking reading I recommend Paul
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