In hoc signo vinces (Was: Revamping the numeric classes)
Mon, 12 Feb 2001 09:33:03 +0000
Marcin Kowalczyk pretends not to understand:
> > Again, a violation of the orthogonality principle. Needing division
> > just to define signum. And of course a completely different approach
> > do define the signum of integers. Or of polynomials...
> So what? That's why it's a class method and not a plain function with
> a single definition.
> Multiplication of matrices is implemented differently than
> multiplication of integers. Why don't you call it a violation of the
> orthogonality principle (whatever it is)?
1. Orthogonality priniciple has - in principle - nothing to do with
Separating a complicated structure in independent, or "orthogonal"
concepts is a basic invention of human mind, spanning from the
principle of Montesquieu of the independence of three political
powers, down to syntactic issues in the design of a programming
If you eliminate as far as possible the "interfacing" between
the integration of the whole is easier. Spurious dependencies are
2. This has been a major driving force in the construction of
entities for centuries. What do you really NEED for your proof. What
is the math. category where a given concept can be defined, where
a theorem holds, etc.
3. The example of matrices is inadequate (to say it mildly). The monoid
rules hold in both cases, e.g. the associativity. So, I might call
both operations "multiplication", although one is commutative, and
the other one not.
In a later posting you say:
> If (+) can be implicitly lifted to functions, then why not signum?
> Note that I would lift neither signum nor (+). I don't feel the need.
I not only feel the need, but I feel that this is important that the
additive structure in the codomain is inherited by functions. In a more
specific context: the fact that linear functionals over a vector space
form also a vector space, is simply *fundamental* for the quantum
mechanics, for the cristallography, etc. You don't need to be a Royal
Abstractor to see this.