In hoc signo vinces (Was: Revamping the numeric classes)

[Jerzy Karczmarczuk]

Marcin Kowalczyk pretends not to understand:

> JK:
> 
> > 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
   the implementation.
   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
language.

   If you eliminate as far as possible the "interfacing" between
concepts,
   the integration of the whole is easier. Spurious dependencies are
   always harmful.

2. This has been a major driving force in the construction of
mathematical
   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. 



Jerzy Karczmarczuk
Caen, France