[Haskell-cafe] What is a number. (Was: Num instances for 2-dimensional types)

[jerzy.karczmarczuk at info.unicaen.fr]

L.A. says:
> complex numbers are just pairs of numbers.
Later :
> Being a number is in the eye of the beholder. :) 

Now, the readers of this forum will with horror witness a
discussion about the meaning of the word "just"...
American people will call it a discussion about "semantics", and
we, European will not understand why this word is used in a pejorative
context... 

"Just pairs" have no natural arithmetic upon them. All the standard set
theory will not help in "making numbers", unless we inject the notion of
equivalence à la Gottlob Frege. And this is surely contextual, but perhaps
"the eye of the beholder" should be somehow objectivized... 

BTW. the missing term of M.M. is DUAL NUMBERS. 

Lennart continues : 

> Everyone agrees that the Haskell numeric hierarchy is flawed, 
> but I've yet to see a good replacement.

There *are* some "mathematical preludes" for Haskell. There were attempts
to emulate the structures of, say, the C.A.S. Magma in FP (I lost the
references...) 

http://www.haskell.org/haskellwiki/Mathematical_prelude_discussion 

The problem is, that Haskell is a huge factory, people responsible NOW
for its evolution have other priorities. But the story has at least 15 
years. Jeroen Fokker did something then, I worked on it at the same
period. Now Jacques Carette has his own system, and Sergey Mechveliani
another one...
But other people don't care, so the efforts are atomized. 

Please, keep cool. 

Jerzy Karczmarczuk