[Haskell-cafe] Thoughts about redesigning "Num" type class

[Jerzy Karczmarczuk]

Le 12/09/2015 10:15, Alexey Muranov a écrit :
> the difference of two points in an affine space is a vector, the sum 
> of a point of an affine space and a vector is another point.

We can give several similar examples, but it might not be so useful for 
the implementation of whatever. In the affine space we can interpolate 
between points: p0 + 1/2*(p1-p0) is the middle point. Writing it just as
1/2*(p0+p1) might be considered as an abomination, since you should not 
add points... Who should allow the first and forbid the second?

Programming languages is not mathematics. Types are not mathematical 
domains. Classes are not categories... Whatever we/you propose, there 
will be many unhappy people around. Math is full of subsumptions, whose 
automatic implementation might be difficult, inefficient or ambiguous.

If you have an additive group, you have automatically a module over 
integers. Sorry, *positive* integers. So, we might second David Thomas, 
we should have Rigs (semi-rings). But why a *class*?? And, if you define 
such object by hand, you may be accused of introducing redundancies, 
that it should be inferred...
And whatever people say, e.g., Wren Romano:

> There are far more objects which have
> addition/multiplication without subtraction than there are objects
> with addition/subtraction without multiplication.
... several simple-minded people (myself included)  will find not so 
useful and/or clumsy the introduction of specific structures just to 
forbid the subtraction. Nobody will fight against these purists, it is 
simply a difficult issue.

All this is a can of worms, and physicists live with it already for 
centuries
[Time : add please today to yesterday... Subtract works.  They were 
happy, because there were no computer formalisators around, who could 
bother them...].

Please read the *easy* article of John Baez (2009) : 
http://math.ucr.edu/home/baez/torsors.html
And then look upon torsors in general, and recognize that is is simply 
horrible.
==

All the best.

Jerzy Karczmarczuk
/Caen, France/