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/