Definition talk:Summation

I sense some trouble here. I have added a handwavery template on a proof yesterday because it used ellipsis for summation. Now, I find that our very definition of summation has the same problem. We really ought to give an inductive definition (perhaps by well-founded recursion on the finite subsets of $\N$) as a precise definition to fall back on.
There are some nasty theorems waiting out there once we're done defining (for example that the sum over a finite set $I$ is well-defined, as in independent of the bijection $I \to n$ chosen). I'll put it on my to-do list for when I get back from my vacation (which should be around July 22). — Lord_Farin (talk) 13:57, 3 July 2013 (UTC)
You mean for example using $\R$ instead of any abelian group? Fine for me, but it will imply that we will have to duplicate all those proofs for abelian groups. But okay, it is probably worth the effort making a distinction. --barto (talk) 12:57, 19 October 2017 (EDT)
We aim to be accessible to people who may not have studied abstract algebra, and are familiar only with the standard arithmetic structures $\N$, $\Z$, $\Q$, $\R$ and possibly $\C$. Same as with polynomials, which evolved suddenly from a simple definition of $a_n x^n + a_{n - 1} x^{n - 1} + \cdots + + a_1 x + a_0$ to something so abstract that it was incomprehensible to anyone not studying abstract algebra at masters level. While it may be the opinion of some that somebody so stupid as not to be completely familiar with the more abstract concepts does not deserve to be allowed to benefit from $\mathsf{Pr} \infty \mathsf{fWiki}$ (such sentiments have in the past been expressed), it has been suggested by others that it would be a good idea to allow for basic concepts to be couched in the language of the more familiar and everyday structures, as well as the deeper and more abstract frameworks. --prime mover (talk) 16:52, 19 October 2017 (EDT)