This, I think, is the main problem with the article.
I also found the opening line curious:
> > I find it rather weird that mathematics is usually taught to people on a historical basis.
I think this is much more common in philosophy than in maths.
you're wrong. every "weird" technical hypothesis in a theorem is a leak. even in your super simple example there's a caveat: the sum has to be finite (that mean no you can't perform any sum of any set of vectors).
edit: i got downvoted so here's a better example that i just ran across: manifolds with boundary can have empty boundaries. where does this come up? exact 1-forms on S^1 integrate to zero. why? because by Stokes. how? S^1 has a boundary but it's empty.
All signs refer directly to abstractions in the mind that can be compiled into a variety of useful forms. Yours does not take precedent in this case, as it is plain to see that sigma is under specified due to the huge range of concepts it’s practical usage covers. Your attempt to prescribe that which should be prescribed is nothing but authoritarianism.
When I prove, say, Cauchy's theorem about groups, it doesn't matter which concrete group I'm operating on to prove it. If I write a sigma to sum some vectors, it doesn't matter if the vectors are coordinates in 3D space, functions defined on an interval, or just numbers. Mathematical abstractions are pretty much the only ones that don't leak.
What, that you have to understand summation to understand what the sigma refers to? Well, duh. That doesn't make the sigma a leaky abstraction, because the sigma is not an abstraction at all! Not even a trivial one. The sigma is notation. And it's hardly news that you have to understand the concepts behind a notation to understand notation itself.
The author seems to misunderstand what an abstraction actually is.