that was interesting!
May I quote your post:
"An "Infinite" series is defined to be the limit of a sequence of partial sums. Note in particular that an infinite series is NOT defined to be a sum. Sums are defined only for pairs of summands"
"If, and only if, that sequence has a limit, then we define the "sum" of that series to be that limit. In that case, we also say that the sequence converges. Convergence is a necessary condition for the definition of the "sum" of an infinite series".
This is a very interesting observation.
For example, I saw a photo of a boy reflected in a mirror while holding another tall rectangular mirror beside him. There were multiple reflections getting smaller and smaller: converging on a limit.
Now: to call the series "infinite" when it is constructed by partial sums; that would be like saying that if an arrow fired at a target goes half the remaining distance to the target after each succesive moment, you get the arrow going infinitely into a "limit" in mid-air and supposedly never reaching the target.
This Zeno's Arrow "paradox" is resolved by seeing that "moment" was defined in terms of the halving distances; the moments were also being halved.
Congratulations to Paul Martin!
Why? Peter Lynds caused a stir in the physics world by having a paper published about quantum indeterminacy where he noted that the position of an object in RELATIVE motion cannot be precisely determined.
Now someone suggested that if objects teleported in jumps then Lynds would be wrong say. But I suggest that by definition a teleportation jump is "instant" so is TWO positions at "once" so still position is not precisely defined! (As a jumping object has TWO positions by definition in an instant!?
Well, in my opinion your observation "Note in particular that an infinite series is NOT defined to be a sum. Sums are defined only for pairs of summands" is very revelant and looks to be similar in character to Peter Lynds' observation.
How does one escape from "converging series"?
By diverging! In fact, "renormalisation" in physics looks like such an escape perhaps? And "div." and "curl" in Maxwell's equations looks connected too perhaps say.
May I suggest that there appears to be a lot of self-reference involved in the "converging series" concept; that the limit, the infinity, is consciousness?
What is a "sum"? We say "1 = 1 = 2". But in reality, this "real sum" is a partial sum? Because there must be SOME aspect of each of the "ones" that does not equal (or you wouldn't have two ones to add, you would only have one!)
To see "two" items requires a SAME background surely; or you wouldn't be able to count the items as you would get them muddled with two different backgrounds?
IF every-day sums are really "partial"; if the very definition of "sum" is "partial grouping"; then what about "everyday partial sums"?
In my opinion your suggestion of "grainy math" is similar to Peter Lynds' ideas about physics. The grain in math may simply be "that 1 + 1 do not have to equal 2" in that items are not forcibly combined in the universe but freely associated.
The mathematical notion "1 + 1 = 2" is hollow.
The next natural numbers seem to be constructed via "re-normalisation groups" in a Zeno's Arrow manner.
Physics serems to be about all the different ways that "1 = 1 COULD equal 2"; that is, looks like what Chris Langan calls "conspansive duality". DR. Dick refers to "data transmission as part of explanation" and "assignment of definitions"; Stephen Hawling refers to "nuts and bolts", Lee Smolin refers to "flow of information", I referred to "musical chairs, join the dots, know the difference".