I believe I can explain both of these understandings simply as a result of the use of different numeral systems with different bases. I’ll tie it back in with the rest of your post as I go along.
1+1=2 requires a ternary or greater numeral system, usually decimal by human convention in most of the modern day world.
1+1=11 would require a unary numeral system to be true, adapting what would otherwise be “0+0=00” if one were to extrapolate what a base 1 system would look like back through ternary and binary to unary, except 1+1=11 looks more intuitive and meaningful - because using only 0s would make it look like there’s only zero values being dealt with, when the intention here is to be dealing with non-zero quantities. However, it doesn’t actually matter what symbol you use for “1” in a unary system, so long as it’s understood that the symbol being used is equivalent to the quantity “1” in e.g. decimal. “q+q=qq” would even suffice. This is not to be confused with qq representing q*q as it does in conventional algebra. 1+1=11 is simply the result of using the same “carrying” convention that all base numeral systems use: once you exceed the number of symbols for the base you’re using, the next number up resets the unit back to the first symbol and a 1 is carried over to the next position, added to anything that’s already there, if there is anything. In the case of 1+1=11 there isn’t anything there, so the unit 1 is reset and a 1 is carried over to the next position, making “11”. There are other numeral systems of course, like Roman Numerals, but they don’t follow the same conventions - though coincidentally enough I+I=II in just the same way as this unary system that I’m describing. Obviously they change beyond the quantity “3”…
But back to base numeral systems, 2 and 11 are quantitatively equal, just represented in different bases. Quantity is the underlying concept that is being denoted either way, and how you represent quantity is arbitrary, but some ways are easier and/or more appropriate than others depending on the application of the quantity. Binary for open and closed computer circuits is an obvious example. Decimal is just convention that is more or less convenient for everyday mathematics and arises easily from the number of digits on both normal human hands.
This is a side point, obviously, and I hope I’ve explained it clearly enough.
Mathematics has always been easy for me, and it mystifies me too how some people simply cannot grasp it. In a way, we learn like monkeys to think in base ten, when 1+1=11 in unary is just as fine as 1+1=2 in decimal, albeit contrary to convention and with numbers quickly becoming unwieldy as they increase in quantity - compared to higher base numeral systems. Both can be “true” - I think it helps to simply be as “multi-lingual” as possible to understand as many different perspectives on quantity as possible, for the sake of knowing the meaning of what’s going on at all levels of education.
Are we in agreement that number systems simply “refer” to quantities, where the symbols and presentations of the different bases are the “signifiers” and quantity is the “signified”?
This is my understanding of language - in keeping with de Saussure.
Symbols are convenient and distinct shapes that are real visual sensations. Words can take the form of both visual symbols and auditory language.
Less convenient, but equally real things, like trees and cars can simply be associated with these words/symbols: and thus the signified is represented by the signifier. Language comes to be associated with all things, representing all real phenomena in a form that is not what it represents, but is accepted to denote what it represents. This is why languages can vary so hugely across history and geography - it’s essentially arbitrary, the only thing that matters is that it’s socially accepted and useful. You don’t have to physically drive 2 cars up to someone to communicate a quantity of 1+1 cars.
One thing both signifieds and signifiers have in common is that they require bounds/definition in order to apply to something specific. Problems arise when defined finite signifiers are required to give bounds to that which is not definable - or infinite/boundless. What exactly is meant by a finite word when it denotes something that cannot be entirely conceived/grasped/isolated?
I hope you appreciate just how bizarre a notion it is to “define” something that amounts to infinity.
Notice how it’s always necessary to inject something undefinable into any definition of infinite series: for example, [1+1+1+…+1] has this mysterious “…” in it. What exactly is this “…”? It’s not merely 1+1 or 1+1+1, but it’s an instruction to keep adding 1 without bound on how much you do it. “Keep doing this indefinitely” is a definite instruction, but it is a definite instruction to do something an undefinable number of times. Therein is hidden the undefined element of what is otherwise a very precise definition. Even the conventional format contains within it the same mysteriousness: " i=1 ∑₀∞ 1ᵢ " has all these finite terms, but that one infinite term ∞, which is that same instruction to keep adding 1 without bound on how much you do it. Again, for a definite instruction amongst all those other definite symbols, it’s a definition instruction to do something undefinable. That “undefinable element” remains no matter how precise and defined you want to make any “signifier”, as it logically must do when the “signified” is an undefinable. Replacing the (1+1=) “2” with the “11” changes the signifier, as does calling [1+1+1+…+1] by a different symbol infA that hides the “…” mysteriousness just one more layer deeper. The undefinable “signified” will necessarily still be contained within any attempted definition denoted by the “signifier”.
I like this “conflation of map with terrain” wording that you’ve used - it mirrors this signifier and signified terminology that I’m using. The map can take whatever seeming finitude that you want to use, but in defining it as meaning infinite terrain, the definition will necessarily somewhere be hiding or de-emphasising an undefinable somewhere in there - no matter how layered and sandwiched it is amongst finites.