Yafet, you begin with a musing on 1 1 = 2, the very definition of unassailable established truth in popular language, and how you would explain it to a child.
For a very young child you would assume this 1 and that 1 refer to distinct but interchangeable objects, so that the 2 in 1 1 = 2 refers simply to the counting number with which you would name the result of grouping them. But then you stumble across the problem of how to explain to the child as she grows older the deeper and more profound truth you see in 1 1 = 2 as a mathematical proposition.
I spent three years before the mast teaching algebra (and inevitably remedial arithmetic) to failing ninth graders, many of whom were very bright but angry working-class children destined for the scrap-heap of life. These youth had a deep hunger for a chance at life, not yet completely beaten, turning hopefully to me for help. My commitment was that I would find a way to teach them math that would make sense to them. Sadly, I found I could not. Time and time again I would lead a child who was stuck on something step by step forward from the last thing they understood (usually adding on their fingers.) Always we quickly came to some unprovable axiom which I was unable to persuade them to take on faith.
I searched the literature, and went off to three summer institutes on my own nickel, looking for someone who could show me the key. I found programs that worked based on building strong personal relationships (not an option in a modern public school environment), programs that assumed a basic understanding of arithmetic and the elements of algebra, and programs that tried to make learning fun, but nothing that met my need for a curriculum or "intervention" that would make sense. The best advice I got from the most successful teachers was to forget all that starry-eyed stuff and just make the kids buckle down and memorize and drill until they could do math without thinking!
The meaning even of 2 2 was a problem. Not when it occurred by itself, but when it occurred in some situation where the two 2's are used in different ways. Trying to understand it with the quantity model foundered on order of operations, where only good memorizing skills worked.
So my question to you is, why do you think there is a deeper meaning here? Why would you assume my ninth graders were wrong when they challenged me with "mister, that doesn't make any sense"?
Perhaps it doesn't make sense, except in very limited circumstances. But if it makes no sense as a general proposition, then the entire edifice of mathematics is a fatally flawed jury-rigged device, parts of which have been bent and beaten into a form that is with great effort useful to physics.
If this proposition is too horrifying for you to even think about, by all means ignore this comment. You've worked long and hard at mastering math as it is, a great achievement, and I don't want to ruin your life or your career.
But if you share a gnawing sense of something wrong and you long for a mathematics that actually makes sense, take a slow patient look at Rob MacDuff's deceptively simple submission "Mathematics of Science".