While reading some US political comment about "probability being less than zero," I was just thinking about how confusing negative numbers are to concrete thinkers. How do you explain negative space, concretely? How about so-called complex numbers (square roots of negative numbers)? The answer in concrete terms... requires us to reflect upon the nature of our cognitive apparatus, the information system which is a combination of our wetware and the culture which is contemporary mathematical language.
The number line is what we refer to as a frame of reference... when describing enumerable things, such as space. How is this handled by the information system? The system notes a point in space, and arbitrarily annotates it with a mark, 'zero', arbitrarily picks out another point, 'one', and thereby initiates a vector: a linguistic and now mathematical construct which is extendable to an arbitrary degree (to infinities, and beyond?) - note that the formation of this concept was contingent upon a primal cognizance of space (classically referred to as the transcendental aesthetic - a loose interpretation).
But then what if we wish to refer to some space behind us, oh, it appears we say it is 'negative one', because that is conventional language. (Wow. Much abstract. Many dimensionality.) And then we potter on and perhaps need to find the square root of some locus in negative space, and thereby come to invent the construct of complex numbers. (Wow. Many abstractions. So impressed.) But none of this is strictly necessary if we simply revisit our foundations, and move the mark 'zero' further back from where we had initially placed it.
So it has for a long time, seemed to me that contemporary language relies on crutches upon crutches of monkey-patching. And our Standard Theory is encumbered by these semantics such that they grate upon our apparati, burning resources, and bleeding our civilisations of their productivity. Mathematics is not inherently hard to learn, but our language for describing mathematics is, today.
I have often wondered what we would discover if we rewrote the semantics of Standard Theory in a refactored fashion... attempting to cut through all the pointer-salad that give us our canon on standard mathematical notation. But, I have never really found the time to study the problem carefully.