Maths is what happens when you take nonsense very seriously.
For some odd reason a subset of all the possible non-sense maps really well to the real world.
For some odd reason a subset of all the possible non-sense maps really well to the real world.
The reason Maths works is because it is based on intuitions (counting, space) that are presumably shadows of the physical world cast on the evolving brain via coarse perception. The effectiveness of Mathematics in Physics is probably not unreasonable at all.
For the sake of argument I can conveniently forget what "1" means, and then you can try and explain it to me like I am from another planet.
Semantics are a non-trivial matter.
Note how you are reaching for complex/abstract ideas (like a "topos" and "natural numbers") to explain simple/intuitive ones (like 0 and 1).
"toposes" and "sets" (of "natural numbers") are socially constructed in the subculture of Mathematicians.
Outside of that shared experience, they are pretty meaningless.
I guess you are on the "discovered" side, and I am on the "invented" side of Mathematics ;)
You're super-close to a deep realization: all words are only sensical to certain subcultures. IOW there's no absolute meaning to any word. In that perspective, we're both right; "1" is meaningless and 1 is categorical, and it's just a question of pointers vs. names.
Another quick and deep corollary is that reality is socially constructed; whenever a quorum of humans is mutually intelligible during a conversation, then they are agreeing on the local nature of reality. Humans can't construct global maps of reality, though, since they can't observe the Universe all at once. Indeed global maps of reality are forbidden by the Kochen-Specker Theorem.
Edit: To clarify for the audience, I have no appeal to higher categories here, and the definitions of category theory are not some sort of categorization or classification process, but axiomatic definitions akin to set theory.
"defining categories" and their respective categorisation rules is the process of classification. They don't account for the classifiers themselves.
You are super-close to a deeper realisation even.
If any notation is meaningful (even one that uses symbols like ∞), then it's Turing-recognisable.
Type-0 Chomsky grammar. In formal languages syntax is semantics.
In so far as understanding (comprehension?) goes, you could say that I subscribe to the axiom of unrestricted comprehension. It's rather un-Mathematical doing so.