[Kednicma]

To be extremely precise: 1 is the successor of 0. Anywhere that there is a natural numbers object, there's a semantics for this statement. What is socially constructed here is the choice of topos which hosts the natural numbers object!

[ukj]

SUCC(0) is no explanation of 1, until you explain 0.

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 ;)

[Kednicma]

Humans are what, about 99.9% similar to each other on average? So that gives me 3 nines belief that yes, if I can understand, then you can understand; we're just humans after all. I think that you're being sarcastic; once we define categories, Cartesian closed categories, and elementary topoi, then categorical set theory is a straightforward sequence of diagram schemata. But I understand your point!

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.

[ukj]

Heh, how did I know you are going to appeal to infinity-categories at some point?

"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.