There have been many cultures in which the "uniqueness" of everything is way more important than the idea of grouping like things together in "sets". To then add "ordering" on to that concept would be entirely foreign. A narrative is not an "ordered" set of statements, rather a continuum of expression. I think that "sets" is an idea which requires a "discrete" approach to the world, which does not arise in cultures from "steady state" environments in which every day is the same as yesterday.

I don't follow the reasoning that "sets are impossible to define in a non circular manner" implies "sets are intuitive". And even if they are intuitive, that doesn't imply that the cardinality of sets is also intuitive.

I for one have not stopped caring about the problem of numbers, and I am sure I am not alone. It's not a problem that sees much in the way of publication, probably because there hasn't been that much progress and it's not a study likely to get your Phd. It's the sort of problem that sits on the back burner until some genius comes up with a new insight.
Part of the problem with this thread is that there are different meanings being attached to the symbol "number". The "1" in 1 sheep is probably intuitive, the 1 in {0,1} is probably not, yet both might reasonably be called "numbers". As for the "number line", I think that "things laid out in a line to see how many I've got" is innate, and may even be so for animals such as cats and birds. Naming the thing at any point in that row by the "number" that I count to get there seems to be a level of abstraction which requires "teaching".

That depends on your definition of proof, and of the system of logic being used. A simple (simplistic?) binary logic may produce a domain in which proofs are either true or false but not both, whilst a more interesting logic may suggest that a proof ( or any statement) is either true, false, true and false, neither true nor false, or not determinable. Using such a logic (or any other consistent set of states) is perfectly valid maths, and can give rise to some interesting results, in fact some of these even turn out to be of use to physicists and other students of the "real world", even though a "real" mathematician is disinterested in such mundane matters. Applied maths is just what it says on the box - the (often unwarranted) process of assigning "real world" measurements to mathematical structures and then taking the result of a mathematical operation on those structures and interpreting the values as though they applied to the "real world".
In fact I also disagree with your statement defining maths as an abstract set of axioms and rules, which seems to me to cover only part of the game. There is such a thing as mathematical "elegance", which most mathematicians would recognize as integral to the game, but which I cannot easily define - just that some systems are more "elegant" than others. I suspect that all human maths is "blinkered" by our nature (primates - carbon based - etc.) and would not be surprised if a different kind of mind produced a maths which we could not easily comprehend.
If you are interested in maths, you should really try to read Russel's Principia - but take care - the game of maths is much more addictive than any video nasty.

The UK will continue to exist. Scotland will probably not become a republic, but the Kingdom will include two independent countries, one principality and the province of Northern Ireland. The crown was united long before the parliaments.
There would be no problem in creating a shared "British" military, in fact the usual designation is "British armed forces".
Don't forget, once independence is in place, there will be a brand new political landscape in Scotland, without the unifying theme of "independence", the SNP might well fragment over other issues.

They were available, but our ancestors had enough sense to leave them alone.

I live about 30 miles south of the border, and strongly believe that we should become part of Scotland until we can re-establish the kingdom of Northumbria with a king at Bamburgh. We will then demand compensation from the english for all the coal and iron they stole and take the Australian Government to court for copyright infringement by the Sydney bridge which is a blatent copy of our bridge over the Tyne.