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ginnungagapon Why Discrete Math Is Important

Every mathematical object (ok, this is false but that's not the point here) can be constructed in ZFC (the standard axiomatic framework for set theory) so you can construct the real numbers in terms of sets (if you want more precise informations on this construction look up Dedekind cuts).

However this is irrelevant to, say, analysis, you could define the real numbers as the unique (up to isomorphism) complete, ordered, archimedean field and do analysis just as well, so I'd say that you are right in some sense and some formulation, but it's a bit of a stretch to consider analysis as starting from discrete maths.

I also don't see how set theory fits into discrete maths, apart from the basics it seems pretty far from the common structures studied in discrete maths.

ginnungagapon Advances in number theory inspired by physics

That also has a formal mathematical interpetation as a distribution rather than a function now and the theory of distributions found many applications in PDEs.

An interesting thing is that before Dirac, Green was already informally using distributions in the 1830s to solve differential equations with what we now call Green's functions.

Closely related to Green's functions are the fundamental solutions to an elliptic operator, consider the Laplacian and its fundamental solution in Rⁿ (n≥3), what's the Laplacian of the fundamental solution? A Dirac's delta!

ginnungagapon New Number Systems Seek Their Lost Primes

What they are called might not be the point, but calling them prime in a context in which prime has a precise meaning while they are not can be confusing. Especially since the difference between prime and irreducible is subtle and it's the whole reason we have non unique factorization domains, which is the main topic of the article

ginnungagapon New Number Systems Seek Their Lost Primes

Wikipedia isn't a good source, but it uses 3 as an element of Z[√-5] as an example so it's quite relevant: https://en.m.wikipedia.org/wiki/Irreducible_element

For a more serious reference any abstract algebra book covering rings and UFDs should do, for example it is in Dummit & Foote.

ginnungagapon New Number Systems Seek Their Lost Primes

All four of those factors are prime in the new number ring, giving 6 a dual existence

Those factors are irreducibles, but they aren't primes, which is the reason why the uniqueness of factorization fails.

A nonzero non unit element of a ring is called prime if p|ab implies p|a or p|b.

A nonzero non unit element of a ring is called irreducible if p=ab implies that a is a unit or b is a unit (invertible element).

Primes are irreducibles in an integral domain, but the converse is true in unique factorization domains and Z[√-5] is not one.