"I always took it that for granted that you can't define x/0 without opening the door for mathematical nonsense"
You were correct.
"Defining 1/0 = (any arbitrary symbol) works just fine for any value of x other than zero."
Not true. Let's choose 'A' as the arbitrary symbol.
Then 1/0 = A.
Divide both sides by A and you get 1/0 = 1.
Divide both sides by 2A and you get 1/0 = 1/2.
Divide both sides by 3A and you get 1/0 = 1/3.
Divide both sides by 4A and you get 1/0 = 1/4.
Divide both sides by 5A and you get 1/0 = 1/5.
Divide both sides by 6A and you get 1/0 = 1/6.
. . .
Not very consistent. I'd say it's nonsense.
"It's ironic that people like Paul argue that math is completely abstract, and yet still think of "a very small number" as being entirely different from "zero"."
Sorry, I don't see the irony.
"In truth the only thing that differentiates "zero" from "a very small number" is that things of very small magnitude are supposed to exist while things of zero magnitude do not."
"Things"? Things supposed to exist? No mathematician talks about such "things" (except for those extreme Platonists like me, and I'm no mathematician). The only things mathematicians talk about are concepts, not real things.
"But the same mathematician who tells you that will completely evade the question of what is the mathematical meaning of "exists". "
No mathematician would tell you that. But they probably would evade the question of existence anyway. The concept of existence in mathematics is pretty esoteric. Existence theorems are among the most difficult to grasp, in my experience. But in any case, the mathematical concept of existence is completely different from, and unrelated to, the physical or ontological concept of existence.
"Abstract thought is the shortest route to confusion."
Did you just make that up, or can you quote a reliable source? I think abstract thought is the only route to human success.