Good to talk to you again. It's been a while.
"Paul and I had this nearly identical discussion a while ago"
Yes, and I failed to get you to catch on to what I was saying. Maybe another go at it will help.
"Paul is talking about it in terms of the education of mathematics as an abstract science of patterns."
No, that's not quite right. I am talking about math as the rigorous study of thought itself. The fundamental question pursued by mathematicians is, What can we say for sure about what we think? The "for sure" is the hard part.
"...mathematics is about reality since what else could it be about?"
That's not quite right either. Mathematics is not about reality. It is about abstract objects of thought. Tim's example of numbers is a good one.
"There is a whole history that can trace how things become abstract..."
Close, but not quite. The history traces how things became abstracted. The things themselves did not become abstract. Nor did they become abstractions. The abstractions came into being simply by the exercise of thought. Real things did not somehow get transmuted into abstractions. The abstractions of the real things became new "things" which are not real in the sense that they are not part of physical reality. They are simply ideas or objects of thought.
"...and that this forms a chain of references that ultimately reduce any abstract thing to a particular thing."
The chain does not reduce the abstract thing to a particular thing. The chain merely identifies the history of the thought process that led up to someone entertaining the notion of the abstract thing. Yes, such a history might exist and might even be discovered. But the thing you miss about abstraction, Harv, is that one may consider a real thing and then form an abstract idea that generalizes or expands on that real thing so that the abstraction "goes beyond" or you could say it breaks the connection to, the original real thing initially prompting or suggesting the thought. This has reached a point now where mathematicians routinely consider abstract objects that have nothing to do with reality. The infinitely tall hierarchy of orders of infinities is a good example.
"I think, as you said, as humans age (or as a mathematical society ages) they tend to forget the old references and just treat them as assumptions (or axioms)."
This doesn't accurately characterize how mathematicians treat the "old references". If the "old references" were unacknowledged assumptions, they treat that as an error. They correct the error by turning the assumption into an axiom or a theorem if they still want it in their system. The history of where that notion came from originally is of little to no interest to the mathematician. The only interest is in whether the notion can be unambiguously defined in a way that is consistent with some particular mathematical system.
"Of course, the reason that they are concerned about those axioms versus some ridiculous axioms is because of the chain of references that establish those axioms as meaningful and the ridiculous ones as not meaningful"
What you say may be true if the "they" you refer to are scientists or applied mathematicians. Pure mathematicians, IMHO, seem to be more interested in what you would call "ridiculous axioms".
"This is why we don't see modern mathematicians postulating ridiculous axioms."
The reason you don't see that is because you don't look at the details in the journals of modern mathematicians. If you did, you would see that most of the new axioms postulated would seem ridiculous to anyone but a specialist in that particular narrow field. I don't mean just obscure, I mean RIDICULOUS.
"They aren't meaningful to entertain for their profession."
They are meaningful enough to get funded.