"The reason you're getting nonsense is because you are defining 0/0, or A*0. "
I did not define 0/0. The only numerator that appears in my post is 1, so I never mentioned 0/0 much less define it. Secondly, I didn't define A*0. Instead I used the arithmetical definitions of zero and multiplication which says that the product of zero and any number is zero. So from conventional arithmetic, A*0 = 0, 3A*0 = 0, and so on.
"I still think that is really the only problem. Can you prove me wrong?"
The problem is that the quantity 1/0 is not well-defined. As I demonstrated, it is equal to many different distinct values. That's a problem.
"If math is completely abstract, then we can substitute every instance of the number '0' with, say, '&'. We can also replace any instance of "the smallest number you can think of" with, say, '#'.
Exactly what are the unique properties of '&', as compared to #, $, ^, !, or any other purely abstract symbol?"
The number '0' is much more than simply the symbol for its numeral. There is no problem changing the symbol and using, say '&' for zero. But you must still remain consistent with the definition of the number zero. It is that definition which gives zero, or any other purely abstract symbol for a defined mathematical concept, its unique properties. The properties have nothing to do with the symbol; only the definition.
"If mathematicians can't define 'real', how can they be sure they're not dealing with 'real' things? In other words, where is the formal proof that mathematics doesn't deal with real things?
They can't be sure. For all they know, they might just be dealing with real things and not know it. As the extreme Platonist that I am, I think that they are indeed dealing with real things and that there is nothing real outside of Platoland. What we think is physical reality is nothing but an illusion.
"I think abstract thought is the only route to human success.
That is only true if you consider philosophy the only thing in which humans can be successful. I personally think philosophy is the only thing in which humans can't possibly be successful"
I was thinking of success in terms of survival, happiness, and material well-being.