I think this is one of the many cases where you emerge out the top of an ideath to come right back to where you started at the bottom, but for different reasons.
I grade school many are taught that math === arithmetic. Then they get to interesting stuff in high school and college and realize arithmetic was just a tiny part of the story. So many at that phase are emphatic about refuting what they were told early on about arithmetic being math. Then when you calm down from that notion, you can get back to realizing that while arithmetic is only a small part of math, it is still very much a part.
Yes, really. "Arithmetic is the branch of mathematics dealing with integers or, more generally, numerical computation" -- http://mathworld.wolfram.com/Arithmetic.html
> Mental arithmetic is to number theory as being able to type is to programming.
To paraphrase you, number theory != mathematics. (But number theory ⊂ mathematics, just as arithmetic ⊂ mathematics.)
> That you do not understand this, shows that you too have been damaged by the standard teaching of maths within education - pre university level.
That's a bit rude, and also inaccurate.