So is theoretical Computer Science
that is how you do math,
No, it's how you do part of maths. It's how you convey ideas to others and it's how you codify things to check you're right. Last time I did some maths (as in trying to prove something new), it was me, a friend, lots of coffee and most of the time was spent talking or sketching things. the rest was spent manipulating symbols.
Manipulating symbols is a tool you use in maths to formalise your patterns. But saying that maths IS the manipulation of symbols is like saying woodworking is using a nailgun.
you are taught math or CS on under-graduate levels, which is what we are talking about
Actually it wasn't clear to me that we were talking about that. TFA (actually pretty well written though from the comments almost no one read it) was very much not about being taught maths or CS at university.
, you are learning all the basics of various forms of abstracts and the insights they bring.
Yes and manipulation of symbols is a tool to do that. But that doesn't mean that maniuplation of symbols is the sum-total of it. That's the way it's taught in schools and it massively sucks.