The connection is between a formal language - that is, one that we create by stipulating the syntax and semantics (logic is such a language) and natural language - which came about organically.
I don’t know if there is a distinction between a symbol and a word, in that a word is a symbol. “Dog” is a symbol I speak to represent my little pug when I say “my dog is the cutest in the world.”
But it’s a foregone conclusion that we don’t find circles in nature - we find circular objects for sure. A circle is a mathematical object we invented to describe the circular objects we find in nature. And in a very real sense “circle” has always existed in that a logical object (as in, not physical) that is a line whose points are all equidistant to a central focal point has always been a truth, even without people around to see it.
Addition isn’t something we invented but, in terms of math, is it an operation indicated by the operator +. And it works exactly as other operators in logic do - and, or, if>then, and the biconditional. (and many more). But it’s also a foregone conclusion that’s been discussed long and hard that logical operators are imperfect - they don’t exactly capture what goes on in language, or, reality. They are models. 2+2=4 is not the same as bringing two groups of two apples together to make a group of four apples. It doesn’t even correctly describe the state of affairs, as it loses a lot in the abstraction from apple groups to a proposition of mathematics.
Now, I’m not a mathematician (though phil. math had be really considering going back to get another degree in math so I could be) so I can’t adequately lay out the math-y side of it all, so please forgive me. But the rub goes down to even simpler than we’ve been discussing, to the foundation of arithmetic. That is a puzzle that has been confounding philosophers for thousands of years and there is a great deal of contention about why arithmetic works. Mostly, we just ignore that contention because it does APPEAR to work - in works perfectly to the vast vast vast majority of people’s eyes.
But for me, the ontological problem, the semantic problem, the epistemic problem, the incompleteness problem all seem to be best explained by taking math as a form of logic as I’ve described.