I do believe you really mean formal logic, and I'll add that in my view (among others) that is merely a subset of computation.
No. Formal thinking involves more than just logic, and I get the impression that by "formal logic" you mean symbolic logic. Eventually, notation and convention comes with the territory but that isn't the focus of my point. As I said earlier, even simple addition is a formal system and to own that you have to develop formal thinking. Without that, all you have is monkey-see-monkey-do, and I will admit that there is no monkey even close to our ability at monkey-see-monkey-do, but when the circumstances become more complex and more hidden, monkey-see-monkey-do will not work.
I am starting to get the impression that your take on "understanding" mathematics is quite different than mine. I'll try to explain. You have a friend that plays the violin and out of natural curiosity you ask her how she does it and she explains it to you. We'll assume that she was able to put "playing the violin" into words very nicely and when you leave you feel that you understand how to play the violin. But Joe, I don't care how nicely put was her explanation nor how vivid her analogies. The question is "Do you actually understand how to play the violin?" The answer I think is quite simple. If you can play the violin then you understand how to play the violin. If you cannot play the violin then you do not understand how to play the violin. If after her explanation, you rush home and start trying to play the violin, then at least we can say that you are trying to understand. But baring any playing or attempt to play, even after your friend's vivid explanation, it is my stance that you still understand nothing about playing the violin.
I don't care how vivid an example of a mathematical concept might be, the example itself is not mathematics. Mathematics is a product of the mind, not the world. And I am only going to be convinced that mathematics is taking place if I see the mindfulness I am speaking of. Just like I will never be convinced that you understand how to play the violin, unless, you play the violin.