"If you have a math background (as I do)"
Ah, there's the resolution then. I would never argue that a strong math background is not easily converted into a very effective computer programmer with little further experience. Please do not go around telling people that they "don't need experience" based on that result; they will not hear "I can get a math background" instead.
What you say sounds obvious, but please let me remind you that you've almost certainly seen people who could not leap directly to those conclusions correctly. I can name names of people where I work with years of experience who still engage in "flail at the problem blindly until it seems to go away". In fact just the other day we observer one programmer created a perfect loop; patched a bug, created a new one, patched that, created another new one, another programmer jumped in to fix that, and then the original programmer patched yet another fix resulting in the exact same source code we had at the beginning of this process, only with another new line. (I am not sure what the degree status of this person is but it doesn't seem to have taken.)
(And I would agree you can learn a lot without formal training, but at some point you're still going to have to learn what that formal training covers. You also have to learn more anyhow.)