Well, perhaps we disagree, but I don't think mathematics is about "proofs". I think proofs are a by-product of folks doing mathematics. They're the output of a long chain of mathematical thinking and can take many forms. I would agree that being able to follow contemporary mathematical writing is important, but one should learn the formalisms after they have a grounding in mathematical thinking.
The formalisms aren't the point. For example, I wouldn't stress out if a student couldn't follow Newton's * Principia Mathematica*. I've tried and I can't, really.
This is why the way we introduce "proofs" in high school in the US (via tedious and seemingly arbitrary geometric arguments) is so destructive. It's all formalism and, honestly, very little mathematics.
I can and have explained the Pythagorean theorem to folks with virtually no mathematical training. The only prerequisite is the ability to find the area of a rectangle and perhaps a little but of high school algebra.
The formalisms aren't the point. For example, I wouldn't stress out if a student couldn't follow Newton's * Principia Mathematica*. I've tried and I can't, really.
This is why the way we introduce "proofs" in high school in the US (via tedious and seemingly arbitrary geometric arguments) is so destructive. It's all formalism and, honestly, very little mathematics.
I can and have explained the Pythagorean theorem to folks with virtually no mathematical training. The only prerequisite is the ability to find the area of a rectangle and perhaps a little but of high school algebra.
We're not talking about diagram chasing, here.