You won't do a calculus class for mathematicians without also heaps of real analysis and/or measure theory. It's a different story for a field like engineering, but that's not what the blog post is about. If you haven't studied the proofs, you haven't studied advanced mathematics.

As for Bayesian vs. Frequentist, it's another vim vs. emacs style debate most of the time - which is most appropriate to use, as opposed to which is right and which is wrong. Quite a lot of the time, it just doesn't matter.

The kind of math we engineers study in typical BS/MS curriculum is quite basic.

I just wanted to point it out. The thread title is "Learn Advanced Mathematics Without Heading to University", which could be read with the implication "Learn advanced mathematics, at the university level, without going to university". Under this context, someone replied that this is possible because s/he learned it, but can't prove what s/he's learned. Of course not being able to prove the useful theorems you've learned doesn't render them useless (although perhaps less useful as you're less likely to know precisely when they can be applied and how to extend them to new situations), but in the context of the discussion it seems like an important distinction to make.