My one paragraph assumption on his work is this:
He created a new language which created a layer of abstraction from math-as-it-is-currently-done. He has created the connections (the proofs) via this abstracted layer. This is easier to do because he has more flexibility in his abstracted layer. Unfortunately, like trying to go from three dimensions back to two dimensions, it might not be possible to do so without breaking something in the process.
In short, he might fit maths into his system, but his system might not fit into maths. Does this make the maths wrong or him? Or both right? That’s why having peers look at this stuff is important smile emoticon