Yanis Benson: I feel the same.

But when I try to step ahead and and try something/explain it to others I always get beaten by standard unarguable Math which says there is no way you could invent such a way of explanations so one shouldn't even venture in this land.

I myself of course have some math background(though I wasn't digging too deep, as I left university at 3rd year and rarely used anything complex since back then), but my thought process seems to be much more influenced by programming, which, in this case, shows itself as tendency to build up and ruin abstractions. The way you normally invent an abstraction in math and use it's methods now and then, I invent abstractions every now and then. I guess exactly this is the part lacking to build up this language. And the thing making it hard to introduce is this tendency of mathematicians(as far as I know) to stick to numbered and concrete abstractions. It actually amazes me how snobbish mathematicians are in this field.

Petar Maymounkov: Yes. The issue is that when Mathematician think about math they find the notion of "double-checking" everything from axioms to present interests inhumanly difficult: and it is. But this is the wrong place to look.

That path (axioms to modern interests) is the path of history. But the path through which Math is understood by a person (the consumer of it) is not historic, rather it is through their mother tongue (whatever it may be) which as Chomsky tells us is a cyclic infinite formal system. Not acyclic.

That's all there is to it.

Because Math doesn't care to check back and "defragment", it is sticking to this acyclic system of axioms+logic which is simply—ironically in math's own terms—a local view on a large manifold (the formal system of truth), which is slightly incorrect at the fringes, the way a tangent space is not accurate away from the focus. And these inaccuracies manifest themselves into ever more tightly phrased problems of the Clay Math Institute.

They are Math language closing in on its own contradiction. Beautifully,

the process of abstracting a few very hard open problems is—surprise—Occam's razor in reverse (for physicists, this is “unification in reverse”).

I think the picture is simple and obvious.

Song He: Wonder if you guys continued to explore this a bit further .. I kinda of feel the limitation of the "local view" of our current math and logic system .. Finding a way out would be very exciting ...