TL;DR: If you want to do mathematics in a programming language as you suggest, then you need programming language that can express proof. Mathematics without proofs isn't mathematics. The ergonomics issues present in all of the existing languages for describing mathematical proofs are one way to motivate the work described in the article.

> I'm not too fond of academic-anything.

There's a difference between recognizing your personal tastes and being dismissive of entirely reasonable enterprises.

> Most papers are bad, period. Most jounals are bad, period.

Most software is bad, period. Most documentation of software is bad, period. Especially when written by 20-somethings straight out of university, which describes the vast majority of paper writers in academia.

...what's your point?

> I think Linus is more of a computer science then someone who comes up with a new cryptography algorithm

What benefit is there in re-defining extremely well-established terms? It doesn't actually serve to convince anyone of anything, and confuses conversations.

> I'm not talking about formal verification when I talk about computer science/programming methods for representing ideas.

So then you dismiss mathematics as a discipline? Because writing some code and some test cases is not mathematics. Modern Mathematics means proofs. Period.

Proof assistants are languages we've invented for describing proofs. They are literally exactly what you've asked for in your original post.

And they're pretty difficult to use. So exploring different approaches for describing mathematical structures and proofs makes a lot of sense.

> Also, formal-verification isn't an academic endevour. You've got a CPU in your system that is all formal verification of VHDL or Verilog.

Yes, I acknowledged that we've made tremendous progress!

But formal verification is still extremely resource intensive and extremely rare in software.

> You've got planes flying in the air by Boeing that are almost verified completely.

Where did you even get this idea? It isn't true.

Every line of code might've been run through Coverity or whatever. Some static analysis is very different from verification of functional correctness properties.

> Also, in math your proof is your program so this is an entierly different issue.

Yes, this is exactly my point! To do math like programming, you need a language for describing proofs. All of the languages that exist for doing this are difficult to use. Maybe visual languages could help out here.

> I'm saying they can benifit from the abstractions they stand on to create further experimentation for themselves.

You're tying to argue that the success of modern programming practices indicates that we can do the same in math. But 40 years of proof assistants remaining niche demonstrate this simply isn't true. It's a good idea, but it's not easy to get right.

> I'm not talking about formal verification. I'm taking about the ways we express our ideas.

In mathematics, proofs are the critical piece of output. Expressing proofs is step zero toward "expressing our ideas" in mathematics.

If you want to "do mathematics" in a language, as you suggest here, your language needs to be able to express proofs.

> Computer Science != formal verification.

And math != hacking out code.

> I think your "formal verification is largely an academic endevour" takes the cake on that one.

Okay. When was the last time you wrote production code that had a functional correctness proof? What percentage of software written yesterday would you guess or one day will have a functional correctness proof?

Or, to borrow you point about children, when was the last time a high school student wrote a proof in a proof assistant (i.e. did math in a programming language)?