Programming language researchers have been jumping into the "abyss" for decades now, and nobody has heard of them since. The most popular languages these days were started as hobby projects by professional developers (Ruby, Python, Perl) or funded by big companies that wanted to build out a platform (Java, C#).
In my humble (non-tenured) opinion, proofs belong in math class; model checking belongs in engineering. If you tried to prove any non-trivial thing about your life, you would quickly find it impossible.
Imagine trying to prove that you were going to go to the store and return with a gallon of milk. It sounds simple, but consider: how can you prove that you won't have a car crash or a heart attack? Even if none of those disasters happen, the store might be closed. The kind of milk you want might not be in stock. The traffic might be so heavy that it takes you an hour to drive there. If you can only get skim milk, does that still count as picking up the milk? If traffic takes an hour and the milk is warm when you get back, does that count as success?
Similarly, if you try to prove that your browser will successfully display a web page, you slam head-on into a mountain of difficulties. What if the RAM is bad? How about the hard disk? Will the network lose some packets? What if the web page is so complex that it takes 10 minutes to render on our puny CPU? What about the libraries I depend on? Is there a bug in there? What if the web page is rendered in an "ugly" way (a subjective term). Can I prove that that won't happen? Of course not.
As Einstein said: "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality."
The best I can do is set up a bunch of models and validate my program against them. Static type checking is one such model. Unit tests are another set of models. Tools like lint, sparse, cppcheck provide yet more tests. Another set of tests is giving the program to users and seeing if they like the responsiveness, the user interface, and the overall design.