How does the code, by itself, describe the reason why the code was written that way?

Lots of ways. Good naming of files/instances/classes/variables. Good well structured acceptance/unit test suites. Good well factored code.

How does the code, by itself, describe the business needs that that section of code is supposed to fulfill?

Acceptance tests are one way. Good well structured, well named code is another.

How does the maintenance programmer tell if those business needs have changed?

How do comments help here?

How do I use a programming language, in and of itself, to signal that a given, efficient but obscure algorithm is correct, and that the "obvious solution" is not the right one?

Have a test suite that fails for the obvious, but wrong, solution? Make the complex code easier to understand so that the traps are easier to spot?

Okay, I admit it, I'm playing devil's advocate a bit. But only a little bit. Comments are not always a bad thing - especially if you lack a decent test suite.

However, there is a very bad tendency for people to use comments as the weapon of first resort when tricky code is encountered. Something isn't obvious - just add a comment.

A much better attitude in my opinion is to use comments as a last resort. Before that I will spend time trying to make the code clearer, separate responsibilities better, add explanatory tests, etc.