Probably your friend meant by "language algebrically defined" is that it has a real formal specification in some formal language like Z notation and that you can prove that an implementation is conformant to that specification, and eventually prove some strong properties about programs written in that language.

Also specifications written in plain English are ambiguous but generally more understandable than any strict formalism. Read the definition of natural integers in terms of sets, and you will see what I mean :)

I don't think any significant language useful in the real world will come close to that anytime soon. This is a worthy and necessary goal but that also can degenerate in pointless and pedant formalism like some works in mathematical foundations.

Many languages come with a test suite though. But how can one prove that a test suit is comprehensive enough so that two conformant implementations will behave identically in all cases?

Perl has a significant test suite too. One advantage of perl is that it has a unique implementation unique modulo platforms. It evolves over time but perl has a very good history of backward compatibility.

-- stefp