Given your user name, that's a fitting perspective. I have a very different view: programming language design is applied psychology, not mathematics. Programming languages are the ultimate human-computer interface. It may be tempting to ignore the human side of that interface: after all that side is messy, incomprehensible, impossible to write neat proofs about, and almost as hard to publish tidy convincing papers about. Without the human side of programming languages, however, PL design is a fairly pointless endeavor — without the humans, we wouldn't need programming languages in the first place.
I fear that, for these rather tempting reasons, academic programming language research has fallen into the trap of treating PL design as a form of mathematics, ignoring the human side of the field, and as a result has contributed far less to the practical practice of programming in the past several decades than one might naïvely expect and hope. Academic PL has, for example, almost completely disregarded the wild success and popularity of dynamic programming languages. Instead, prominent PL researchers glibly dismiss dynamic languages as "unityped", ignoring that there is, in fact, a rich life of types in every language if only you shift your notion of types to match what the users of those languages actually mean by the term. Shouldn't more academics be interested in how and why this approach is so successful to see what can be learned from that success, rather than just dismissing it out of hand because it doesn't fit neatly into the theory?
We have seen a recent cross-pollination of ideas between dynamic and static programming languages resulting in the likes of TypeScript, Go and Julia—and these hybrids have been wildly successful. But none of this is coming from academic PL circles. It's coming form the people who view languages as tools and want tools that fit the hands of their human users, rather than designing beautiful but alien tools that don't fit human hands. Even heavily static languages like Rust that have become very successful, have done so largely by going against the grain of static language research.
So yes, academic PL does seem to have largely become a subfield of mathematics, but to that extent it has, it has also become as detached from actual programming as modern literary theory is from the practice of writing books that people want to read.
> I have a very different view: programming language design is applied psychology, not mathematics.
It's both, and probably more.
As you note, programming languages are the ultimate human-computer interface. They have to serve two masters - the computer's unforgiving hardware, and the inevitably fallible humans attempting to harness the computer's power. Each master requires different types of proof/evidence of the language's suitability to the task.
I fear that, for these rather tempting reasons, academic programming language research has fallen into the trap of treating PL design as a form of mathematics, ignoring the human side of the field, and as a result has contributed far less to the practical practice of programming in the past several decades than one might naïvely expect and hope. Academic PL has, for example, almost completely disregarded the wild success and popularity of dynamic programming languages. Instead, prominent PL researchers glibly dismiss dynamic languages as "unityped", ignoring that there is, in fact, a rich life of types in every language if only you shift your notion of types to match what the users of those languages actually mean by the term. Shouldn't more academics be interested in how and why this approach is so successful to see what can be learned from that success, rather than just dismissing it out of hand because it doesn't fit neatly into the theory?
We have seen a recent cross-pollination of ideas between dynamic and static programming languages resulting in the likes of TypeScript, Go and Julia—and these hybrids have been wildly successful. But none of this is coming from academic PL circles. It's coming form the people who view languages as tools and want tools that fit the hands of their human users, rather than designing beautiful but alien tools that don't fit human hands. Even heavily static languages like Rust that have become very successful, have done so largely by going against the grain of static language research.
So yes, academic PL does seem to have largely become a subfield of mathematics, but to that extent it has, it has also become as detached from actual programming as modern literary theory is from the practice of writing books that people want to read.