A colleague sent me an interview with Kalani Thielen about trends in programming languages.

I’m fascinated by this interview. Kalani is obviously an expert. But what should we think of the following?

The function, or “implication connective” (aka “->”), is an important tool and ought to feature in any modern language.

I have a problem with this kind of jargon. Why? Because if concept programming teaches us anything, it’s that a computer function is anything but a mathematical “implication connective”. *Programming languages are not mathematics.*

#### Programming languages are not mathematics

Computer science courses often spend a lot of time teaching us about various aspects of mathematics such as lambda calculus. So we are drawn to believe that programming languages are a sub-branch of mathematics.

And indeed, this is the view that Kalani expresses:

The lambda calculus (and its myriad derivatives) exemplifies this progression at the level of programming languages. In the broadest terms, you have the untyped lambda calculus at the least-defined end (which closely fits languages like Lisp, Scheme, Clojure, Ruby and Python), and the calculus of constructions at the most-defined end (which closely fits languages like Cayenne and Epigram). With the least imposed structure, you can’t solve the halting problem, and with the most imposed structure you (trivially) can solve it. With the language of the untyped lambda calculus, you get a blocky, imprecise image of what your program does, and in the calculus of constructions you get a crisp, precise image.

The statement that I strongly dispute is the last one: *In the calculus of constructions, you get a crisp, precise image*. The Calculus of Constructions (CoC) is the theoretical basis for tools such as Coq. These tools are intended to assist with computer-generated proofs. So the very thing they talk about is represented crisply by the CoC. But if I want is to represent the behavior of `malloc()` (a function whose primary role are its side effects) or how CPUs in a system communicate with one another (physical interactions), then CoC is of no use. It doesn’t give me a crisp, precise image.

In other words, high-level languages with first-class functions, implicit garbage collection or dynamic typing are really cool, but they give me a blurry, imprecise picture of how the computer actually works. The reason is that a computer is not a mathematical object, it’s a physical object. An “integer” in a computer is only an approximation of the mathematical integer, it’s only an approximation of the mathematical entity with the same name.

Trying to hide the differences only makes the picture more blurry, not more crisp. For example, you can use arbitrary-precision arithmetics instead of integers with a fixed number of bits. And now, your “integers” start consuming memory and have other side effects. In any case, these arbitrary-precision numbers are not “native” to the computer, so they are “blurry”.

#### Programming languages are languages

With concept programming, I’ve consistently argued that programming languages are, first and foremost, languages. Trends in programming languages are a problem of linguistics, not mathematics. The goal should not be to make the language more precise, but to make it more expressive. Nobody cares if you can solve the halting problem regarding programs in your language, if to achieve that objective you have to give up expressiveness.

Let’s make an analogy with real-world languages. Imagine that we decided that it’s important to make English “precise”. We’d set the goal that any sentence in Precisenglish could be provably true or false. First, you’d end up with a language where it’s impossible to write “This sentence is false”, since it’s impossible to prove this sentence true or false. Now, this particular sentence may not seem indispensable. But what about questions? “What time is it?” wouldn’t be a valid Precisenglish sentence… Isn’t that a hefty price to pay for precision?

The same is true for programming languages. You can impose constraints on the language that make it easier to prove things. And then, simple things like side effects become really complex. In Haskell, the simple task of writing something to the console requires complex constructions such as monads…

#### Mathematics and programming both derive from languages

It’s interesting to observe that mathematics is also a form of language. It’s a precise, if restricted, language that helps us reason about symbols, build theorems, prove things. So it makes sense that mathematics and programming languages are related: they both are forms of languages. But it’s not because programming languages derive from mathematics. It’s because programming languages and mathematics both derive from languages.

In my opinion, progress in programming languages will happen if we decide to give up mathematics and embrace linguistics. When we try to start focusing on how we *translate* concepts in our head into computer approximations.

A good language by that definition can adapt to mathematics as well as to other domains of knowledge. The ideal programming language should be capable of representing mathematical properties. It should be possible to write a subset of the language that is precise, just like mathematics can be seen as a precise subset of human languages. But it should also be possible to have a subset that is not necessarily as precise because it addresses other domains. There should be a subset that deals with I/Os not using monads or other abstractions, but representing what’s going on in the machine as precisely as possible. There should be a subset that deals with parallelism, or computations on a GPU, or floating-point arithmetic.

And you basically have a definition of what I’m trying to achieve with XL.

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