[bmitc]

You don't need OOP plus another paradigm to do automatic differentiation though. I've implemented automatic differentiation, albeit "simple" versions and only forward-mode at this point, but there's really nothing special about the implementations.

Logic programming, on the other hand and for example, needs something much more substantial to be implemented as a library in an existing language, such as backtracking, unification, or the full-on Warren Abstract Machine.

If someone has a clear example of differential programming that is different than just using automatic differentiation as a technique or library, then that might help.

> doesn't let me have a variable hold half of one object and half of another or let the language derive the code that gave me that object at runtime

I'm not sure what you mean here. Could you elaborate?

[oddity]

I don't need OOP to do a hash table lookup and then an indirect function call with the receiver as the first argument either but that ignores that there's more to a paradigm than the algorithm I use for facilitating it. You can embed unification of expression trees as a library in C++. People implement backtracking all the time in almost every language. Talking about differentiable programming as if it's just autodiff is missing the point of what a programming paradigm is.

There's a mechanism, yes, but that's just a means to an end of efficiently enabling a different way of approaching programming. In the case of differentiable programming, that's continuous code and continuous data enabling program search that doesn't have to use purely discrete methods (like logic programming). If that sounds like autodiff and backprop, then yes, that's because that's a good way to implement it. Tensorflow and PyTorch are DSLs embedded in Python and C++ both useable and used for more than just implementing neural networks, but most people aren't happy calling a library a language until it has a parser and a file extension.

> I'm not sure what you mean here. Could you elaborate?

Most programming languages assume that a variable can only contain one value, or a composite value of values. Differentiable programming lets code be smoothly transformed from one to the other while being meaningful at all points between. In an object oriented case, this would be like having a variable contain an object that behaves like some known object A or object B selectively depending on which choice maximizes the success of the program at any given moment.