[WCSTombs]

> imagine that instead of having axis names live as single-letter variables within einsum, our arrays themselves had these names attached to their axes?

I already replied to a couple of other comments with this, but you're exactly describing the Python library Xarray: https://docs.xarray.dev/en/stable/ I've found it to work quite well in practice, and I've gotten a lot of mileage out of it in my work.

[taliesinb]

Thanks for highlighting XArray in your other comments. Yup, XArray is great. As are Dex and the various libraries for named axis DL programming within PyTorch and Jax. I never said these things don't exist -- I even mention them in the linked blog series!

But I do think it's fair to say they are in their infancy, and there is a missing theoretical framework to explain what is going on.

I anticipate name-free array programming will eventually be considered a historical curiosity for most purposes, and everyone will wonder how we put up without it for so long.

[WCSTombs]

Thanks for pointing that out, and sorry for being redundant. I skimmed the blog but missed that.

Overall, I agree with your larger point. I think this it's a natural progression and pretty analogous to assembly language leading to higher-level languages with a much better developer experience. In that case we went from fixed registers to developer-named variables, and now we can go from fixed numerically-indexed array axes to ones indexed by developer-chosen names.

[taliesinb]

No worries! Yes, exactly. It's also similar to doing arithmetic with and without units. Sure, you can do arithmetic without units, but when you are actually working with real-world quantities, but you easily get yourself into a muddle. Unit-carrying quantities and algebraic systems built on them prevent you from doing silly things, and in fact guide you to getting what you want.

[WCSTombs]

Yeah, I'm totally with you on the units, too. For me it's even more frustrating, because it's such an obvious and ubiquitous problem, and still relatively unacknowledged. There are third-party solutions (like Pint in Python), but in my experience by default scientific software projects tend to begin with just NumPy basically, and introducing more advanced (IMO) tools like Xarray and Pint is an uphill battle against ingrained habits.