I have a little git project called 'cetis' where I throw cute/dangerous ideas for a next-gen language, and actually the idea of eval/apply as the "Maxwell's equations of software" is in there due to a talk by Gerald Jay Sussman which is transcribed. He says something to this effect: "Look at what these equations say. They say that a changing magnetic field causes a changing electric field, and a changing electric field causes a changing magnetic field. Look at what we have -- eval calls apply, apply calls eval."

So, cool, but why not push this metaphor further? That's the dangerous idea.

The dangerous idea is that Maxwell's equations don't use voltages. We love voltages. They're useful everywhere. In Einstein's special relativity, once you invent voltages (and their magnetic counterpart, the vector potential) all four Maxwell equations stop being separate, they all take the same form. So you tack the charges ρ onto the current j to get the 4-current Jⁿ = (ρ c, j). You tack the voltage V onto the vector potential a to get the 4-potential Aⁿ = (V/c, a). Then the entire set of Maxwell's equations becomes a single wave equation of your voltages emanating from your currents:

    ∂ᵢ ∂ⁱ Aⁿ = μ₀ Jⁿ 

There are four of them, one for every n, but they're all the same form. If you understand one, you understand them all. There are a couple conventions -- these are summed over i, and if you really wanted to know the kinematics you would want to know the fields -- and those are instead:

    Fᵤᵥ = ∂ᵤ Aᵥ − ∂ᵥ Aᵤ

The space-to-time components of the field matrix are the electric field, the space-to-other-space components of the F tensor are the magnetic field; the minus sign guarantees that there are no space-to-same-space or time-to-time components; those are all zero.

Now, before this becomes a rant, here is the idea. Is there a way in which apply can look like eval, and eval can look like apply? If Einstein is a lesson, we might have to invent a means of programming where you look directly at invariants which you want to create.

The goal is that you should just specify an inhomogeneity, a little piece of data and some information on how it should be. The computer then constructs the invariants through some sort of "computational wave" -- and from this the computer can derive the actual field, the instructions needed to actually run the process on a computer.

There are some tantalizing suggestions and possibilities. The Meta II compiler (recently extended in JS as the OMeta project, which is definitely on my source-code reading list) works by realizing that a bunch of different stages in compilation are all just the same idea of pattern matching.

Perhaps one day we'll do what we do with photomultiplier tubes: we just visualize the voltages that we'd need to focus and accelerate electrons into the metal plates, and not even pay attention to the lower-level picture of how these fields are being generated to do the right thing.