Lisp as the Maxwell’s equations of software – DDI

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On my first day of physics graduate school, the professor in my class on electromagnetism began by stepping to the board, and wordlessly writing four equations:

He stepped back, turned around, and said something like [1]: "These are Maxwell’s equations. Just four compact equations. With a little work it’s easy to understand the basic elements of the equations – what all the symbols mean, how we can compute all the relevant quantities, and so on. But while it’s easy to understand the elements of the equations, understanding all their consequences is another matter. Inside these equations is all of electromagnetism – everything from antennas to motors to circuits. If you think you understand the consequences of these four equations, then you may leave the room now, and you can come back and ace the exam at the end of semester."

Alan Kay has famously described Lisp as the "Maxwell’s equations of software". He describes the revelation he experienced when, as a graduate student, he was studying the LISP 1.5 Programmer’s Manual and realized that "the half page of code on the bottom of page 13… was Lisp in itself. These were “Maxwell’s Equations of Software!” This is the whole world of programming in a few lines that I can put my hand over."

Here’s the half page of code that Kay saw in that manual:

What we’re going to do in this essay is understand what that half page of code means, and what it means that Lisp is the Maxwell’s equations of software. However, we won’t literally work through the half page of code above. Instead, we’ll do something much more informative: we’ll create a modern, fully executable equivalent of the code above. Furthermore, to make this essay accessible, I won’t assume that you know Lisp. Instead, I’ll teach you the basic elements of Lisp.

That perhaps sounds over-ambitious, but the good news is that it’s easy to learn the basic elements of Lisp. Provided you have a little facility with computer programming and comfort with mathematics, you can learn how Lisp works in just a few minutes. Frankly, it’s much easier than understanding the elements of Maxwell’s equations! And so I’ll start by explaining a subset of the Lisp programming language, and getting you to write some Lisp code.

But I won’t stop with just showing you how to write some Lisp. Once we’ve done that we’re going to write an interpreter for Lisp code. In particular, we’ll create a interpreter based on a beautiful Lisp interpreter written by Peter Norvig, which contains just 90 lines of Python code. Our interpreter will be a little more complex, due mostly to the addition of a few conveniences absent from Norvig’s interpreter. The code is still simple and easy to understand, provided you’re comfortable reading Python code. As we’ll see, the benefit of writing the interpreter is not just that it gives us a running interpreter (although that’s no small thing). It’s that writing an interpreter also deepens our understanding of Lisp. It does that by taking what would otherwise be some rather abstract concepts in our description of Lisp, and giving them concrete, tangible representations in terms of Python code and data structures. By making concrete what was formerly abstract, the code for our Lisp interpreter gives us a new way of understanding how Lisp works.

With our Python Lisp interpreter up and running, we’ll then write a modern equivalent to the code on the bottom of page 13 of the LISP 1.5 Programmer’s Manual. But while our code will be essentially the same as the code from page 13, it will have the considerable advantage that it’s also executable. We can, if we wish, play with the code, modify it, and improve it. In other words, it’s a living version of the Maxwell’s equations of software! Furthermore, with our new understanding it becomes an easy and fun exercise to understand all the details on page 13 of the LISP Manual.

This second part of the essay is based primarily on two sources: the first chapter of the LISP 1.5 Manual, of course, but also an essay by Paul Graham (postscript) in which he explains some of the early ideas behind Lisp. Incidentally, "LISP" is the capitalization used in the LISP Manual, but otherwise I’ll use the modern capitalization convention, and write "Lisp".

The great Norwegian mathematician Niels Henrik Abel was once asked how he had become so good at mathematics. He replied that it was "by studying the masters, not their pupils". The current essay is motivated by Abel’s admonishment. As a programmer, I’m a beginner (and almost completely new to Lisp), and so this essay is a way for me to work in detail through ideas from masters such as Alan Kay, Peter Norvig, and Paul Graham. Of course, if one takes Abel at his word, then you should stop reading this essay, and instead go study the works of Kay, Norvig, Graham, and the like! I certainly recommend taking the time to study their work, and at the end of the essay I...