beanz Magazine

Lambda Calculus

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It’s a programming language unlike any you’ve seen before. Check out this symbolic system designed for mathematical calculations.

Programming languages are cool. There’s so many and most of them are really different from one another! We’ve covered a lot of really interesting ones so far: languages that treat pictures as programs, languages designed to run on mobile devices, languages designed to do computing fast, and many others.

Now, we’re going to talk about a programming language completely unlike everything else we’ve seen but one that’s still exerting influence today: the lambda calculus.

The lambda calculus is old; old enough that it predates computers. It was invented by Alonzo Church, a mathematician, in the 1930s. You may not have heard of Church before, but you’ve probably heard of one of the students that worked for him: Alan Turing! There’s a neat connection here between Turing’s work and Church’s lambda calculus.

You see, Turing came up with what we now call “Turing machines” as a way of figuring out what you could do with a machine that calculates. That may be surprising, but even though computers didn’t yet exist, the idea of a machine that could do math had been around since the early 1800s when people like Ada Lovelace wrote the first algorithm for a very hypothetical computer called the Analytical Engine.

Church, on the other hand, tried to answer the question of the limits of computers by inventing a symbolic system, the lambda calculus, for doing calculations. It’s a kind of “algebra” of computation, where each expression you could write down in the lambda calculus executes to perform a complex calculation. In other words, it’s a language for programs you can write down that can be executed to calculate a number. It’s a programming language, though a special one that was meant to be calculated by hand instead!

That’s enough of a preface. What’s it actually look like? Well here’s how you write the numbers 0 through 3 in the lambda calculus:

And here’s an if-else-statement in the lambda calculus:

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