Epilogue: Computer Science, Lambda Calculus, and Y Combinator

Who invented lunchboxes?

In the last page, a villager from Lambda Village asked the following question:

To answer this question, we need to ask another question: Who invented lunchboxes?

Answer: Lunchboxes were invented in the 1930s by a mathematician Alonzo Church.

However: Alonzo Church did not call it a “lunchbox” when he invented it. Instead, he called it “Lambda calculus”.

Let’s now talk about lambda calculus.

Lambda calculus and lunchboxes

Lambda calculus, invented by Alonzo Church, works exactly like a lunchbox—although they look different.

Here is an example of lambda calculus. The leftmost letter, λ, is a Greek letter “lambda”. That’s why it’s called “lambda calculus.”

The above lambda calculus expression is exactly the same as the following lunchbox:

If you run the above lunchbox, it will become . Press Run :

Similarly: If you “run” the earlier lambda calculus expression, the result will also be B.

Of course, there are more complicated lambda calculus expressions. Take a look at this one:

The above lambda calculus expression is exactly the same as this lunchbox:

If you replace with sushi , and replace with a sandwich , it will be the same as the Y Combinator lunchbox we used on previous pages.

So: The previous lambda expression represents Y Combinator.

We have been talking about lunchboxes in this course. But we really were talking about lambda calculus. I invented lunchboxes to make lambda calculus more approachable for non-programmers.

Back to the villager’s question

Now, let’s revisit the villager’s question:

Earlier, we said that: Lambda calculus works exactly like a lunchbox.

So we can rephrase the above question as follows:

And here’s the answer: Lambda calculus can do any calculations that can be done by today’s computers. Therefore, any calculations that your computer or smartphone can do are also possible for lambda calculus.

And because lunchboxes are the same as lambda calculus: If something can be calculated by today’s computer, it can also be done using a lunchbox.

Legacy of lambda calculus

Lambda calculus, invented by Alonzo Church in the 1930s, had a big impact in the evolution of modern computer science.

Specifically: Lambda calculus influenced many programming languages. Today’s popular programming languages usually have a feature that originated from lambda calculus.

Example: Python is one of the most popular programming languages today, especially in the field of AI.

The python language has a feature called “lambda”, which is similar to lambda calculus.

Here’s an example of python’s “lambda” feature: Take a look at the Python code below (we’ll use green for the Python code).

The above Python code is pretty much the same as the lambda calculus expression below:

If we represent this using a lunchbox, the final result will be .

Similarly, if you run the earlier Python code, the result will be 'B' as well.

Summary: Today’s popular programming languages like Python have a feature that originated from lambda calculus.

Building a complex thing from simple things

We’re almost done, but let me say a few things about computer science and Y Combinator.

One of the most important ideas in computer science is to build a complex thing from simple things. After all, computers are made of very simple things: 0’s and 1’s.

In this course, we learned about lunchboxes (or lambda calculus). And lunchboxes are simple. There are only a few rules you need to know:

But we learned that: Even though lunchboxes are very simple, they can do any calculations that can be done by today’s computers.

Why is that possible? Because, as we’ve seen so far, by combining simple lunchboxes, you can build a complex lunchbox that can do complex calculations.

For example: We learned that we can use the Y Combinator lunchbox to do complex calculations like factorials.

Again: Computer science is all about building a complex thing from simple things. I think lunchboxes—especially Y Combinator—are a great example of this idea.

We didn’t have time to talk about any other topics in computer science. But in every sub-field of computer science, you’ll learn to build a complex thing from simple things.

For example: If you study artificial intelligence, you’ll learn to build a “neural network” —a complex thing that can be used to process images—by combining simple things called “neurons” .

Hopefully, this course gave you an idea of what learning computer science is like. If you do end up learning computer science, you’ll be surprised that most innovative ideas in computer science are based on lots of simple ideas.