Lambda Calculus

* journal

[2026-01-01 Thu]

1. Definition

Lambda calculus describes computation using ONLY:

Variables → placeholders x
Functions → rules with a placeholder

\begin{equation} \lambda x\ldotp body \end{equation}

read: a function that takes x and produces body

Application → using a rule on something

\[ f a \] read: apply f to a There are no numbers, booleans, or objects.

1.1. Example: Simplest lambda

\begin{equation} \lambda x \ldotp x \end{equation}

This means: Given something, return the same thing. In everyday terms this is like your identity, or "do nothing" If i apply it:

(\x -> x) "Bob"

The result will "Bob" We remove values.

1.2. Example without math

You think a chair is an object. Lambda calculus says:

A chair is whatever lets you sit

If you can sit on it, it is a chair. No essence beyond use. This is functional ontology.

1.3. When types start to matter

Let

\begin{equation} likes: Person \to Person \to SocialRelation \end{equation}

Because likes is curried, we can partially apply it.

Although both 1.3.1 and 1.3.2 both yield propositions, they are different kinds of objects. One describes an agents behavior, the other tests a population.

1.3.1. Anna likes everyone

\begin{equation} likes\ Anna : Person \to Socialrelation \end{equation}

This describes annas outwards behavior. There is no lambda needed here

1.3.2. Everyone likes Anna

\begin{equation} \lambda x \ldotp likes\ x\ Anna \end{equation}

This reads: "give me a person, and ill tell you whether that person likes Anna. Its type is Person → SocialRelation. This is further discussed in (see: Type Theory in Computer Science, Linguistics, Logic)

1.4. So what the fuck even is a lambda?

λ just defines a behavior

2. Significance

Lambda calculus says:

A thing is nothing but how it behaves when used

Everything is a way of responding to application. A number is a behavior instead of a primitive object. This matters because life is not mostly about truth. It is about responding to stimuli, adapting, and transforming situations. Under this, you are not a thing with properties but a bundle of behaviors applied to situation. Identity becomes a function & character composition. This is tied to the buddhist view of interdependence(see: Pratītyasamutpāda) and Anatta(No-self). A “chair” isn’t a chair in itself; it’s a combination of wood, nails, and the function of supporting sitting. Its chairness is relational and situational. Similarly, the self isn’t a fixed entity. “You” are a flow of interdependent processes. Both decenter essence and emphasize relational existence.

2.1. Historical

Before lambda calculus, mathematics described structures. Alonzo church described computation. This confused pretty much everyone and gave us programming languages, … Church used lambda calculus to formalize effective computability, leading to the Church-Turing thesis.

3. Elsewhere

3.1. References

3.2. In my garden

Notes that link to this note (AKA backlinks).