Based on the exclamation mark in the title, we can conclude that the lambda calculus is fact an unintentional esolang.

"but its a card game" this line read was perfect

This is hands-down the best explanation of lambda calculus I've ever heard. Good job!

functional programming is applied category theory, where a coconut is just a nut

8:20 it should be `(a successor) b` not `a (successor b)` [which technically = (b+1)^a] I can't believe this wasn't immediately apparent in the extremely clear and human readable syntax of lambda calculus smh

And when you implement an evaluation algorithm, you get LISP.

I am a junior in college, this single video has helped me more then any lecture i have experienced in this semester. Thank you

This whole paradigm could be represented with the new esolang I'm calling "Threadr". You know those fancy thread first/last functions (-> and ->> respectively) in Clojure? Those are the only two things that are allowed in the language other than basic math and lambda definitions/application!

There are so many fun things in terms of Turing completeness! Microsoft PowerPoint, HTML and CSS (if used together), SUBLEQ, and heck there was a sigbovik paper that was Turing complete.

Genuinely glad to see you are covering more computer science related stuff. I've been fascinated with lamda calculus and how it can be used to do math. Very sorry about missing the premiere, anyone who forgives me will be sent a 50% discount on their next purchase of Tux Cola.

I will need to watch this x times and play with it for a long time to wrap my head around this.

You can turn any algebraic datatype into a lambda calculus representation: values of the type are represented as functions that perform one level of pattern matching. For example if you have a standard functional singly-linked list type, then your pattern matching needs to know what to do with a cons, and what to do with an empty list, so it takes two functions (which I'll call f and x). In that case, the list [1,2,3] is the function: (^f. ^x. f 1 (f 2 (f 3 x))), and the empty list is (^f. ^x. x) i.e. the same as false and Church zero. The ADT for a boolean is just a choice between true and false, and translates to the same as the ones you describe in the video, assuming you put true first and false second. This suggests some other ways to do numbers in lambda calculus, other than Church numerals. E.g. you can make a linked list of booleans (forward or backward), or put 2^n booleans at the leaves of a perfectly-balanced binary tree to make a fixed-size number, or make a giant tuple of 64 booleans.

Guess i'll have to stay up until 2am to watch this master piece

My head hurts

i did not understand a single word except "turing machine"

8:46 multiplication is MUCH more clever if you know how to do it. just take lambda a, lambda b, lambda f, lambda x: a (b f) x, which is behaviorally identical to the bluebird combinator: lambda f, lambda g, lambda x: f (g x). exponentiation is just applying one number to the other, even simpler!

I spent the past couple months with pure functional programming (lambda calculus and similar) as a special interest so *happy noises*

I knew this would be a great video the moment I saw the title. I love this channel!

Have you heard about Concatenative Calculus? It's like lambda calculus (actually more like combinatorial calculus), but juxtaposition denotes composition instead of application and instead of Polish Notation it results in RPN. It's also way easier to pass and return multiple values and extend with non-pure functionality like I/O than functional programming. Making composition the main thing of the system makes so much sense: unlike application it's an associative operation and a composition of a list of functions is just a list of transformations to be done in order, which is how people normally think about algorithms. Its associativity also makes it extremely easy to factor out frequently occuring sets of commands to new named functions.

I’m so excited for this vid! Will be watching tonight :)

I found this video at the end of my semester and it's AMAZING

I still come back to this video a lot

very good video, amazing explanation, and engaging humor! :D

3:53 Doesn't carl know that a turing machine is a card game?

3:24 personal attack incoming

Man, this is so fucking abstract but so fucking cool. Love it

good video man,the links are so good .I was doing SICP but had to stop due to something ,i so wish i completed it.

i've been working on a modified lambda calculus that completely sidesteps the need for renaming variables, and it's by producing a very different restriction: all function definitions must be pure juxtapositions (such as: lambda a b c d = a (b d) (c d)). you can recover all lambda calculus behavior by using placeholder variables wherever you would want a constant to appear in your expression, such as: (lambda p a b c = p a (p b c)) pair; this produces a tree-like pair nesting from three provided arguments.

The insane viewcount to like ratio tells me this is the video I needed! I am 5minutes in and even if the rest of the video is a black screen with white noise, this will still be a masterpiece.

Lambda Calculus: Turing complete before Turing complete was a thing.

WHY DO I LOVE THIS SO MUCH

You are better than Church at explaining this. I don't know he felt the need to remove the intuition behind everything he wrote, compare that to Turin who is actually fun to read. Fun fact, Church's students agree that he was really boring as a teacher, just barely reading his books and copying the proofs

Church numerals is isomorphic to the set theoretic definition of numbers.

Just saying dude, I absolutely love your videos. Please keep it up! Gonna try this is python lmao

2:05 This is not applying a function to another function, this is applying a function to the result of a function.

I dunno if anyone has ever used this one esoteric programming language called Microsoft PowerPoint 🤔

im scared

This is excellent (thumbs up!). But it is very frustrating the way the steps in reductions replace each other instead of being on the screen at the same time.

even tho i have seen these things before, this simplified explanation still baffled me

KZfaq... the place where some guy explains to you in 5 sentences what your professor couldnt in multiple hours of lectures.

First time i feel confident that i actually understand lambda calculus

still somehow less cursed than pure prolog (i.e first-order logic/horn clauses)

Since you already made add/multiply functions, it’s also possible to create exponentiation, tetration, pentation, hexation, …

me when new truttle1 video

Don’t feel bad that your spice tolerance is low. Once, my cousin said something along the lines of, “My spice tolerance is low, and this food isn’t even spicy to me.” Despite my better judgement, I tried the food, and my mouth burned.

Basically IF() function from excel

No idea what this video was about, but I enjoyed it.

Can you cover dlang templates; I wrote an appendable list in it

That Curry Tangente ist Just an adhd mood

can you make a vid about JS F*ck? It's a pretty fun esolang that's also pretty popular.

I had to reduce my speed to .25x to understand your example about the NOT function, but it's clear now

And mulitiplication defined exponentiation!

Here for the premiere!

nice trailer lol

WHY DO THESE CHARACTERS CANT STOP MOVING WHILE THEY TALK THATS INSANEEE DUDE

b-but wherr's the truth machine.....

I have learned lambda calculus in combination with semantics and logical programming. And then logic.

lambda calculus the yes

2:47 Evil Rush be like

Can you please please please do a video on plankalkül sir

best trailer

How would you do subtraction/division?

How to simulate floating point numbers in lambda calculus?

The highs are too high and the lows too low. Equalize sound a bit more including music. really good video!!! :))

I think I would argue that function composition is not the same as applying functions to functions, though I guess you could argue that the composition operator is some form of lift from a function f: A -> B, to a morphism (f o) taking the category of arrows into A to the category of arrows into B. Also not sure I agree with "everything that's Turing complete is a programming language" Honestly a decent overview of Church encoding otherwise

turing machinen’t

what about the Akkerman function

1:34 mmh desmos so based

2:10 Technically this isn't applying a function to another function; this is just the *composition* of two functions. You're applying f to the number g(x). Other than that, great video!

Interesting to see a programming language that predates computers.

really nice

Funny. I still have no idea what you're talking about. For some reason I think it would be better if you would leave ALL the steps on the screen at the some time so that I can follow along sequentially.

Nice AWS Lambda image

So basically numbers are constant functions?

Nice vid!

did you change the color of the one?

What about fixpoint

That lip-sync looks so cursed

omg i rmemeber the fucntion machines

b-but truttle, you didn't show how to do hello world in lambda calculus xd

holy crap Carolina Reapers are hotter than pepper spray.

my brain hurt

isn’t this just math haskell

truttle posted :P

Is redstone Turing complete?

I wonder how equality works in lambda calculus

Haskell be like:

I can define a turing machine off the top of my head, but it's not pretty and involves a heterogenous 7-tuple. (Starting State, All States, Accepting States, Input Alphabet, Initial Tape Contents, Tape Alphabet, State-Tape Transition Function)

hrrrnnnggg there were a lot of obfuscates feet in this one.

The Discord invite in the description doesn't work.

plz dark mod

Okay but what is a Monad??

You must have gone to one hell of a kindergarten.

2:10 no thats not what plugging one function into another means. What you have described is function composition. a function f:A->B is defined as a set of ordered pairs AxB, such that no two elements of f have the same first value (roughly speaking). We define f(x) to essentially mean "find whichever ordered pair has x as the first element, and return the second element". For example, if f={(3, 2), (4, 3), (5, 4)} then to find f(4), we find the ordered pair which has 4 as the first element (namely, (4, 3)) and return the second element. Thus f(4) is 3. There's a lot more nuance but it's not too important for this explanation. If we have a function f:B->C, and a function g:A->B, we define fog:A->C to be the function represented by f(g(x)). Roughly speaking, we plug in x to g, then plug in this value to f. This is what you describe at 2:10. This is completely different from f(g) which is what plugging one function into another actually means. For this we just use our above definition, where find the ordered pair which has g as it's first value, and return the second value. Let's use an example to illustrate. Let's say that g = {(1, 2), (2,3), (3,4), ...}. This function can be represented with the equation g(n)=n+1 when n is a natural number. Now let's say that f = {(g, -1), (2, 1), (3, 2), (4, 3), ...}. We can represent this function with the equation f(n)=n-1 when n is a natural number. we say that f(g(x)) = g(x)-1 = n+1-1 = n, when n is a natural number. This is not what plugging one function into another means though, this is just function composition. f(g) = -1, since that's the corresponding value in our function.

GEORGE

Do the lambda calculus

3(4x - 3) + 2

Half-Life: Esolang

hmm. math. neat

YAYYYYY