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  @sdal49266 күн бұрын

  Unfortunately I have watched almost all videos about homology and cohomology. I have not seen a video explaining very clearly and understandable.

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  @sdal49266 күн бұрын

  Unfortunately I have watched almost all videos about homology and cohomology. I have not seen a video explaining very clearly and understandable.

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  @jafo19707 күн бұрын

  Maybe if I watch it 100 more times it'll make sense? Feels like religion to me - you're just pulling shit outta your ass and expecting us to believe it's real.

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  @DumblyDorr16 күн бұрын

  Effectively, combinatorial logic with only the iota-operator shifts most of the "work" to the order of evaluation and its manipulation (for a single rewrite rule). To get other combinators from iota, you force nested lazy evaluation of self-application on the right... it's amazing to think what can be expressed mostly with judicious application of parentheses (ask any LISP/Scheme programmer 😄).

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  @columbus8myhw21 күн бұрын

  I thought the natural numbers were defined in a different way in the lambda calculus.

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  @doc711528 күн бұрын

  🤯🤯🤯🤯🤯🤯

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  @yuzhang1366Ай бұрын

  7:56 should be C=S(BBS)(KK)

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  @unixux3 ай бұрын

  KIxy should be K[Ix,y] = K(x,y) = x ? Or KIxy=(KIx)y=Iy ?

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  @yuzhang1366Ай бұрын

  KIxy=(KIx)y=Iy=y

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  @MagicGonads4 ай бұрын

  Since you only need one symbol (iota) coupled with two grouping symbols (parentheses), you can represent a program using instructions that only take two bits each, with a value left over. Then if you want to support side-effects, do a UTF inspired 'if the chunk is 11 then load another chunk to extend the code' for whatever side effects you wanna add to your architecture/CPU. Though maybe we need a proper padding instruction (usually nop or int 3)

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  @khaliliskarous22255 ай бұрын

  This is really beautifully done!

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  @DerMathematicker5 ай бұрын

  8:58 Is it realy [f]_x -> K or is it [f]_x -> kf?

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  @cmilkau6 ай бұрын

  You really only need S and K, which correspond to the two THEN axioms of intuitionistic logic. There is a straightforward way to turn any lambda into a combination of just these two, exploiting the special case I = SKK (Note that the other axioms of intuitionistic logic are just definitions of AND, OR, and FALSE.) If you want to know how, just keep replacing expressions like this: λx. fg →S (λx.f) (λx.g) λx. x → SKK λx. y → Ky Where x,y are variables and f,g are expressions.

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  @imrematajz16246 ай бұрын

  Is this a roundabout way to introduce Lambda calculus?

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  @academyofuselessideas7 ай бұрын

  Excellent! Do you have something new in the works? In one of the comments you mention "to mock a mockingbird" which is the classic book about combinators, but your presentation seems more modern and focused on the application to computation. It would be nice to know what references you used... it would be also nice to see how this compares to the lambda calculus, or how combinators and lambda calculus are the same... Also the argument is that any computation can be done using combinatory logic but the proof is not obvious (although the result is believable as yet one more example of the church turing thesis)... the fact that one can achieve turing completeness with such simple systems make one wonder whether this has something to do with the basic structures of life. Perhaps the laws of physics build simple systems like combinators, which evolve into what we call life...

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  @epgui7 ай бұрын

  You lost me around the two-minute mark. If Kxy -> x and Ix -> x, doesn't KIxy -> Kxy -> x ?

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  @academyofuselessideas7 ай бұрын

  @epgui Part of the confusion arises because they didn't explain how combinators associate... so, it is not clear if one should think of KIxy as (((KI)x)y) or as K(I(x(y)). The way you associate things in this context matters... When working with combinators, the convention is to associate to the left, so Kxy->x would be more pedantically be written as ((Kx)y)->x... Based on that observation KIxy = (((KI)x)y) -> Iy -> y... A fun book about this is "to mock a mockingbird" by Smullyan... they mention that book in one of the comments... Working with combinators take some practice

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  @Bobby_1016 ай бұрын

  In simple words, the First symbol is applied first always (lazy evaluation). And since K takes the 2 objects in front of it and becomes the first one, ignoring anything else (KIx -> I)..So KIxy -> Iy -> y.

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  @academyofuselessideas6 ай бұрын

  @@Bobby_101 Nice explanation!

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  @epgui6 ай бұрын

  ​@@Bobby_101 In other words, everything is implicitly left-associative, and symbols representing functions are treated the same way as symbols representing values? That seems a bit weird syntactically, but if that's the rule then that makes sense.

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  @Bobby_1016 ай бұрын

  ​@@academyofuselessideas Thanks. @epgui Yah it's a bit odd at first, just roll with it to see what results it brings.

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  @paskoolio7 ай бұрын

  The single combinator from which S and K can be defined goes well beyond Baker. I was a student of Corrado Boehm in 1985 (from Boehm trees in lambda calculus) and he taught us about the single combinator in 1985... This video reminded me of his fantastic lectures on combinatory logic, thank you.

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  @uku41717 ай бұрын

  I wonder what was in Schõnfinkel's lost texts

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  @DeclanMBrennan7 ай бұрын

  So iota is Turing complete? Mind blown. 🙂

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  @academyofuselessideas7 ай бұрын

  @DeclanMBrennan The iota combiator together with the machinery to interpret it is turing complete (this is like saying that not only you need a program but also a compiler that runs the program)... that sounds mind blowing but then you realize that there are a bunch of things that are turing complete...I believe that so far, the smallest turing complete machine Wolfram 2-state 3-symbol turing machine... this is interesting because it kind of hints that we can get that type of computation with very simple systems, which kind of leads to how intelligence and life are plausible (the argument is that something as simple as the wolfram 2-state 3 symbol machine is simple enough that it could have happened by chance without any intelligent designer)

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  @DeclanMBrennan7 ай бұрын

  @@academyofuselessideasI was using "Turing complete" in the sense of a "Turing complete language" but that's a fair point. Thanks for mentioning the Wolfram TM- looking forward to reading up on it. I believe Conway's Game of Life may be Turing complete as well and somebody with way too much free time made hardware within Game of Life to execute its own Game of Life - a cool example of a simulation within a simulation.

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  @DeclanMBrennan7 ай бұрын

  @@academyofuselessideas I just had a quick look at your wonderful channel which I didn't know existed. I feel like a kid in a mathematical sweet shop. 🙂

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  @academyofuselessideas7 ай бұрын

  @@DeclanMBrennan Game of life inside game of life sounds pretty cool... The observation that there are many simple turing complete machines is also part of Wolfram's argument that all physics are automatas (kind of like what we observe is some sort of emergent phenomena caused by very tiny automatas)... I like the philosophy behind some of those ideas but I don't know enough about them to give an informed opinion though

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  @academyofuselessideas7 ай бұрын

  @@DeclanMBrennan I am glad to have you on board!

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  @charlesrosenbauer313510 ай бұрын

  As far as displaying large combinator systems goes, it would probably be easier to appreciate their structure if drawn as a tree as opposed to with tons of nested parentheses.

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  @jackozeehakkjuz4 ай бұрын

  It would be nice to see it in action in a similar way to the legendary post "to dissect a mockingjay".

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  @Godfather-qr6ej2 ай бұрын

  ​@@jackozeehakkjuz I could not find that post? Are you referring to the book, to mock a mockingbird?

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  @jackozeehakkjuz2 ай бұрын

  @@Godfather-qr6ej Sorry, yes. To mock a mockingjay is a 1985 book by Raymond Smullyan. However, my intention was to point to the 1996 post by David Keenan called "To dissect a mockingjay", which is partly based on Smullyan's book. I hope you can find it now. I already edited my original comment to fix this mistake.

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  @jackozeehakkjuz2 ай бұрын

  @@Godfather-qr6ej I tried to post the link here a couple of times but I think youtube keeps deleting it.

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  @Godfather-qr6ej2 ай бұрын

  @@jackozeehakkjuz yes I found "To dissect a mockingjay" can I ask you a question? where are combinators usefull? I've heard about them from people in lambda calculus as a way to make a tiny turing complete machine. and from APL users who use combinators in some practical way I dont know how?

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  @AkamiChannel10 ай бұрын

  What is the paper that was shown? Nice i3 window manager btw.

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  @r4fa3l5910 ай бұрын

  Oh my god! This is spectacular!

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  @eugenemosh365811 ай бұрын

  27:43 -- impressive!! Only two symbols and () and Fibonachi encoded!

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  @shreyasbhokare7192 Жыл бұрын

  wow, didn't expected that this would solve all my doubts about taking maths major. Like I knew I wanted to take maths major, but didn't have a solid justification, now I got a lot of insight. Really thanks a lot!!

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  @theshinydunky7745 Жыл бұрын

  I just wanted to draw a 4d shape

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  @mching4473 Жыл бұрын

  Love this video, esp how you keep the properties at the bottom. Feels like a power up for understanding @code_report's APL solution videos.

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  @olekkatan6614 Жыл бұрын

  24:32 there is an odd number of parentheses

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  @samytamim2603 Жыл бұрын

  honestly i feel dumb

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  @academyofuselessideas7 ай бұрын

  @samytamim2603 @samytamim2603 awww... that happens with some math explanations... in my experience, it is best not to think to much about the math that makes you feel dumb and instead focus on the math that you enjoy and understand... even professional mathematicians have no idea of what other mathematicians are doing in a different field... Combinators can be fun but they are somehow esoteric anyways... just do what you find fun!

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  @samytamim26037 ай бұрын

  @@academyofuselessideas I actually got the Ahaaa moments, then got shocked knowing I was missing out these stuff. It was a bit sarcastic xD

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  @academyofuselessideas7 ай бұрын

  @@samytamim2603 now I feel dumb 🙃

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  @samytamim26037 ай бұрын

  @@academyofuselessideas 😆

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  @hansulrichkeller6651 Жыл бұрын

  Never thougt that there were so many interesting points in looking at S(n,k): I learned a lot! Thank you, Alexander!

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  @xactxx7 ай бұрын

  You're welcome!

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  @samernoureddine Жыл бұрын

  I am not sure I follow why you can write any program using S and K. Why is this true?

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  @lukewatson88487 ай бұрын

  It is because other combinators can be expressed in terms of S and K alone, for example: I = SKK (1) B = S(KS)K C = S(BBS)KK where C can be written using (1) as C = S((S(KS)K)(S(KS)K)S)KK and so on...

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  @r75shell Жыл бұрын

  At least first 14 minutes doesn't tell how to verify that stuff works. For example 1:51 in video says KIxy reduces to y. Can I verify it? Or is it just by definition? Why it's not K(Ix)y -> Kxy -> x? At time 3:00 "we have written program" what it does? "This is how combinators logic avoids variables altogether". We write x, y everywhere, isn't it variables? "Our program is simply string of combinators" Well, isn't Ix reduces into x? Isn't program "x" then? Summary: in my opinion this video is very.... very confusing.

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  @mching4473 Жыл бұрын

  Not an expert in this either but it's a mix of both. The first few stuff tells us how notations work. Kinda like the PEMDAS of combinatory logic. So if you have KIxy, it is a signal for you to evaluate it as K(I, x) then y K(I, x) -> I. Then I(y) since I takes in one argument.

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  @padawanrl88347 ай бұрын

  Agree. I don't get how KIxy reduces to Iy. It would help if precedence/order is shown. My initial understanding was: KIxy K(Ix, y) K(x, y) y But apparently it should be Iy?

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  @MrRyanroberson1 Жыл бұрын

  the real challenge: can you make a proper combinator (a combinator which could be written as a function of N arguments that returns a pure juxtaposition of those arguments) that can compute everything? i'm pretty sure it's impossible. It has been proven that there exists a proper combinator X such that X and I=SKK together can compute everything, but one truly universal proper combinator has yet to be found.

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  @antoniolewis1016 Жыл бұрын

  It would have helped a lot if you had parentheses to see which things the K and the I were acting on. It's very confusing.

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  @academyofuselessideas7 ай бұрын

  @antoniolewis1016 Same opinion here... it would have been nice if they had mentioned that combinators associate by default to the left, so KIxy is shortcut for ((KI)x)y

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  @freddyfozzyfilms2688 Жыл бұрын

  left assoc

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  @caselbravo Жыл бұрын

  Fantastic work thanks to everyone involved ! 🕊️

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  @klevisimeri607 Жыл бұрын

  We love it!

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  @sabriath Жыл бұрын

  That bonus bit reminds me of whitespace programming language

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  @user-rj7mg6yu7h Жыл бұрын

  Prof., I like your textbook 《Functional analysis》, the best I've ever seen.

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  @geraldpatrick5552 Жыл бұрын

  😭 【ｐｒｏｍｏｓｍ】

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  @skilz8098 Жыл бұрын

  Next up, an FFT and its Inverse Algorithm using only Combinatory Logic followed by The Wave Collapse Function.

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  @academyofuselessideas7 ай бұрын

  hahaha... this brings an important point that is often left out in these type of results... many things are turing complete but computer languages are not only about producing computations but about making it easy for humans to express those computations... combiators are interesting for what they say about the nature of formal languages but they are probably not going to replace any real world programming language any time soon (though functional programming is valid and useful but even pure programming language go beyond implementing combinators)

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  @malikwaddy6897 Жыл бұрын

  The first 2 mins don't make sense at all

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  @buraianmath11 ай бұрын

  same..

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  @Asterism_Desmos Жыл бұрын

  I am your 250th subscriber

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  @theunknown4834 Жыл бұрын

  Is there any way to discuss this? (discord, reddit etc.)

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  @alexanderfarrugia9299 Жыл бұрын

  Are you in the SoMe2 Discord? We could discuss there.

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  @theunknown4834 Жыл бұрын

  ​@@alexanderfarrugia9299 Yes, will be trying to contact you soon

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  @alexanderfarrugia9299 Жыл бұрын

  @@theunknown4834 I am farrugiamaths there. Message me even if I'm offline and I'll get back to you when I am. :)

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  @theunknown4834 Жыл бұрын

  @@alexanderfarrugia9299 Already did :>

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  @theunknown4834 Жыл бұрын

  @@alexanderfarrugia9299 I finished my implementation and uploaded it on GitHub and sent it to you :)

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  @user-tk2jy8xr8b Жыл бұрын

  Unfortunate naming for C params, btw, better use Cfxy -> fyx, otherwise the second arg is expected to be used as a function (f, g, and h are usual names for functions, whereas x, y, and z - for values (I know functions are values, it's about their intended usage, not the type)), which is not the case

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  @xactxx Жыл бұрын

  g could be a function though. Nothing wrong with being so.

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  @user-tk2jy8xr8b Жыл бұрын

  @@xactxx everything is a function in combinator logic. The naming is about the intended usage

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  @xactxx Жыл бұрын

  @@user-tk2jy8xr8b and what is the intended usage?

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  @user-tk2jy8xr8b Жыл бұрын

  @@xactxx to be applied to anything or not on the right-hand side of the definition

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  @xactxx Жыл бұрын

  @@user-tk2jy8xr8b fair enough. The reason why we used fgx is for consistency with the variables used for B and S. Essentially, B and C are the two halves of S, and using the same variables for S, B and C drives home this point.

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  @theunknown4834 Жыл бұрын

  How do you represent decimals in this form?

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  @xactxx Жыл бұрын

  One would first need to represent integers (for example, as a pair of naturals with appropriately defined operations), then move on to define rationals, as pairs of integers. Then reals could be defined using infinite lists. This video did not define lists (we didn't need them for our Fibonacci program) but lists may also be defined in terms of combinatory logic.

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  @theunknown4834 Жыл бұрын

  @@xactxx This topic is scary lol, still won't stop me from writing it in python

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  @xactxx Жыл бұрын

  @@theunknown4834 I have a working implementation of the constructs in this video. It works, though very slowly, as is expected. But it works.

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  @bartvanpelt588 Жыл бұрын

  @@theunknown4834 writing this in python seems like a nice challenge, if you want a language that can evaluate these combinators with less of a hassle (and builtin lazy evaluation!) I would highly recommend Haskell.

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  @lexinwonderland5741 Жыл бұрын

  yeah, this went WAY over my head, but it gets an upvote anyway cuz it's awesome and obviously brilliant. great job on the video!

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  @xactxx Жыл бұрын

  Thank you!

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  @user-tx4wj7qk4t3 ай бұрын

  That's because it's explained in the most complicated way possible. If he just wrote the lambda calculus notation it would be really simple

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  @theunknown4834 Жыл бұрын

  this video is pretty good, however, I do not think the software used here is helping in understanding as it scrambles all the lines

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  @floydnelson92 Жыл бұрын

  We invent the rules and discover the consequences. But, I think it is discoverable that there are elemental rules and concepts which are more convenient and which provide symmetry such that an alien in a different universe with different sensory information and a different thought process could discover and likely would discover those rules and concepts. Philosophically, I would think any sufficiently large universe, such as ours, would have pattern because it's too improbable that it all be random. As such, there is an underlying math which is discoverable.

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  @synaestheziac Жыл бұрын

  0:52 did I see a subliminal Hilbert in there?

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  @quentincarver926 Жыл бұрын

  might've been cut content. Hilbert proof systems have a correspondence with combinatory logic (curry-howard correspondance)

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  @johnchessant3012 Жыл бұрын

  interesting

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  @alexanderfarrugia9299 Жыл бұрын

  Thank you for your encouraging comment!

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  @NoNTr1v1aL Жыл бұрын

  Absolutely amazing video! Subscribed.

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  @xactxx Жыл бұрын

  Thank you very much for your encouraging comment!