The Mathematician's Weapon (Old Version)

Рет қаралды 301,319

Күн бұрын

PLEASE NOTE: You can find the updated version of this video here: • The Mathematician's We...
A gentle introduction to the study of category theory and abstract algebra, done from the ground-up by exploring the mathematical weapon of abstraction.
This video aims to give an overview of the fundamental tool at the forefront of pure mathematical research: abstraction. By seeing this tool in various contexts, and how it allows us to encounter category theory in a natural and intuitive manner, we'll see how beautiful the abstract can really be.
Note: Sorry for the woeful audio levels and patchy editing, this was my first shot at making a video! I've kept the video up for nostalgia purposes but it should be noted that I have reuploaded the video (link above) with some fixes.
― Timestamps ―
0:00 - Intro
1:49 - Abstraction and Algebra
7:30 - Examples of Abstraction
8:49 - Set Theory
13:25 - Category Theory
21:16 - Outro
― Credits ―
All animation and voiceover created by Eyesomorphic.
Background music: 'Abstraction', composed by Caleb Peppiatt.
― Further Reading ―
Category Theory for Programmers: The Preface, by Bartosz Milewski: bartoszmilewski.com/2014/10/2...
Category Theory for Computing Science, by Michael Barr & Charles Wells (Book)

Пікірлер: 621

@WayOfTheCode 9 ай бұрын

Man this is KZfaq gold. You gotta become a regular. I would be so happy to grab i was in the first 60 views.

@Eyesomorphic 9 ай бұрын

Thanks! I'll definitely try to make some more videos, I've already got an idea for another one. Glad you enjoyed :D

@ultimaxkom8728 9 ай бұрын

@@Eyesomorphic Graph Theory is the abstraction of relationships? **Put down glasses** I've studied to PhD all for this and yet I'm still a single dog. The graph, does not work!

@100c0c 9 ай бұрын

@@ultimaxkom8728 clever 😂

@RigoVids 6 ай бұрын

Brag*?

@MarceloUmeda 9 ай бұрын

Welcome to KZfaq! Fantastic presentation. Absolutely looking forward for a long and prosperous playlist, Eyesmorphic. Thank you for existing!

@Eyesomorphic 9 ай бұрын

That means a lot! Thanks

@encapsulatio 9 ай бұрын

@@Eyesomorphic Please make a course where you teach Type theory but do it by simultaneously teaching ALL the prerequisites and building blocks that we need to know from logic and proof theory. I would be so happy paying for a course if you can make it just as intuitive and visual as this video.

@Gabcikovo 9 ай бұрын

@@Eyesomorphichave a nice day and be wise ☕️🍰🍸

@Gabcikovo 9 ай бұрын

@@Eyesomorphicintuition is hugging you deeply, warmly and passionately

@Colonies_Dev 8 ай бұрын

tbh i thought my real weapon was a combination of logical reasoning and math theorems.. i used a lot of statistics and logic to check how likely a statement is to be true or not given for example, if the person claims that 10 rare conditions were simultanously true, that's probably bullshit. but if it was a smaller set of conditions, or there is proof that some of them are true, the situation changes. or for example, using advanced counting and probabilities to win card games and impress ppl

@VIue_ 9 ай бұрын

Category theory for programmers is an amazing book, and I absolutely reccomend watching the lectures too!

@jacobsjdin4057 9 ай бұрын

Could you elaborate on why it’s an amazing book? I am considering bying

@Eyesomorphic 9 ай бұрын

@@jacobsjdin4057You can access it for free online if you'd prefer, but obviously buying the book and supporting his work is great too. It was a culmination of Bartosz's posts made on the website I listed in the description. He has a lecture series as well that follows the same content

@corneliusgoh 9 ай бұрын

Yes, I was introduced into Category by Bartosz's KZfaq series many years ago, didn't understand all but later from other books/videos then came back to Bartosz whose videos the BEST.

@madlep 9 ай бұрын

This is possibly the best intro category theory video in the category of intro category theory videos

@Treston-ri7of 9 ай бұрын

I have been a haskell programmer for quite a while, shared this video with some friends as I think it makes Category Theory really approachable! Well done!

@user-so5vl9ee6g 9 ай бұрын

Is haskell any good for cryptography?

@stotto75 9 ай бұрын

Brilliant. Clean animations, thoughtful and clear exposition and a great topic. Can’t wait for more videos from Eyesomorphic.

@cstruble2 9 ай бұрын

I can't say it any better, so a great big DITTO!

@TheKivifreak 9 ай бұрын

I’ve watched like 5 introductions to category theory, but your analogy of what algebra is to numbers is what category theory is to composition was the priming I needed. Heck! I learnt algebra before. And I know how I learnt algebra. For me learning algebra involved doing a lot of examples and then creating algorithms for it once I got a good grasp of the subject. So yesterday I started with an exercise heavy book to category theory (spivaks category theory for the sciences) and it’s exactly what I need to understand it. He introduces ologs that are the first real objects I studied seriously that don’t fulfill composition. Thanks a ton on getting me started!

@Eyesomorphic 9 ай бұрын

That's great to hear! It took me at least 10 different books and lecture series to fully get to grips with the principles of category theory, and I guess this video is the culmination of that, and the progression that felt most intuitive to me. Good luck on your quest!

@YuruCampSupermacy 9 ай бұрын

do you have a CS/Software background? if yes can we expect some CS related videos to from you in the future besides pure mathematics? @@Eyesomorphic

@Eyesomorphic 9 ай бұрын

@@YuruCampSupermacy My background is in mathematical foundations for CS, so you can definitely expect some more CS orientated content in the future!

@suicraft8395 9 ай бұрын

This video is a pearl in the middle of the internet ocean. It's fascinating how you explain how some subjects of math are the abstraction of things we already use in Simple math. As a student of computer science i apreciate a lot this work

@Eyesomorphic 9 ай бұрын

Thanks!

@samuelwaller4924 3 ай бұрын

learning about lambda calculus is how the magic of abstraction "clicked" for me. Being able to define problems away is just mind-blowing to me still lol. "This function is a number now because I say it is" is just amazing

@user-yg4br8ut5t 9 ай бұрын

instant subscribe! i got my b.s. in mathematics last year, but real life hasn’t given me much time to keep up with reviewing what i learned in undergrad. this video made me feel like i was in one of my theory classes again! ❤

@Eyesomorphic 9 ай бұрын

That's great to hear!

@jordanzamora422 9 ай бұрын

Instantly subbed, this is one of the most “eye” opening videos on mathematics that I’ve seen so far, you took an “abstract” concept and broke it down in a very digestible way, one could say you used abstraction to minimize the number of elements it takes to convey the underlying structure of “the utility of abstraction within mathematics” lol. Thank you for the video!

@GabriTell 25 күн бұрын

This is not only a great introduction to Category Theory, but also a great introduction to Mathematics in general. 🔥

@fahrenheit2101 9 ай бұрын

Don't let this distract you from the fact that a monad is a monoid in the category of endofunctors.

@vonBottorff 20 күн бұрын

Alas, Haskell Brutalism rears its ugly head again.

@eridisk1826 14 күн бұрын

@@vonBottorffHaskell Brutalism may be ugly but still a beautiful phrase

@NovaH00 9 ай бұрын

When I was 14, my relationship with functions was "it just a name we give to an expression" like we give π to 3.141592..., or e give to 2.7182... When I turned 15, I started to learn programming and it completely change my mind about how functions work. Yes sure it still is a name we give for an expression but the way we can use any kind of input and any kind of output that make me surprised, then I did a little research and I found out that functions in math is also the same, they're freedom, you can do what ever you want; don't just stop at the Natural numbers or Real numbers, start making your own numbers like a set with even numbers, or a set with prime numbers, or a set with your family members and we can make a function on those sets too. That kind of freedom make me love math❤ The way you explain everything reminded me of that feeling, feeling of I can do what I want with math. I hope you can keep up the good work!! And welcome to youtube🎉

@agrajyadav2951 Ай бұрын

We have an absolute sigmachad here 🗿

@riccardoformenti4332 9 ай бұрын

As a math student, absolutely fantastic work!! Would love to see more

@GabriTell 25 күн бұрын

As a first-year Math Degree student, I can that almost all (if not all) Mathematics can be described by the art of Abstraction. For example, if you want to prove "The sum of even numbers is even" you could try to check one by one, but you'd just never end because there're infinity even numbers. Instead, you can consider work with only two objects "A" and "B", which are two representatives of all integers that verify the property of being even (formally, if we consider the universe "Ω" to be set of all intergers: "∃n(A=2n)" and "∃m(B=2m)"). In other words, we've just removed all particular characteristics of individual even integers to stay only with what characterizes an "even integer"; we've made abstraction. Where the magic lies is that if you manage to prove any property with "A" and "B" (such that "A+B" is even), you're proving it for any two even numbers (actually, "killing infinit many birds with one shot"). 🔥

@geoemg7140 9 ай бұрын

I dont regret a single second of watching this video, you completly changed my perception of not only mathematics, but the whole reality. Thank you for this amazing content, you defenitely deserve a sub! 👏

@mertnuhoglu 4 ай бұрын

Unbelievable simplicity. Please keep on making new videos. This is pure gem.

@Eyesomorphic 4 ай бұрын

That's really reassuring! Keep an eye out for the next installment in this series, I took a rather long hiatus but there's a new video in the works currently! Thanks again

@LifeObserver-007 9 ай бұрын

Welcome to KZfaq, Eyesomorphic! Stunningly simple and accessible presentation that couls invite more people into this beautiful area of study. Hope you could go deeper into category theory and present useful results in a similarly accessible fashion. Looking forward to them!

@LarryPanozzo 9 ай бұрын

Great video. A young kid could understand this, and it would likely turn them into mathematicians! The world needs more mathematicians, especially now. Bravo.

@sarveshpadav2881 9 ай бұрын

This was my very first exposure to category theory and it was beautiful!.....Thank you.

@GabrielLima-gh2we 9 ай бұрын

Wow, man, you're really really good at this, your approach to mathematics and your skill to teach in a simple way with many examples is amazing. Please, keep making videos, there's nothing like this on KZfaq.

@personanongrata7976 8 ай бұрын

first comprehensible explanation of rational numbers I've ever heard. I'm 65. May God richly bless you, sir.

@user-cp1dn3ns7k 9 ай бұрын

I can see how thoroughly you planned this video! Both the visualisation and the topic are beautiful!

@max3446 14 күн бұрын

I wish my lecturer just show us this video when I first started taking a class on category theory. I spent way too long trying to understand very abstract concepts without much of an idea as to why I should care - this video would have helped a lot!

@Eyesomorphic 14 күн бұрын

I'm glad to hear it!

@StratosFair 9 ай бұрын

What a beautiful introduction to category theory. As an applied guy who knows nothing about this subject, I enjoyed it a lot ! You got yourself a new subscriber, looking forward to your future videos :)

@mmzzzmeemee 9 ай бұрын

9 minutes in, That I can view 'age' as a function applied to a set 'people' just blew my mind. Insta subscribed

@Eyesomorphic 9 ай бұрын

Glad that it felt intuitive! I spent quite a bit of time thinking of the most natural way to describe sets and functions without using any unnecessary notation, so I'm happy my example 'clicked' for you! Thanks for the subscription :)

@danielyuan9862 9 ай бұрын

A function gives an output from whatever input you defined it on, not just on numbers!

@abdulfatah66 9 ай бұрын

Such an amazing video that explains category theory in a remarkable way. I have been looking for category theory videos but I never found any with this level of articulation and quality. Your hard work to put together explanations and graphics deserve great applause 👏

@bananatassium7009 9 ай бұрын

beautifully reminiscent of 3Blue1Brown imo, looking forward to more! Gave me a really solidly improved understanding of category theory :)

@user-yn9br1uo2q 9 ай бұрын

I hope you're in top tier winners for #SoME3! Brilliant explanation . As a math student, absolutely fantastic work!! Would love to see more.

@Halesnaxlors 9 ай бұрын

Great vid! I can feel something clicked inside my head

@noahmartinez314 9 ай бұрын

Keep making videos man, you are destined to be right up there with numberphile, mathologer, and threeBlueOneBrown

@fedimser 9 ай бұрын

Thank you for amazing video and especially for reference to a course by Bartosz Milewski! I tried to study Category Theory twice before, and gave up both times. Thanks to your video I found Bartosz' lectures (here on KZfaq) on Category Theory, and they are much more accessible than materials I used before - so I'll give it another try. This video was very inspirational for me. Thank you!

@alanpeter2172 9 ай бұрын

Very nice and simple ! Thank you !!! Hopping There will be others videos !!!

@iansnow4698 8 ай бұрын

This kind of math content is definetly worth a channel for!

@wejt5454 9 ай бұрын

such a great video, the best one i’ve seen so far on introducing category theory, i love it so much! also i love the channel profile picture and name

@Eyesomorphic 9 ай бұрын

That's very kind, thanks!

@YuruCampSupermacy 9 ай бұрын

I have never subscribed and turned on notifications for a channel so fast.

@minimath5882 9 ай бұрын

I categorize this video as cool

@Eyesomorphic 9 ай бұрын

Greatly appreciated!

@MrBebopbob 9 ай бұрын

Beautifully simple overview of a difficult area of mathematics. I'm looking forward to a deeper dive in future videos.

@darrellee8194 9 ай бұрын

Apparently this is your very first video (on this channel). Obviously the algorithm knows what I like. The channel name is awesome. Isomorphism explains everything. Looking forward to the N+1 video.

@Nebula_ya 9 ай бұрын

I REALLY like your style, visuals are so perfect for following on

@ItAintNecessarilySo 9 ай бұрын

Thank you for creating this! I've watched several intro to category theory videos and even started some books and courses, but this is by far the best explanation I've seen!

@symeonmandrinos9528 9 ай бұрын

A video with an agile structure and a lean data posture. A job well done!

@harsh.p0 9 ай бұрын

I hope you're in top tier winners for #SoME3! Brilliant explanation 👏🏼

@sig1678 9 ай бұрын

Nice Video! I am glad to see a new great KZfaqr joining the platform and teaching concepts. Thank you for this piece of art.

@david_junior 5 ай бұрын

What a nice channel, I know this is a gold mine. Please post more.

@aladdinovich 5 ай бұрын

unbelievable simple clear explanation of abstraction and category theory fundamentals. I really thank you for your effort, I would absolutely like to see more of these please. Thank you again

@Eyesomorphic 5 ай бұрын

How very kind, do stay tuned for more videos :D

@Benja-px2xi 9 ай бұрын

Please make more videos! Im so lost learning abstract math and this was such a nice way around it

@johnstuder847 9 ай бұрын

Beautiful simple, clear explanation! Thank you for abstracting math education to its essence. I believe a ‘top down’ approach to math is as important as building a bottom up foundation - it helps to build motivation, and provides a framework to hang the whole of math. Would love to see your take on linear algebra, calculus, geometry, etc, and how they are different aspects of the same concepts.

@nelson6814 9 ай бұрын

Spanish: Este video es una maravilla, en serio, poder explicar algo tan "abstracto" de una forma tan digerible es todo un arte. FELICITACIONES!!

@Eyesomorphic 9 ай бұрын

¡Eso significa mucho! Gracias :)

@RogasTV 9 ай бұрын

Thank you for the great video! Explanations are very clear and easy to follow. I wasn’t lost for a second. Great job! This video is similar to my favourite book about the subject: Conceptual Mathematics by Lawvere and Schanuel. Despite great respect I have towards Bartosz Millewski’s work, I would recommend the book over his lectures.

@enriquechaix2069 9 ай бұрын

Excellent video!. The explanation of the concept of abstraction was fantastic. Thank you so much!

@abrarhameem8424 9 ай бұрын

Welcome to youtube!! Really appreciate your dedication since day one.💖

@IDK_OR_DO_I 9 ай бұрын

This video is so well made, the best I've seen in a long while on pure maths. Amazing work!

@viks3864 9 ай бұрын

Hey, this video is seriously incredible. I remember you commented one of my videos a couple weeks ago and I knew recognised your name. Seriously this video is legions in quality above mine and genuinely congrats and good luck in the future!

@Eyesomorphic 9 ай бұрын

I remember your video well, I thoroughly enjoyed it! Keep up the good work, and thanks for the kind words :D

@Andrew-rc3vh 9 ай бұрын

That was a very nice video, and easy to understand. I would love to see some more on this subject in order to develop the various concepts of this theory.

@simonsalazar9336 7 ай бұрын

Man, I'm still SO EAGER for another video! This video is absolutely beautiful, I can't wait for the next one, and the next ones!

@jimgerth6854 9 ай бұрын

Very clear and beginner friendly, waiting for More :)

@robertvarner9519 8 ай бұрын

The BEST introduction to category theory for a non mathematician like me. Can't wait for more videos by you.

@Eyesomorphic 8 ай бұрын

That's very encouraging, thank you!

@dutonic 9 ай бұрын

That intro was sick af

@alonelyphoenix8942 9 ай бұрын

One of the best math videos ive ever seen, so informative! I am truly thankful for being able to see it

@friender6886 6 ай бұрын

Beautiful presentation of category theory. I enjoyed how Alice and Bob's speech colors were mapped to a (for Alice) and b (for Bob) respectively as a way of abstraction which then allowed me to understand the rest of the video, as though you created a rhetorical category to be an analogy to mathematical categories.

@isaacnewtonstolemyjoy 9 ай бұрын

Two minutes in and I know I'm gonna be watching every video you post :D

@carlos25222 6 ай бұрын

"I, too, aspire to break the boundaries seemingly imposed by the word 'mathematics,' not only with logic but with any kind of thought."

@Ikiguyy 4 ай бұрын

This video has really helped me on a personal level, I've been undergoing changes in my thinking pattern recently, influnced by formal logic, and abstraction was a huge part of it. I've being trying to articulate my process of thinking, but I think this video on abstraction really hits the nail on the head. Thank you.

@aikoaiko2008 9 ай бұрын

Apart from the superb technical content, the music chosen is very relaxing. Thank you for this wonderful creation.

@Eyesomorphic 9 ай бұрын

Thanks! Had a friend compose it :)

@calebp6114 9 ай бұрын

Thanks from the composer! I'm hoping to compose some tracks for future videos as well, so let me know if you have any suggestions.

@bismajoyosumarto1237 Ай бұрын

@@calebp6114Could you please upload it as a separate video or upload it on some music sharing website somehow? Or maybe sell it, I'd buy 🥺

@user-le8jx5mb6j 9 ай бұрын

Beautiful! Thank you for showing that beauty of “abstraction”😊

@DavidGonzalezSamudio 5 ай бұрын

Love it!!!!!! I will link your video as required on every first lecture of my courses. I have used Ring Theory to "obliterate" the Integer type in some programming languages (Java, JavaScript)... An Integer (concept) in a knowledge domain is not isomorphic in another domains Integer (construct) =P. Infinity points to a reference in memory in computers, LOL. infinity +1 is -infinity =)....

@adamcuculich8901 9 ай бұрын

Best category theory introduction I’ve come across thus far!

@YuruCampSupermacy 9 ай бұрын

Loved it. Subscribed. Looking forward to future videos, i can't believe your first video is of such high quality. kudos.

@matrixmash9907 9 ай бұрын

Welcome, so glad to see a new and starting math channel :) I have to say, as an avid set theorist myself I absolutely love your presentation! Your discussion on the identity function was thorough. I'd love to see you do a video on set theory in detail or logical paradoxes (Russels and axiom of choice) or something on large cardinals (recursive functions and calls?) if you choose. Either way though, subscribed.

@mcharles6375 9 ай бұрын

Somehow this founds its way to me …. Of course we want you to make more videos. Excellent job, very cool, well orchestrated :)

@robertpfeiffer4686 9 ай бұрын

Outstanding! I just stumbled across this channel and have become a fan from watching this first vid. Great job and best wishes for a long string of successors!

@craftycurate 9 ай бұрын

Beautifully presented and crystal clear! Looking forward to more.

@monadbornslippy 9 ай бұрын

This video blew my expectations. AWESOME. I hope to see many cool videos like this one from this channel now

@AhirZamanSairi 9 ай бұрын

Minus the background music that causes unnecessary additional mental busyness, top notch production 👌 (your proper diction/pronunciation paired with quality explanation are more than enough on their own)

@shaneshan998 7 ай бұрын

Convinced by this creativity of ‘Abstraction’

@siradmiralbanana 9 ай бұрын

You have a very dry and subtle sense of humor that is perfect for math videos.

@Firqin1986 9 ай бұрын

I really like the visuals! The explanation was succinct as well! Please do continue!

@EvaMariaOprea 9 ай бұрын

Loved it! Looking forward to your upcoming videos!

@rasmusbjersing6860 16 күн бұрын

Gold!👌 Looking forward to following this series!

@alexandertownsend5079 9 ай бұрын

Of all the videos I have seen so far in the SoME3 video contest so far, this has been my favorite. I only have 1 small gripe with it, but its not that a big of a deal to anyone except me and maybe a few other people. My gripe is that the dark text and light background is a bit visually jarring to me. My eyes fatugue easily, so I had to take a few breaks during the video to rest my eyes. If possible could you release a dark mode version of this (black background white text or something similar)? Other than that great video. The explanatioms were well thought out and the concrete examples helped quite a bit.

@bluebon5228 9 ай бұрын

Damm it was worth clicking, nice visual

@wh0an 9 ай бұрын

Dang this was beautifully done! Thanks for the time you put into making such a nice presentation.

@valminyaylo9747 9 ай бұрын

This is amazing, simple and comprehensive way to display explanation of the topic.

@rainbowllamas7423 9 ай бұрын

Loved the “always has been” joke. Great video very well made ❤ I love the more philosophical approach to mathematical concepts. Can’t wait to see whatever you decide to do next

@THash-qs5qg 8 ай бұрын

After watching this video, I'll abstract my comment into one word, "Beautiful"! :)

@Komatic5 9 ай бұрын

Very nice video! I really appreciate how well made everything is here, keep it up ^^

@aptrock327 9 ай бұрын

I love this. It beautifully presents what math is really all about.

@sososo3906 6 ай бұрын

This is the most exciting math related video i have ever seen!

@ashvinjagadeesan5217 9 ай бұрын

What a beautiful video. When something is presented simply enough, it’s truth is revealed in all forms. Your section on rationals and reals got me thinking of kolgomorov complexity in a whole new light. Your section on composing functions got me thinking of computable functions and how one can apply category theory to complexity theory. I imagine that was unintended, but welcome for sure. Cheers to you, this is inspiring work. You’ve won a new subscriber. I hope to see more of your videos in the future.

@alejrandom6592 9 ай бұрын

First time I watch a video about category theory that actually makes sense. Thanks for this ❤

@id10tothe9 9 ай бұрын

Really beautiful i loved it! Thanks a lot, can't wait to learn more! Would love to hear more about category theory if you want to consider it (but i see you provided an awesome list to learn more!). I wish you the best going forward ❤

@arthurgames9610 9 ай бұрын

Good luck with the channel! Has a lot of potential!

@jacobsjdin4057 9 ай бұрын

This is a great video, it has really given me a new perspective on category theory, thanks!

@esquilax5563 9 ай бұрын

Best intro to category theory I've seen, you get an instant new subscription from me!

@matthewrberning 9 ай бұрын

Excellent! Thank you for creating this video, it's a lovely way to express these relationships 🤩

@HIVEEX01 9 ай бұрын

This video is beautifully made Everything from starting to defining what abstraction means in mathematics and it's usefulness within the study of either numbers,orders,relationships and compositions I also admire the effort put into creating such video with very high quality Im looking forward to the next 😊

@TheRattle 8 ай бұрын

Beautifully done. Great content, good pace, lovely animations.

@hazeldekat2376 9 ай бұрын

I really enjoyed the video, it was clear and i liked the examples you used. I really hope you will make more!

@giovannironchi5332 8 ай бұрын

I did my undergrad thesis on adjoint functors and my Master degree thesis on the homotopy theory of differential graded categories. Congratulations for the work and the views!