Thank you for watching! I recently hit 10K subscribers and planning a Q&A video. Head over to the Another Roof subreddit to ask your questions. If I get enough questions, I'll make the video -- should be a fun, less scripted one. www.reddit.com/r/anotherroof/comments/wj8hhn/10k_subscriber_qa/ While I'm here, let me respond to some of the common questions related to this video: 1. Couldn't you use [this definition] of ordered pairs? Yes -- the one I use in the video is attributed to Hausdorff is a little non-standard. The Kuratowski definition (a,b):={{a},{a,b}} is more widely used and accepted. I try to make these videos accessible to as many people as possible, and I think the Hausdorff "tagging" description is the most accessible. That's just my opinion -- I make a lot of decisions like this for pedagogical reasons. When teaching a concept for the first time, it's not always best to take the most efficient approach. 2a. "If A is a set, I can always find some element not contained in A" -- what if A contains everything? A fantastic question and I glossed over this because I didn't want to get too bogged down. But the axiom of regularity has a consequence which says that all sets cannot contain themselves. So for any set A, there exists at least one element not contained in A, namely A itself! 2b. So couldn't you use A itself as the "tag" instead of 1 -- yes, a great idea! 3. Why can't we define a function which maps one element to two different things? A "function" is a thing invented by mathematicians to serve a purpose, and it turns out it's just more useful in serving that purpose when it's defined such that no element can map to two different things. Try removing that stipulation and see what sort of theory develops! 4. There exists a bijection from 'n' to 'k' implies n = k. How do we know? Well spotted. Well done to those who called me out on this -- this is a statement which requires proof, and I did record an explanation of this but I felt it was getting too bogged down, especially for viewers unfamiliar with the material. Another hand-wavey pedagogical choice!
Really excited about this channel and where it's going. This video and your previous one are such a good starting point for people interested in "proper math"
I appreciate that. My friends and I often bemoan the lack of decent texts acting as a bridge between school and "proper maths." These videos will never be a substitute for a formal education, but I hope they provide a springboard for the curious
@@AnotherRoof I think the situation is not quite so bleak. I don't having any specific books in mind, but simply googling for sources on set theory, elementary number theory, discrete math, real analysis, etc. give enumerable results suitable for any level of sophistication. I think having a video version to accompany that material should help the self-study crowd even more!
@@theflaggeddragon9472 While it's true material exists, I'd argue it's not the most discoverable. Would you know how to find resources on set theory et al if you didn't know they existed?
@@SleepyHarryZzz This I fully agree with. Many people think math is "over" at calculus. In the US (where I'm from), I get the feeling that the curriculum is designed to stifle people's imaginations and creativity and make people hate math. Having channels like this one should be really helpful exposing people to "what's next" in math. Sort of a side issue, but I hear from people a lot about how to improve the curriculum is to make it more applied. I half agree (having people learn math though games, programming, project-based problem solving, etc. could be great), but the applications we see in our books are extremely dry, boring, and unrealistic. The classes I had in high school that were by far the most engaging for most students were the one-off lessons where we would do some fun problem-solving or puzzle that the students would spend the class trying to solve. This is exactly the kind of collaborative and imaginative experience that mathematics is about and hardly anyone knows about it. It's really quite sad to see how hollow and mechanical the standard US math curriculum is.
Very, very ironically - I think this is somehow a good idea. It might take slightly longer to get the idea of counting to sink in if you start from first principles, but you will greatly decrease the time it takes to get from third to fourth principles and from the fourth layer of bricks to the fifth, it will go even faster. By the time your kid is 7 or 8, they will be equal to their peers but with a far better foundation to accelerate from and avoid the trap a lot of students in late high school get to, where they are speeding along in math but find out the foundation is made of sand and they have to go back to basics to shore up the foundation underneath the mental edifice of math they already understand to put up a new floor. By the same token, if I had taken my "learning how to learn and memorise effectively" courses seriously in primary school I wouldn't have stalled out in the early years of university when my natural ability to remember stuff suddenly became overwhelmed with the volume of information and I had no good studying skills to fall back on because talent had always managed to substitute up to that point.
@@blumoogle2901 an interesting idea but probably won't work in practice. It'd be like teaching someone to code by starting with 1s and 0s. Without some amount of abstraction it'd be basically impossible to get off the ground.
@@blumoogle2901 I'm fairly certain "New Math" is what you are looking for. It didn't succeed, and it didn't go as far as you seem to want to go. Working from the bottom up is exactly what makes learning hard - instead, abstraction should let teaching go from easy material to hard material, or top down. Starting from the bottom and developing things from scratch is better suited for a technical class where you design operating systems or motivate a single massive theorem. As for your courses in primary school, it's unfortunate that such a mature topic was taught to children, expecting they would care. You can't really directly motivate children to "study better" by telling them how; they won't see the point in paying attention. You have to push them in and out of the classroom to exercise creativity and engage their curiosity on their own. Unfortunately, much of our current curriculum is rote memorization, so different approaches (for each subject) must be taken to make kids interested in what they are studying. Children who were pushed to learn an instrument come to resent it if pushed too far - but children who are made to love music tend to love it all their lives. For math, if we are teaching very basic ideas, it's important that children should see the immediate importance of what they are learning - for math doesn't get interesting/beautiful until much, much later on.
I'm an elementary school teacher, so I'm always thinking about how to unpack things that are intuitive to me but new to my students such that I can teach them clearly and effectively. I won't exactly be sharing this video with any elementary school students learning how to count, but I can absolutely appreciate how easy you make it look to get complex ideas across clearly and engagingly. Kudos!
perhaps if you have a student on the autism spectrum that is having difficulty with it, you could show them the videos to show that there are clear rules to it, and these rules make no assumptions.
@@JakubS I don't understand the assumption that a person on the spectrum would have trouble with the fact that there are assumptions you make when simplifying. (And I'm about in the middle on the spectrum, so don't go with the "But you're not autistic" argument. I've seen that happen before, and it doesn't go well for the person making the argument.)
@@nikkiofthevalley (as someone who is also on the spectrum) they didn't say someone with autism will always have a difficulty with it, they just said that it could be useful if a student with autism questions it. Given that we don't aways think the way neurotypical people do it is not uncommon for us to be confused about how or why to follow something that seems arbitrary to us, and the video could be helpful in showing why those rules work as they do. I don't believe they had any intention of saying someone with autism "couldn't understand" it or anything of the sorts, in fact the doubt would come from a desire of wanting to understand it more instead of just taking someone's word for it.
I love the mathematical bricks. Building knowledge one brick at a time. Also opens the door for some good jokes about getting hit by hard truths, where those truths are things like the brick of fermat's last theorem
I am slowly writing a set of teacher reference (in French) and in each, I have dependents and dependencies. When I will have a few hundred of them, I will collect this data and make a dependency graph. This will be a useful tool, because during the year, a teacher could easily take another path to accommodate to pupils. It will also be something great to introduce students to the material they will see and why in that specific order. When I will be teacher, I'll probably "waste" some hours explaining the usefulness and structure of mathematics, because it shows its beauty! These videos will probably be a good inspiration to make the classes more dynamic. Students are just hyped by physical props, because it's synonymous to a more lighthearted curriculum! (It also helps kinetic students)
This kind of "step by step proving something we took for granted" is something I absolutely LOVE about math. Would be great to see this turned into a series, where you prove even more basic mathematics from the ground up!
I know this channel is new but it's amazing to see such a small channel with great production quality, clear explanations, and a bit of humour and personality. Really excited to see where you go.
For the curius ones, a good way of encoding ordered pairs is (x,y) = {x, {x, y}} The set {x, {x, y}} contains the pair {x, y} and the first element of the ordered pair (the element x). In this way you don't have to worry about which labels you should choose.
Two things: (1) I love the wall-building visual because that's literally what you're doing: Starting with the very fundamentals (What is a number? What is counting?) to build a solid mathematical understanding from the ground up. As a high school math teacher, I see constantly that my students struggle with complex math almost entirely because they don't understand the fundamentals, and that we can't explain it to them properly because WE don't fully understand it either. This is beautiful content and I look forward to much more of it. (2) I also love that you're using inexpensive, everyday objects for your visuals. Many teachers depend too much on technology or kits or fancy purchased manipulatives (all of which are FANTASTIC in their place!!) and struggle to teach without them. It's ALSO important for us to be able to communicate concepts to students using materials that are familiar and accessible to them so the focus is on the concept and not on the fancy toy. I love to see it!
I love the idea that this episode builds on the previous one. It's not just a discussion about some mathematical topic, but an extention of the previous one and that's honestly something I don't see often on TH-cam. It would be awesome to see another part of this journey!
@@AnotherRoof Each video being another brick in the "wall". We just have to remember that although some people like to use a wall to divide, it was originally used to protect. And later a wall was also used to support a roof. That roof, is it supposed to just arbitrarily separate or to simply shelter? In the same way, we have axioms that support proofs. All of this shows why the people that maintain that wall and roof (teachers, etc) are so important since they can so easily corrupt its intended purpose.
Wouldn’t an efficient “tag” for 1 and 2 be A and B itself so we would have {{c,A},{c,B}} since by regularity our elements a,b,c can’t be A themselves. I know we aren’t using regularity in this video but in this sense it helps us use a ‘best’ tag. Still in middle of the video but loving it so far!
@@AnotherRoof The don’t worry about it approach works too! I have a degree in theoretical mathematics so this topic is close to home for me, but definitely for newcomers to this the don’t worry about it approach works best. Love your videos, looking forward to what’s next!😊
Then you have the issue of what if A=B, and you lose ordering again. The way I was taught ordered pairs was like this (x,y) = {x,{x,y}} That way the order is set because when you see an ordered pair which is {A,B}, either A is an element of B, or B is an element of A, and that tells you which comes first
@@Double-Negative That’s a great point, but for the purpose of just comparing sizes we wouldn’t be concerned with A=B because then trivially they are the same size since they are the same set. But yes technically you are correct and that uses the rigorous definition of an n-tuple, in this case where n=2.
@@Happy_Abe Does regularity imply that while A = { A } is not a legal set, A = {{A}} **is** a legal set? Otherwise, I don't see how tagging a set with itself doesn't cause problems.
I had such a hard time in foundations courses in college because so many things "just worked" already. My intuition, especially when it's correct, is a huge crutch. Your examples and counter examples are so reasonable in a way I never heard in school and I love the building blocks as a constant visual reference.
I really like how you never take a step without explaining why it's justified, while at the same time avoiding getting caught in the weeds. You strike a great balance of being thorough, concise, and entertaining throughout. Also, I'm commenting in hopes that the almighty algorithm will shine on you once more. Your channel has some serious potential and I hope more people see what you're making here because it's truly great.
@@AnotherRoof can't wait to see more from this channel, love the videos and style in which they present these complex subjects. Hope to see math being built from the ground up.
Another amazing video! I love seeing things about ‘foundational’ math, it forces you to confront your intuition, then rewards you at the end by rigorously justifying it. Very satisfying! And speaking of satisfying, that moment at 27:25 was great. The tiles sliding out, demonstrating the actual, real, no-analogy-it-really-is equality between counting and the actual number… so exciting!
Man, your videos are fascinating! As someone who loves math but doesn't really know where to go to learn more, this is an amazing resource. Thank you for making this
I love the highly abstract and low-level bits of math and the visualisations you use. In university I had a series of lectures leading to the definition of a function and when I understood it, I was amazed at how beautiful and general the definition of something seemingly so obvious and basic is.
What a channel man. This dude is such a perfectionist in a good way. I am pretty sure that he has optimized the quality of math lessons, great job, I have no words
I've been studying mathematics and computation casually for a while and every problem I found (others have found) in our usage of computation, writing or using or distributing or understanding software (universal algorithms), for instance, all eventually leads to the knowledge in your videos. Regardless of which angle. All roads lead to how to count I guess. And your videos do a wonderful job of highlighting these concepts from their own merit.
Nice explanation! The way I think about disproving the |Z| > |2Z| thing is doing it in reverse. Make a function Z -> 2Z where we multiply every integer by 4, thus hitting just "half" the elements in 2Z. By the same reasoning this would mean |Z| < |2Z|. Clearly |Z| > |2Z| and |Z| < |2Z| can't both be true, and these functions show instead that |Z| >= |2Z| and |Z|
I'm in my 70's and I enjoy watching math videos of subjects I have studied, or perhaps should have. One thing that came to mind, and I see you have listed a correction, is sets. I recall counting numbers being defined as follows: 0 = {}, 1 = {0}, 2 = {1} ... I don't recall if this is an important difference from what you present, or was demonstrating something I have forgot.
I didn't get maths at school but I like it now (from a distance). I watch other channels but they always go beyond my understanding at some point. Your last video and this one are absolutely brilliant - really accessible, but thorough. I've watched them a few times to get them straight in my head. Fantastic work.
I’m a high school math teacher & I’m learning so much from your videos. I’m a big fan of rigorously scrutinizing our first principles. A request, could you make a video on regularity principle & it’s connections to Gödel’s two theorems?
Me too, glad you're finding them useful! I definitely want to discuss Gödel's theorems, although there have been some good videos on them already, so if I can find a unique approach it'll be something I do in future. Enjoy the summer holidays!
You are so damn good at this! This is what the internet was made for: the most talented and likeable teacher for the world to learn from. Can't wait for the next one.
Great video, just like the previous one! Happy new French subscriber here. At 15:52, the nitpicker in me would say that you need to prove that there is a universal rule to find those tags and identify which signifies "first" and which signifies "second" :P It would seem reasonable that you could make one, so fair enough, but I haven't actually ever seen that concretely written down. The construction I usually take is to define (a, b) := { a, {a, b} }. This is good enough to prove the usual characterization of pairs, i.e. (a, b) = (c, d) if and only if [a = c and b = d]. But as I mentioned this is mere nitpicking, the content is obviously solid and the explanation is crystal clear. Excited for the next videos!
I feel you on the not wanting to make erronious assumptions even as a kid. I remember when we learned addition but it was 3 rows high instead of two (5+3+8=). Other kids had no problem, but for me it was TOTALLY different! Since much of math is presentation, I think it is legitimate to consider a change in presentation a change in the rules.
Yeah, going from how to add a + b to how to add a + b + c is a real step, which involves what's known as the associative property (that "(a+b)+c" is the same as "a+(b+c)" so can be written as "a+b+c" without needing any additional rules added). And then if you're involving numbers with multiple digits, you're also relying on the commutative property (that "a+b" and "b+a" are the same thing, so between the two, you can add things in any order). Going to three rows high works with addition, but not with subtraction, and gets very messy with multiplication...
I had to pause this at several points to digest what was happening, and finally getting that “aha!” was truly satisfying. I appreciate you making this so easy to understand and logical. I’m self studying Mathematics, and have found it confusing at some points; This video has helped me to finally understand some of the concepts that eluded me. Thank you!!
These videos are so fantastic. I feel like I learn so much and yet for a while, as the video goes on I feel like I know less and less. Everything I thought I knew gets gradually stripped away. Then it gets rebuilt in a new way that is much more interesting and stronger than before.
It took me a few beats to get the Sir Jection joke 😂💀 it's a testament to your teaching skill that you guided my intuition to the correct conclusions so many times. Your videos are excellent!!
Imagine my surprise when I realized I couldn't count. Can't wait to start teaching the kids set theory in order to help them understand how to count. Thanks! :)
Your two videos are beautifull, as a mathmatitian myself I love how you are explaining this and could be use for people who are studing math or are just curious. I wish I had this when I was taking number theory. I hope you can make a video about the Axion of Choice, and you can continue to construct the Whole Numbers, Rational and Irrational Numbers. Great work.
It's neat seeing how many assumptions are inherent in even very basic exercises like counting and comparing the sizes are two very small sets. It makes me really appreciate how amazing our brains are that they can automatically recognize all the reasonable assumptions to make without our conscious minds even being aware of it. It saves so much time and effort compared to if we actually had go through all this logic all the time -- imagine having to take all the steps he went through in this video every time you had to count something! Of course, this often causes trouble, since our subconscious minds don't always make the _correct_ assumptions, and sometimes make assumptions where _no_ assumptions should be made, or generalize things too far, such as automatically applying stereotypes to individuals without even realizing it. And intuition can certainly lead us astray sometimes, like he talked about in the prior video. But even so, while our brains our far from perfect and our pattern recognized skills sometimes take things too far and cause problems, our brains are still pretty darn amazing.
Excellent video, just like the last! Very glad your prior video found its way into my recommendations (also, the "hELp" puzzle was amusing). Just wanted to point out a problem with your encoding of a pair (a, b) as { {a, 1}, {b, 2} }: What if either _a_ or _b_ happens to be equal to 1 or 2? Then the resulting encoding is degenerate. The encoding (a, b) := { a, {a, b} } does not have any such problems, and is where I thought you would be going in this video. *EDIT:* Of course you mentioned this like 10 seconds after the point at which I paused to write this comment!
This is not what I expected from a video called "How to count". I'm so glad I watched it though, because I've been studying lambda calculus for awhile now and this filled in a few gaps on the pure math side of things, e.g., why is zero indexing in a for loop correct. I had an argument on the CS side but no equivalent argument on the pure math side, but you've provided it! It's because to count to five, one has to enumerate the elements starting from zero, not one.
20:53 « If you wanna take a break ». I actually just decided to take a quick break at that very moment. This man is not just a very educational mathematician, he is actually a wizard
That's really cool and you explained it perfectly, i love this kind of stuff. One question: With ordered paires, the pair (2,1) would be {{2,1},{1,2}}={{1,2}} and that's weird. I once read a definition saying (a,b)={{a},{a,b}}, or something like that
The ordered pair (2, 1) would not be {{2, 1}, {1, 2}}. The definition you provided does not support your conclusion. The only weird aspect of the definition (a, b) = {{a}, {a, b}} occurs when a = b, but even this is not actually very weird: while you do get {{a}} in that case, this is not an issue, because a is never equal to {a}.
@@angelmendez-rivera351 The video offers a different definition of ordered pairs (due to Hausdorff, if Wikipedia is to be trusted), in which (2,1) would, indeed, be {{1,2}} (and (1,2) would be {{1},{2}} ) but only because the video handwaves past the part of Hausdorff's definition where "1" and "2" are arbitrary things not in either set.
Nice video as always! But did anyone noticed the letters in the background when he was mentioning cake (it was hard not to be distracted by the cake 🍰). I have no idea how to dishpifer it though.... Here is what I got: "ACROSTIC"
so I started watching this video, then you convinced me to watch the previous. then about half way thru this video I said to myself "this was a lot of rigorous math, I need a small break" so I took a break. then I come back, resume the video and its the part where you say "you should take a break here" that was perfectly timed
11:23 well, not always. There's no OTHER way to choose the pairs in comparing {c} vs {c} which distinguishes it from {c} vs the empty set. :-B 14:15 huh, i think i'd rather keep to the way i was taught: get the order within the construction: {a, {a,b}} ... since your approach gets a little messy when a = 2 and b = 1. Or when a = 1, and/or when b = 2. Although the former way can get messy too, when a=b or when a is a set. Grr, defining math concepts from scratch is indeed hard. ^^
Went through all those thoughts in writing and landed on the Hausdorff {{a,1},{b,2}}, the one I felt was most accessible! All are messy in certain scenarios :P
@@AnotherRoof What if you used the sets A and B as labels? using {{a,A}, {b,B}} you cannot have a = A or b=B because of regularity, and you also cannot have a=B and b=A, also because of regularity, right?
@@NAMEhzj I can't see anything wrong with that (the set A can't contain A becuase we've defined it such that A means the set A, always. So if it did contain A that would mean it contained itself which means it's not a proper set.)
"I can't teach you how to matter because... you don't matter." Words cannot describe how much I love this type of humor. That isn't even the best joke in the video either...
33:31 As drawn on the brick, we call that a "frog diagram" for pupils in primary school. For example, if the need to multiply by 6, they know they can decompose "x6" in "x2x3". So, instead of doing "x6", they do the simpler two step operation.
And BTW, it quite ironical that a lot of pupils did function decomposition that way and ZERO teachers I saw leverage it when explaining function (de)composition ^^ Another one I liked to explain when tutoring is how one can draw a quadratic by rewriting it as a(x-b)²+c as b is the abscissa, c the ordinate and a is the vertical stretching factor. Indeed, it's the composition of: f(x)=x-b g(x)=x² h(x)=ax k(x)=x+c q(x)=k∘h∘g∘f=k(h(g(f(x)))) (skipped i and j as those letters usually denote complex and quaternions)
I absolutely love, love, LOOOOOOOOOOOOOOOOOVE this channel. Manim is fantastic, and I love the animations people make with along with of course 3b1b's videos, but your unique physical style is so playful. Your humor is charming, and you really embrace the exploratory and fun of entering higher math. Great content, and I'm so happy you make these videos!
Love the channel! I am hoping you will slowly and rigorously build maths from the axioms up only to finally throw Godel at us and give us some sweet Lovecraftian existential dread.
The bricks remaining in frame, representing the accumulated knowledge, are really good. It's like keeping frequently used tools on the workbench instead of on the shelf.
I really like this series of videos, just to add that the best way to tag the element of A is probably to use {a,A}, and the fact that A can't be an element of itself!
The clarity of these explanations is fantastic - I just stumbled into this channel and now I'm learning set theory, which is something I never thought I'd say!
Woah, this is such an underrated channel! As a fellow educational TH-camr, I understand how much work must go into this- amazing job!! Liked and subscribed :)
One thing I learned from French counting (and Rewboss In Germany). They have individual names for 11-16 and then they say ten seven, ten eight and ten nine! English count the other way round (three ten, four ten,...).
These two videos (What is 3 and this one) are some of the best representations of explanatory logical thinking I've ever seen. You've got a good sense of humor, and it adds a pleasant spice, but you're not shrieking comedy into your videos in a forced way. You're not only knowledgable and passionate about the subject material, you've also come up with a great, simple way to express all of your ideas. There's an element of what you're doing that reminds me of a magic show. And I think math should be somewhat like that, building and building until the cord is pulled and you see what you've been building this whole time and it sharpens into focus and you find yourself laughing at the a-ha moment. It's very good, very powerfully simple stuff. Well done on both counts, if you keep going like this you're going to build an impressive channel. I watch videos on physics and maths for fun, but as much as I love them it's easy for the eyes to glaze over at certain points, and that literally never happened once with your videos. I dunno just top marks, man, well done.
This kind of thing explains why so many seemingly 'obvious' off-by-one or just bad in general counting bugs in computer programming aren't exactly so simple.
I learned so much from this 2 part series! Definitely something I don't want to learn in a school setting (It'll be hell), but perfect as an edutainment video like this.
Fantastic stuff. Loved the little sleight of hand bits mixed in, added to the entertainment without detracting from the lesson. Thanks for the refresher!
As a french, i agree that our counting system is strange but it tell about how we count before. We have different words from 1 to 16 because of hexa decimal base And 4 20 (80) is an inheritance of base 20 In the same way, English have different words from 1 to 12 : base 12 Anyway i'll finish the video thank you for your work
Most languages have distinct words for 11 and 12, owing to 12 being a nice base to work in. English has words like "dozen" and "score" (meaning 20, as in Abraham Lincoln's famous "four-score and seven years ago"; "quatre-vingt-sept ans dernier"), it's just funny that French has soixante-dix and quatre-vingt as holdovers, where I don't think any other language has retained such terminology. What's funnier is that despite the use of quatre-vingt, you still have just "cent" for 100 (as opposed to something like "cinq-vingt"), and everything from there is base 10 as in other languages.
Slight question. How do you rigorously define the idea that a function must map each element *once*? At this point we have yet to define counting, so we cant say for instance "The subset of tuples with A as their first element has size 1"
This is a fantastic question. Because I want to keep my videos tight, I'll invariably have to gloss over certain things but I'll do my best to answer here. The ZF axioms are built upon a foundation of logic where we can enquire whether elements belong in a set. We don't need to use the concept of "size 1" if we do it this way. A function F from set A to set B is a set of pairs such that: (i) If (a,b) is in F, then a belongs to A and b belongs to B. (So the pairs are structured as 'from A' and 'to B'). (ii) For all a in A, there exists b in B such that (a,b) is in F. (So every element of A is assigned a partner in B). (iii) For all (a,b) and (a,c) in F, then b = c. (So any pair starting with a only has one partner in B but nowhere have we used "one"). Hope this helps!
cool! this very much feels analogous to the proof used to make sure counting is well defined, saying that if you can count N and count K then N=K. this kind of proof comes up a lot.
First of all, amazing channel, just found this stuff, love it, hope you keep doing more math fundamentals because it is a fascinating subject. Also this topic makes me think of that one part in the Wayside School series where a teacher is trying to tell him he doesn't know how to count, so, to prove her wrong, he counts to 3. "A thousand, a million, three." "That's wrong." "But teacher, he got to the right number!"
It's so nice to have a educational video that focuses on the core mechanisms of the subject before anything else. A flowing ground up build with simple props that neatly represent nothing more and nothing less than they need. Clear description of the tools available and their relevance and the limits and variations of the functions. Most the videos I see do a quick outline of the principal or logic bullet-points before focusing on the jazzy conclusions. To me logic requires presentable proofs otherwise it's just clever or overcooked conjecture -which can be very entertaining but don't provide solid answers let alone an introduction to usable tools that I can use and prove through related applications in my own self learning. I look forward to your future videos
This is sooo goood!! 😱🤩 I can't believe you just started this channel! Even though I don't generally subscribe to a lot of channels I subscribed to support your channel growing!
WOW, this was amazing, I wish I saw this when I was younger. The props you use in the problem make it more real (like an engineering or computer problem). Very excited to see how far you can take this concept. Best way i've seen this explained by far.
To avoid some ordered pair related problems, I would suggest instead using set of {{a,b},a} to encode pair of a and b (b is second here). It's a bit less obvious that this one would work, but it avoids using any arbitrary elements, which may be in conflict with ones we pair. The singleton problem can be solved by checking the pairing with opposite order, but I won't go into further detail on that one, go figure...
I'm so pleased to have found your channel so early. Your content is great and leverages things I've already learned to introduce new concepts. Love it! Very clearly explained. Keep going!
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Thank you for watching! I recently hit 10K subscribers and planning a Q&A video. Head over to the Another Roof subreddit to ask your questions. If I get enough questions, I'll make the video -- should be a fun, less scripted one. www.reddit.com/r/anotherroof/comments/wj8hhn/10k_subscriber_qa/
While I'm here, let me respond to some of the common questions related to this video:
1. Couldn't you use [this definition] of ordered pairs? Yes -- the one I use in the video is attributed to Hausdorff is a little non-standard. The Kuratowski definition (a,b):={{a},{a,b}} is more widely used and accepted. I try to make these videos accessible to as many people as possible, and I think the Hausdorff "tagging" description is the most accessible. That's just my opinion -- I make a lot of decisions like this for pedagogical reasons. When teaching a concept for the first time, it's not always best to take the most efficient approach.
2a. "If A is a set, I can always find some element not contained in A" -- what if A contains everything? A fantastic question and I glossed over this because I didn't want to get too bogged down. But the axiom of regularity has a consequence which says that all sets cannot contain themselves. So for any set A, there exists at least one element not contained in A, namely A itself!
2b. So couldn't you use A itself as the "tag" instead of 1 -- yes, a great idea!
3. Why can't we define a function which maps one element to two different things? A "function" is a thing invented by mathematicians to serve a purpose, and it turns out it's just more useful in serving that purpose when it's defined such that no element can map to two different things. Try removing that stipulation and see what sort of theory develops!
4. There exists a bijection from 'n' to 'k' implies n = k. How do we know? Well spotted. Well done to those who called me out on this -- this is a statement which requires proof, and I did record an explanation of this but I felt it was getting too bogged down, especially for viewers unfamiliar with the material. Another hand-wavey pedagogical choice!
Really excited about this channel and where it's going. This video and your previous one are such a good starting point for people interested in "proper math"
I appreciate that. My friends and I often bemoan the lack of decent texts acting as a bridge between school and "proper maths." These videos will never be a substitute for a formal education, but I hope they provide a springboard for the curious
@@AnotherRoof I think the situation is not quite so bleak. I don't having any specific books in mind, but simply googling for sources on set theory, elementary number theory, discrete math, real analysis, etc. give enumerable results suitable for any level of sophistication. I think having a video version to accompany that material should help the self-study crowd even more!
@@theflaggeddragon9472 While it's true material exists, I'd argue it's not the most discoverable. Would you know how to find resources on set theory et al if you didn't know they existed?
@@SleepyHarryZzz ... or even which words to search for.
@@SleepyHarryZzz This I fully agree with. Many people think math is "over" at calculus. In the US (where I'm from), I get the feeling that the curriculum is designed to stifle people's imaginations and creativity and make people hate math. Having channels like this one should be really helpful exposing people to "what's next" in math.
Sort of a side issue, but I hear from people a lot about how to improve the curriculum is to make it more applied. I half agree (having people learn math though games, programming, project-based problem solving, etc. could be great), but the applications we see in our books are extremely dry, boring, and unrealistic. The classes I had in high school that were by far the most engaging for most students were the one-off lessons where we would do some fun problem-solving or puzzle that the students would spend the class trying to solve. This is exactly the kind of collaborative and imaginative experience that mathematics is about and hardly anyone knows about it. It's really quite sad to see how hollow and mechanical the standard US math curriculum is.
Currently teaching my 1 year-old how to count. So far we've got to 3, but I guess we should scrap that progress and restart at Extensionality!
😂😂😂
Very, very ironically - I think this is somehow a good idea. It might take slightly longer to get the idea of counting to sink in if you start from first principles, but you will greatly decrease the time it takes to get from third to fourth principles and from the fourth layer of bricks to the fifth, it will go even faster. By the time your kid is 7 or 8, they will be equal to their peers but with a far better foundation to accelerate from and avoid the trap a lot of students in late high school get to, where they are speeding along in math but find out the foundation is made of sand and they have to go back to basics to shore up the foundation underneath the mental edifice of math they already understand to put up a new floor.
By the same token, if I had taken my "learning how to learn and memorise effectively" courses seriously in primary school I wouldn't have stalled out in the early years of university when my natural ability to remember stuff suddenly became overwhelmed with the volume of information and I had no good studying skills to fall back on because talent had always managed to substitute up to that point.
@@blumoogle2901 an interesting idea but probably won't work in practice. It'd be like teaching someone to code by starting with 1s and 0s. Without some amount of abstraction it'd be basically impossible to get off the ground.
@@blumoogle2901 I'm fairly certain "New Math" is what you are looking for. It didn't succeed, and it didn't go as far as you seem to want to go. Working from the bottom up is exactly what makes learning hard - instead, abstraction should let teaching go from easy material to hard material, or top down. Starting from the bottom and developing things from scratch is better suited for a technical class where you design operating systems or motivate a single massive theorem.
As for your courses in primary school, it's unfortunate that such a mature topic was taught to children, expecting they would care. You can't really directly motivate children to "study better" by telling them how; they won't see the point in paying attention. You have to push them in and out of the classroom to exercise creativity and engage their curiosity on their own. Unfortunately, much of our current curriculum is rote memorization, so different approaches (for each subject) must be taken to make kids interested in what they are studying.
Children who were pushed to learn an instrument come to resent it if pushed too far - but children who are made to love music tend to love it all their lives. For math, if we are teaching very basic ideas, it's important that children should see the immediate importance of what they are learning - for math doesn't get interesting/beautiful until much, much later on.
Wait until they successfully count to five, then show the start of this video so they can say "yeah, nailed it!"
I'm an elementary school teacher, so I'm always thinking about how to unpack things that are intuitive to me but new to my students such that I can teach them clearly and effectively. I won't exactly be sharing this video with any elementary school students learning how to count, but I can absolutely appreciate how easy you make it look to get complex ideas across clearly and engagingly. Kudos!
I teach secondary math, and seeing an elementary teacher excited about a very mathy video gives me great joy. Thank you for this!
Y'all are wholesome as heck ^.^
perhaps if you have a student on the autism spectrum that is having difficulty with it, you could show them the videos to show that there are clear rules to it, and these rules make no assumptions.
@@JakubS I don't understand the assumption that a person on the spectrum would have trouble with the fact that there are assumptions you make when simplifying.
(And I'm about in the middle on the spectrum, so don't go with the "But you're not autistic" argument. I've seen that happen before, and it doesn't go well for the person making the argument.)
@@nikkiofthevalley (as someone who is also on the spectrum) they didn't say someone with autism will always have a difficulty with it, they just said that it could be useful if a student with autism questions it.
Given that we don't aways think the way neurotypical people do it is not uncommon for us to be confused about how or why to follow something that seems arbitrary to us, and the video could be helpful in showing why those rules work as they do.
I don't believe they had any intention of saying someone with autism "couldn't understand" it or anything of the sorts, in fact the doubt would come from a desire of wanting to understand it more instead of just taking someone's word for it.
I love the mathematical bricks. Building knowledge one brick at a time. Also opens the door for some good jokes about getting hit by hard truths, where those truths are things like the brick of fermat's last theorem
The physical props are top notch indeed!
I am slowly writing a set of teacher reference (in French) and in each, I have dependents and dependencies.
When I will have a few hundred of them, I will collect this data and make a dependency graph.
This will be a useful tool, because during the year, a teacher could easily take another path to accommodate to pupils.
It will also be something great to introduce students to the material they will see and why in that specific order. When I will be teacher, I'll probably "waste" some hours explaining the usefulness and structure of mathematics, because it shows its beauty!
These videos will probably be a good inspiration to make the classes more dynamic. Students are just hyped by physical props, because it's synonymous to a more lighthearted curriculum! (It also helps kinetic students)
I'd also love to see an oriented graph showing which brick necessitates witch other to be proven.
@@scialomy In 3 years, for lower secondary school if all goes well ^^
@@programaths Vos élèves vont comprendre les vids en anglais?
This kind of "step by step proving something we took for granted" is something I absolutely LOVE about math. Would be great to see this turned into a series, where you prove even more basic mathematics from the ground up!
You'll pleased to know that this *is* a series! Currently scripting part 3 so stay tuned for more!
@@AnotherRoof HELL YEAH!
I know this channel is new but it's amazing to see such a small channel with great production quality, clear explanations, and a bit of humour and personality. Really excited to see where you go.
100% agree with this
I didn't even realize it was a new channel when I watched the first video. It's destined to get way bigger.
For the curius ones, a good way of encoding ordered pairs is
(x,y) = {x, {x, y}}
The set {x, {x, y}} contains the pair {x, y} and the first element of the ordered pair (the element x).
In this way you don't have to worry about which labels you should choose.
Two things:
(1) I love the wall-building visual because that's literally what you're doing: Starting with the very fundamentals (What is a number? What is counting?) to build a solid mathematical understanding from the ground up. As a high school math teacher, I see constantly that my students struggle with complex math almost entirely because they don't understand the fundamentals, and that we can't explain it to them properly because WE don't fully understand it either. This is beautiful content and I look forward to much more of it.
(2) I also love that you're using inexpensive, everyday objects for your visuals. Many teachers depend too much on technology or kits or fancy purchased manipulatives (all of which are FANTASTIC in their place!!) and struggle to teach without them. It's ALSO important for us to be able to communicate concepts to students using materials that are familiar and accessible to them so the focus is on the concept and not on the fancy toy. I love to see it!
disliked. you didn't finish telling everyone about me. i was knighted in 1705 for discovering what we now know as sir jections
It's him!
is your name alfred with an a because a is greater than b?
I love the idea that this episode builds on the previous one. It's not just a discussion about some mathematical topic, but an extention of the previous one and that's honestly something I don't see often on TH-cam.
It would be awesome to see another part of this journey!
The journey doesn't end here -- currently writing my next video so stay tuned to see the next continuation!
@@AnotherRoof Each video being another brick in the "wall". We just have to remember that although some people like to use a wall to divide, it was originally used to protect.
And later a wall was also used to support a roof.
That roof, is it supposed to just arbitrarily separate or to simply shelter?
In the same way, we have axioms that support proofs.
All of this shows why the people that maintain that wall and roof (teachers, etc) are so important since they can so easily corrupt its intended purpose.
Love this type of stuff. Math is so pedantic and thats why its great.
Please do more videos how we build math from the ground up.
I can't believe the quality of your videos. One would think you'd been making videos for years and years. This is incredible.
Wouldn’t an efficient “tag” for 1 and 2 be A and B itself so we would have {{c,A},{c,B}} since by regularity our elements a,b,c can’t be A themselves. I know we aren’t using regularity in this video but in this sense it helps us use a ‘best’ tag.
Still in middle of the video but loving it so far!
Yes, absolutely, and I did consider this for the exact reasons you gave, but I preferred the "don't worry about it" approach in the end!
@@AnotherRoof The don’t worry about it approach works too!
I have a degree in theoretical mathematics so this topic is close to home for me, but definitely for newcomers to this the don’t worry about it approach works best.
Love your videos, looking forward to what’s next!😊
Then you have the issue of what if A=B, and you lose ordering again.
The way I was taught ordered pairs was like this
(x,y) = {x,{x,y}}
That way the order is set because when you see an ordered pair which is {A,B}, either A is an element of B, or B is an element of A, and that tells you which comes first
@@Double-Negative That’s a great point, but for the purpose of just comparing sizes we wouldn’t be concerned with A=B because then trivially they are the same size since they are the same set. But yes technically you are correct and that uses the rigorous definition of an n-tuple, in this case where n=2.
@@Happy_Abe Does regularity imply that while A = { A } is not a legal set, A = {{A}} **is** a legal set? Otherwise, I don't see how tagging a set with itself doesn't cause problems.
I had such a hard time in foundations courses in college because so many things "just worked" already. My intuition, especially when it's correct, is a huge crutch. Your examples and counter examples are so reasonable in a way I never heard in school and I love the building blocks as a constant visual reference.
I really like how you never take a step without explaining why it's justified, while at the same time avoiding getting caught in the weeds. You strike a great balance of being thorough, concise, and entertaining throughout. Also, I'm commenting in hopes that the almighty algorithm will shine on you once more. Your channel has some serious potential and I hope more people see what you're making here because it's truly great.
Except 34:30 is not justified.
Is this channel going to build mathematics from the ground up? I LOVE IT
Subscribe to find out! I have a few more planned which build from here!
@@AnotherRoof can't wait to see more from this channel, love the videos and style in which they present these complex subjects. Hope to see math being built from the ground up.
As a programmer, the first few minutes were like living my life. Explicit instructions must be given
Another amazing video! I love seeing things about ‘foundational’ math, it forces you to confront your intuition, then rewards you at the end by rigorously justifying it. Very satisfying!
And speaking of satisfying, that moment at 27:25 was great. The tiles sliding out, demonstrating the actual, real, no-analogy-it-really-is equality between counting and the actual number… so exciting!
Can't tell you how many takes that took. Going to post a blooper of it to my Patrons later this week!
This is the dopest mathematical channel of the last 3 years.
Man, your videos are fascinating! As someone who loves math but doesn't really know where to go to learn more, this is an amazing resource. Thank you for making this
I lost it when he brought out the building blocks, literally. This is brilliant. Couldn't have enough of it.
You've like immediately became one of my favorite channels, looking forward to your future work!
I love the highly abstract and low-level bits of math and the visualisations you use. In university I had a series of lectures leading to the definition of a function and when I understood it, I was amazed at how beautiful and general the definition of something seemingly so obvious and basic is.
What a channel man. This dude is such a perfectionist in a good way. I am pretty sure that he has optimized the quality of math lessons, great job, I have no words
I've been studying mathematics and computation casually for a while and every problem I found (others have found) in our usage of computation, writing or using or distributing or understanding software (universal algorithms), for instance, all eventually leads to the knowledge in your videos. Regardless of which angle. All roads lead to how to count I guess. And your videos do a wonderful job of highlighting these concepts from their own merit.
Nice explanation! The way I think about disproving the |Z| > |2Z| thing is doing it in reverse. Make a function Z -> 2Z where we multiply every integer by 4, thus hitting just "half" the elements in 2Z. By the same reasoning this would mean |Z| < |2Z|. Clearly |Z| > |2Z| and |Z| < |2Z| can't both be true, and these functions show instead that |Z| >= |2Z| and |Z|
I really enjoy the conversational tone and tactile nature of your videos. I especially love the bricks in this one. Looking forward to more!
TH-cam algorythm has blessed me today. Looking forward to all the future videos!
The production value of these is remarkably high for someone who has made as few videos as you. Looking forward to the next one!
Another video this fast n just as long?? I'm going to love this channel
I'm in my 70's and I enjoy watching math videos of subjects I have studied, or perhaps should have. One thing that came to mind, and I see you have listed a correction, is sets. I recall counting numbers being defined as follows: 0 = {}, 1 = {0}, 2 = {1} ... I don't recall if this is an important difference from what you present, or was demonstrating something I have forgot.
I didn't get maths at school but I like it now (from a distance). I watch other channels but they always go beyond my understanding at some point. Your last video and this one are absolutely brilliant - really accessible, but thorough. I've watched them a few times to get them straight in my head. Fantastic work.
I’m a high school math teacher & I’m learning so much from your videos. I’m a big fan of rigorously scrutinizing our first principles.
A request, could you make a video on regularity principle & it’s connections to Gödel’s two theorems?
Me too, glad you're finding them useful! I definitely want to discuss Gödel's theorems, although there have been some good videos on them already, so if I can find a unique approach it'll be something I do in future. Enjoy the summer holidays!
You are so damn good at this! This is what the internet was made for: the most talented and likeable teacher for the world to learn from.
Can't wait for the next one.
Watching your videos takes me back to watching Open University broadcasts on BBC2 in the 1980s
As a back-end developer with a strong SQL background I now understand the intricacies of the theory behind my job better, THANK YOU!
It is obvious :D
Great video, just like the previous one! Happy new French subscriber here.
At 15:52, the nitpicker in me would say that you need to prove that there is a universal rule to find those tags and identify which signifies "first" and which signifies "second" :P It would seem reasonable that you could make one, so fair enough, but I haven't actually ever seen that concretely written down. The construction I usually take is to define (a, b) := { a, {a, b} }. This is good enough to prove the usual characterization of pairs, i.e. (a, b) = (c, d) if and only if [a = c and b = d].
But as I mentioned this is mere nitpicking, the content is obviously solid and the explanation is crystal clear. Excited for the next videos!
I feel you on the not wanting to make erronious assumptions even as a kid. I remember when we learned addition but it was 3 rows high instead of two (5+3+8=). Other kids had no problem, but for me it was TOTALLY different! Since much of math is presentation, I think it is legitimate to consider a change in presentation a change in the rules.
Yeah, going from how to add a + b to how to add a + b + c is a real step, which involves what's known as the associative property (that "(a+b)+c" is the same as "a+(b+c)" so can be written as "a+b+c" without needing any additional rules added). And then if you're involving numbers with multiple digits, you're also relying on the commutative property (that "a+b" and "b+a" are the same thing, so between the two, you can add things in any order).
Going to three rows high works with addition, but not with subtraction, and gets very messy with multiplication...
I had to pause this at several points to digest what was happening, and finally getting that “aha!” was truly satisfying. I appreciate you making this so easy to understand and logical. I’m self studying Mathematics, and have found it confusing at some points; This video has helped me to finally understand some of the concepts that eluded me. Thank you!!
Big fan of the various examples of how natural candidates for definitions fail. Keep this up! Excited to see how far you go into set theory.
These foundational math topics are so fascinating, and I love your delivery and humor. Looking forward to the next one!
I’m in top set at my school. I never thought a video on how to count would confuse me.
Underrated channel, your videos are so exciting.
These videos are so fantastic. I feel like I learn so much and yet for a while, as the video goes on I feel like I know less and less. Everything I thought I knew gets gradually stripped away. Then it gets rebuilt in a new way that is much more interesting and stronger than before.
It took me a few beats to get the Sir Jection joke 😂💀 it's a testament to your teaching skill that you guided my intuition to the correct conclusions so many times. Your videos are excellent!!
I think I see where this channel is headed, and I love it! New favourite ❤
Imagine my surprise when I realized I couldn't count. Can't wait to start teaching the kids set theory in order to help them understand how to count. Thanks! :)
Your two videos are beautifull, as a mathmatitian myself I love how you are explaining this and could be use for people who are studing math or are just curious. I wish I had this when I was taking number theory.
I hope you can make a video about the Axion of Choice, and you can continue to construct the Whole Numbers, Rational and Irrational Numbers.
Great work.
It's neat seeing how many assumptions are inherent in even very basic exercises like counting and comparing the sizes are two very small sets. It makes me really appreciate how amazing our brains are that they can automatically recognize all the reasonable assumptions to make without our conscious minds even being aware of it. It saves so much time and effort compared to if we actually had go through all this logic all the time -- imagine having to take all the steps he went through in this video every time you had to count something! Of course, this often causes trouble, since our subconscious minds don't always make the _correct_ assumptions, and sometimes make assumptions where _no_ assumptions should be made, or generalize things too far, such as automatically applying stereotypes to individuals without even realizing it. And intuition can certainly lead us astray sometimes, like he talked about in the prior video. But even so, while our brains our far from perfect and our pattern recognized skills sometimes take things too far and cause problems, our brains are still pretty darn amazing.
Excellent video, just like the last! Very glad your prior video found its way into my recommendations (also, the "hELp" puzzle was amusing). Just wanted to point out a problem with your encoding of a pair (a, b) as { {a, 1}, {b, 2} }: What if either _a_ or _b_ happens to be equal to 1 or 2? Then the resulting encoding is degenerate. The encoding (a, b) := { a, {a, b} } does not have any such problems, and is where I thought you would be going in this video.
*EDIT:* Of course you mentioned this like 10 seconds after the point at which I paused to write this comment!
See 15:52 :)
This is not what I expected from a video called "How to count". I'm so glad I watched it though, because I've been studying lambda calculus for awhile now and this filled in a few gaps on the pure math side of things, e.g., why is zero indexing in a for loop correct. I had an argument on the CS side but no equivalent argument on the pure math side, but you've provided it! It's because to count to five, one has to enumerate the elements starting from zero, not one.
20:53 « If you wanna take a break ». I actually just decided to take a quick break at that very moment. This man is not just a very educational mathematician, he is actually a wizard
Ah yes, my favorite song. It's just another brick in the body of mathematical knowledge by Pink Function
That's really cool and you explained it perfectly, i love this kind of stuff.
One question: With ordered paires, the pair (2,1) would be {{2,1},{1,2}}={{1,2}} and that's weird. I once read a definition saying (a,b)={{a},{a,b}}, or something like that
The ordered pair (2, 1) would not be {{2, 1}, {1, 2}}. The definition you provided does not support your conclusion. The only weird aspect of the definition (a, b) = {{a}, {a, b}} occurs when a = b, but even this is not actually very weird: while you do get {{a}} in that case, this is not an issue, because a is never equal to {a}.
@@angelmendez-rivera351 The video offers a different definition of ordered pairs (due to Hausdorff, if Wikipedia is to be trusted), in which (2,1) would, indeed, be {{1,2}} (and (1,2) would be {{1},{2}} ) but only because the video handwaves past the part of Hausdorff's definition where "1" and "2" are arbitrary things not in either set.
@@rmsgrey Fair enough. I was just confused because I did not realize OP was talking about the video's definition, but the other one.
Nice video as always! But did anyone noticed the letters in the background when he was mentioning cake (it was hard not to be distracted by the cake 🍰). I have no idea how to dishpifer it though.... Here is what I got: "ACROSTIC"
so I started watching this video, then you convinced me to watch the previous. then about half way thru this video I said to myself "this was a lot of rigorous math, I need a small break" so I took a break. then I come back, resume the video and its the part where you say "you should take a break here"
that was perfectly timed
11:23 well, not always. There's no OTHER way to choose the pairs in comparing {c} vs {c} which distinguishes it from {c} vs the empty set. :-B
14:15 huh, i think i'd rather keep to the way i was taught: get the order within the construction: {a, {a,b}} ... since your approach gets a little messy when a = 2 and b = 1. Or when a = 1, and/or when b = 2. Although the former way can get messy too, when a=b or when a is a set. Grr, defining math concepts from scratch is indeed hard. ^^
Went through all those thoughts in writing and landed on the Hausdorff {{a,1},{b,2}}, the one I felt was most accessible! All are messy in certain scenarios :P
@@AnotherRoof What if you used the sets A and B as labels? using {{a,A}, {b,B}} you cannot have a = A or b=B because of regularity, and you also cannot have a=B and b=A, also because of regularity, right?
@@NAMEhzj I can't see anything wrong with that (the set A can't contain A becuase we've defined it such that A means the set A, always. So if it did contain A that would mean it contained itself which means it's not a proper set.)
The former way does not get messy even if a = b, since a = {a} is false for all sets a.
"I can't teach you how to matter because... you don't matter."
Words cannot describe how much I love this type of humor.
That isn't even the best joke in the video either...
33:31 As drawn on the brick, we call that a "frog diagram" for pupils in primary school.
For example, if the need to multiply by 6, they know they can decompose "x6" in "x2x3".
So, instead of doing "x6", they do the simpler two step operation.
And BTW, it quite ironical that a lot of pupils did function decomposition that way and ZERO teachers I saw leverage it when explaining function (de)composition ^^
Another one I liked to explain when tutoring is how one can draw a quadratic by rewriting it as a(x-b)²+c as b is the abscissa, c the ordinate and a is the vertical stretching factor.
Indeed, it's the composition of:
f(x)=x-b
g(x)=x²
h(x)=ax
k(x)=x+c
q(x)=k∘h∘g∘f=k(h(g(f(x))))
(skipped i and j as those letters usually denote complex and quaternions)
I absolutely love, love, LOOOOOOOOOOOOOOOOOVE this channel. Manim is fantastic, and I love the animations people make with along with of course 3b1b's videos, but your unique physical style is so playful. Your humor is charming, and you really embrace the exploratory and fun of entering higher math. Great content, and I'm so happy you make these videos!
Love the channel! I am hoping you will slowly and rigorously build maths from the axioms up only to finally throw Godel at us and give us some sweet Lovecraftian existential dread.
Love this video, I'm trash at math and I understood this so easily.
The bricks remaining in frame, representing the accumulated knowledge, are really good. It's like keeping frequently used tools on the workbench instead of on the shelf.
@@lforlight they've become a staple of my channel!
I really like this series of videos, just to add that the best way to tag the element of A is probably to use {a,A}, and the fact that A can't be an element of itself!
I love your props; first time in pushing 2 decades that I feel like I actually get wtf set theory is.
This is what we learned in class 11 but definitely not this way. It was an awesome experience. And you made a masterpiece imo.
The clarity of these explanations is fantastic - I just stumbled into this channel and now I'm learning set theory, which is something I never thought I'd say!
You deserve more recognition, you're redoing all math only using the axioms
I already learned this in university, but you're such a good teacher I was still entertained!
Woah, this is such an underrated channel! As a fellow educational TH-camr, I understand how much work must go into this- amazing job!! Liked and subscribed :)
Glad I stumbled on this new channel. It's people like you that make math fun again.
One thing I learned from French counting (and Rewboss In Germany).
They have individual names for 11-16 and then they say ten seven, ten eight and ten nine! English count the other way round (three ten, four ten,...).
These two videos (What is 3 and this one) are some of the best representations of explanatory logical thinking I've ever seen. You've got a good sense of humor, and it adds a pleasant spice, but you're not shrieking comedy into your videos in a forced way. You're not only knowledgable and passionate about the subject material, you've also come up with a great, simple way to express all of your ideas. There's an element of what you're doing that reminds me of a magic show. And I think math should be somewhat like that, building and building until the cord is pulled and you see what you've been building this whole time and it sharpens into focus and you find yourself laughing at the a-ha moment. It's very good, very powerfully simple stuff. Well done on both counts, if you keep going like this you're going to build an impressive channel. I watch videos on physics and maths for fun, but as much as I love them it's easy for the eyes to glaze over at certain points, and that literally never happened once with your videos. I dunno just top marks, man, well done.
19:03 when you said you call it a map I thought you were serious because I program in Java a lot so it made sense lmao
This kind of thing explains why so many seemingly 'obvious' off-by-one or just bad in general counting bugs in computer programming aren't exactly so simple.
There are two problems in computer science. We have one joke and it isn't funny.
Yes. Yeeeeessss. YASSSSSSS! MORE!
Thanks bud! Super stoked for this channel.
Thanks for this, it actually got me from my depression 😅
Absolutely love this. I've always wanted to understand maths from the axioms but assumed it was too complicated. You explain it so clearly!
I learned so much from this 2 part series! Definitely something I don't want to learn in a school setting (It'll be hell), but perfect as an edutainment video like this.
Currently doing discrete maths at uni and this video was useful thanks!
Fantastic stuff. Loved the little sleight of hand bits mixed in, added to the entertainment without detracting from the lesson. Thanks for the refresher!
I really liked the Brinks. Allowed me to visually remind myself of what you where talking about.
As a french, i agree that our counting system is strange but it tell about how we count before.
We have different words from 1 to 16 because of hexa decimal base
And 4 20 (80) is an inheritance of base 20
In the same way, English have different words from 1 to 12 : base 12
Anyway i'll finish the video thank you for your work
Most languages have distinct words for 11 and 12, owing to 12 being a nice base to work in. English has words like "dozen" and "score" (meaning 20, as in Abraham Lincoln's famous "four-score and seven years ago"; "quatre-vingt-sept ans dernier"), it's just funny that French has soixante-dix and quatre-vingt as holdovers, where I don't think any other language has retained such terminology. What's funnier is that despite the use of quatre-vingt, you still have just "cent" for 100 (as opposed to something like "cinq-vingt"), and everything from there is base 10 as in other languages.
Slight question. How do you rigorously define the idea that a function must map each element *once*? At this point we have yet to define counting, so we cant say for instance "The subset of tuples with A as their first element has size 1"
If f maps x to y, and f maps x to z, then y = z. This is the rigorous definition.
This is a fantastic question. Because I want to keep my videos tight, I'll invariably have to gloss over certain things but I'll do my best to answer here. The ZF axioms are built upon a foundation of logic where we can enquire whether elements belong in a set. We don't need to use the concept of "size 1" if we do it this way. A function F from set A to set B is a set of pairs such that:
(i) If (a,b) is in F, then a belongs to A and b belongs to B. (So the pairs are structured as 'from A' and 'to B').
(ii) For all a in A, there exists b in B such that (a,b) is in F. (So every element of A is assigned a partner in B).
(iii) For all (a,b) and (a,c) in F, then b = c. (So any pair starting with a only has one partner in B but nowhere have we used "one").
Hope this helps!
cool! this very much feels analogous to the proof used to make sure counting is well defined, saying that if you can count N and count K then N=K. this kind of proof comes up a lot.
First of all, amazing channel, just found this stuff, love it, hope you keep doing more math fundamentals because it is a fascinating subject.
Also this topic makes me think of that one part in the Wayside School series where a teacher is trying to tell him he doesn't know how to count, so, to prove her wrong, he counts to 3. "A thousand, a million, three." "That's wrong." "But teacher, he got to the right number!"
Your explanations are really compelling. The props and gentle humour are really engaging.
Vsauce: how to count past infinity
Combo class: how to count in fractional and irrational bases
Another roof: how to count
This is the most intuitive explanation of Sets, Relations and Functions I've seen!
It's so nice to have a educational video that focuses on the core mechanisms of the subject before anything else. A flowing ground up build with simple props that neatly represent nothing more and nothing less than they need.
Clear description of the tools available and their relevance and the limits and variations of the functions.
Most the videos I see do a quick outline of the principal or logic bullet-points before focusing on the jazzy conclusions.
To me logic requires presentable proofs otherwise it's just clever or overcooked conjecture -which can be very entertaining but don't provide solid answers let alone an introduction to usable tools that I can use and prove through related applications in my own self learning.
I look forward to your future videos
This is sooo goood!! 😱🤩 I can't believe you just started this channel! Even though I don't generally subscribe to a lot of channels I subscribed to support your channel growing!
WOW, this was amazing, I wish I saw this when I was younger. The props you use in the problem make it more real (like an engineering or computer problem). Very excited to see how far you can take this concept. Best way i've seen this explained by far.
This video very clear and focused! No noise, like in Wikipedia about the subject! Thanks!
To avoid some ordered pair related problems, I would suggest instead using set of {{a,b},a} to encode pair of a and b (b is second here). It's a bit less obvious that this one would work, but it avoids using any arbitrary elements, which may be in conflict with ones we pair. The singleton problem can be solved by checking the pairing with opposite order, but I won't go into further detail on that one, go figure...
I can't wait for your channel to rack up more and more random callbacks and inside jokes. I better be seeing those pencils everywhere.
Great video
Great video. For the first time, I feel like I have an intuitive grasp on where the different sized infinites come from.
I'm so pleased to have found your channel so early. Your content is great and leverages things I've already learned to introduce new concepts. Love it! Very clearly explained. Keep going!
Amazing video, this explained all of the assumptions taken in my college algebra class that I didn't even think to question
Great videos! May the youtube algorithm be in your favor!
I hope your students appreciate how well you explain things. I sure do!