I find the youngest child to be really impressive, she kept her cool when confronted with unexpected and complex thoughts, and had surprisingly accurate intuitions.
she already could handle more than me… you can have more infinities than infinity? is that like when i told my mom i loved her infinity plus 1 and thought i had her beat? lmao you mean she could’ve said i love you times infinity plus another infinity…? jeez maybe i did love her more lmao
@@dark_sunset I love how people say that, when it is clearly easier to just have an expert tell something to a kid and have them react to it, than writing dialogue that sounds authentically kidlike and getting a childactor to remember it. Also if it were scripted wouldn't you write a kid character who is intuitively wrong more rather than less often?
Heck yeah, it's a super endearing quality you see a lot in mathematicians. I figure you only become a professional math doer if you really, really like doing math.
Every mathematician is like this, almost down to a t in personality. It's just the effect really understanding math has on people, no matter who they might be
You can tell how much a person understands a subject by the way they explain it to people. The fact that she can explain such a complex idea in simpler terms shows how much she understands about the subject.
She communicates complex concepts so clearly with no filler sounds, clearly thinking at top speed the whole time. I'm no fan of mathematics, but somehow I'm feeling an interest through her passion
Absolutely. There is a kind of magic that happens when you hear a person talk about something they are passionate about. You could be totally uninterested in the topic before, but suddenly something sparks when you hear them talk, and you want to know more. It's fascinating and beautiful.
@@AracneMusic we kinda are a good passion detector , don't it? you just know that they put their lifetime into the subject before making the every sentence to make us understand. Definitely beyond fascinating ,i would say.
As a fan of maths, she communicates extremely complex concepts very precisely with no filler sounds. The points she made about the axiom of choice blew my mind. People talk about how counterintuitive it is but the examples she chose are so perfect yet not the natural go-tos that I've seen people mention. I've mainly heard the Banach-Tarski Paradox like the student mentions. Doing induction on the reals is so mind-boggling. It's like drawing infinitely many, infinitely small dots on a page to colour it in. And then the circle thing? Chef's kiss.
the youngest child's "can infinity get bigger" was a surprisingly deep question the brings you into some significantly more advanced topics, i wouldve been completely stumped trying to answer that in a way that would address the question and also make sense to the child
It's the equivalent of asking "can there be something that isn't a part of everything." Infinity isn't a description of a product, it's a description of a process that has no end. To restate: it isn't simply that there are "infinite integers" it is that if you were to try and create the set of integers, that process would not end. And so, any description of infinity needs to include the interval of calculation to make comparisons between the processes (and an injected stopping point). eg, if you were limited to calculating the numbers of "X" category between -10 and 10, 1 per second and had 10 seconds, you might come up with 0-9, or 1.1 1.2 1.3 ... 2.0 or -10, -9.9 ... -9 which we can prove are different % of the infinite sets of integers v. natural numbers v. real numbers. It depends on the process used.
As a mathematician I often tell people that the (countable) infinity is not the biggest you can think of and sometimes they even get angry with me and tell me that if something never ends, there can't be anything bigger. Especially people on social networks quickly get aggressive. I think people should learn the concept of infinity as soon as possible. Maybe at the age of that first girl. I think at that age students would already understand the Cantor trick. At the same time the would learn the concept of proof by contradiction, which may be the most essential tool in maths.
@@skyscraperfan I still don't get how some invites are bigger than others. Is it more so that you can get to a bigger number faster? But the "size" of each set is the same? I saw someone draw a circle centred on a point x, and someone else drew a bugger circle centred on the same point. Then someone claimed that for every point in the small circle, there would be the same point on the big circle (injective). But then they tried to say that the bigger circle had more points and was therefore a bigger infinity. That's just false to me. For every point in the big circle, there is also a point in the small circle. If you couldnt always find another point, it wouldn't be infinity. Both of those circles have the same "amount" of points to me. Is a countable infinity perhaps something that is more tangibly infinite. Whereas uncountable infinite has so many "linking" points that you count it far enough. But each set is the same size
It cannot be omitted that this woman has a tremendously impressive ability to teach. She was able to walk a very wide range of people through the topic of infinity, adjusting the flow and terminology to the interlocutor, regardless of their age or degree. As a teacher myself, I can only offer my admiration and congratulations.
@@masneomlock5344 wow Though somebody already told me that it's something called kippah and then i googled and found out that yamakah and kippah both are kinda the same thing. Thanks though
So true, I still regret not having a good professor my whole grad life, in high school, I was so much addicted to Maths and all the wonders it had in it, but later on I started hating Maths just because the professors sucked out all my interests....All I had to do was learn and mug up the questions and their solutions delivered by the professors because that was only what used to show up in exams, no creativity, nothing...
@@bumblebeeflies20 that's interesting, I'm curious what the highest courses you took were. I've found that my upper level math courses were all solely proof-based and required lots of creativity and critical thinking rather than just computation
No? At the experts state, they were literally stating theorems and ideas previously used in mathematics to explain infinity, just like in the other stages
Because there are experts. They know a lot about the subject as much as the host themselves, so there can only be discussions and sharing of ideas, not actual teaching.
The way she spoke to the young girl was on point. She explained things in a way that could make anyone understand it, while building up the complexity at a rate she could keep up with. Granted, the kid seems to be a very intuitive individual with a good ability to connect the dots. Overall, a great video.
@@midchib9236 Pronouns are important and HE should be afforded the correct pronoun based on what is in his pants. You may not care about science, but it's important. He is a mathematician with a johnson.
@@emmapasqule2432 lady, she has a female skull shape which is congruent to female ratios of shoulders to hips. its a biological woman wearing ugly clothing with a bad haircut. are you stupid?
I’m actually surprised it wasn’t another “been trying to find patterns in the primes” kind of conversations because generally speaking if you study mathematics and physics for long enough you reach similar conclusions about the nature of mathematics itself and how…arbitrary it is. Usually mathematicians are oblivious to this fact and just focus on the particular interesting pattern they are looking into (like finding Waldo in the digits of pi), rather than identifying the nature of patterns themselves and why they exist.
It's interesting that when she explains infinity to kids, it's about instincts. When she explains it to teenagers, college students, grad students, it becomes mathematics. And when she reaches the final level, it goes back to instincts.
It was very interesting to watch Emily's eyes during each conversation. Eye contact was strong early on because everything that was said was so routine. Later on, they would drift away as they reached more difficult to explain ideas. Really fascinating.
My first instinct was that she has autism. The eye contact, the incredible depth into a single subject, and even her hands on her knees seemed like she wanted to stim, but couldn't. Whenever I explain something deeply (I wish to be on her level someday), I also avoid eye contact. When listening intently as well
having studied math myself and partially crossed over courses with soon to be teachers, and i gotta say that's kind of unsurprising to me. Math teachers aren't mathematicians. They are teachers. They stop in their pursuit of mathematical knowledge to be able to put time into getting better as an educator. But at some point, you reach a level of mathematics, where the level that the teacher is familiar with is no longer sufficient such that the teacher has enough excess knowledge to be able to explain things very well. Basically the higher level you go, the more important math will be and the lower (younger) you go, the more you need to be an educator and well versed in pedagogics. High school is somewhere in the middle of that. Yes, being a good educator is important, but if the teacher doesn't know their Taylor Series for example, then it gets a lot harder to explain what a derivative is, no matter how well he could explain it if he DID know their Taylor Series. SO yeah ... not easy being a teacher if you wanna be a good one.
I like how all of these levels can be summarized by a single question each. Level 1: What's the difference between a really large finite number and infinity? Level 2: What happens if you try to do basic maths with infinity? Level 3: How can one kind of infinity be larger than another? Level 4: What kinds of weird logical consequences are there to the fact that infinity exists? Level 5: Why are we asking these questions about something we literally can't imagine? I have never actually had to study maths beyond high school, and videos like this make me feel like I'm missing out. Maybe I'll pick it up for fun once I'm retired. It'll be super interesting to see what mathematicians will come up with until then.
Well, we can imagine infinity just fine. We make bad conclusions with it, but that is different than not being able to imagine it. Most people reach erroneous conclusions about most things they imagine.
There is no use in knowing what infinity means to real life. I think it's much better for retiring individuals to study aspects of medicine, so that they could understand their doctor appointments, and what they're conditions means. It is much more interesting to study what is already there to what is theoretically possible
Do it now! Like in the video most concepts are intuitive and you don't have to do the math. Then in most cases it is much more useful to know that a concept exists rather than knowing how to do it exactly.
@@josephdahdouh2725 Not everything you do has to be useful, and things that don't have real world applications can still be interesting. If you have the time and you find it interesting, why not learn about anything you want to?
I think the most interesting part of this was the conversation with the fellow expert. When speaking with the graduate student, the vocabulary and concepts reached a point one could no longer connect with. But speaking with a fellow expert made it human again, in how they became philosophical and how they were in fact vulnerable humans in a world of.... infinite possibilities, but with finite knowledge and capabilities. It brought it all back to earth as it were.
My favorite part was the technical discussion with the grad student, but then, I'm also a graduated grad student that imbibes all math and science I can on UA-cam, so I'm quite biased here lol
I guess this is what Terry Tao describes as “post-rigorous”. They both have such deep knowledge that they get each other without using rigorous language.
Interesting. I wasn’t aware of this field when I studied but that was early 2000s and I suppose this wasn’t as advanced as it is today. I would have confused ‘category theory’ with taxonomisation! 😂
Emily Riehl is amazing. Just discovered her through this video. She communicates so clearly, and 100% does not sound/feel like a 'nerd' at all. She almost tricked me into thinking I was smart enough to understand all the concepts in this video. Which... I fell apart at level 4. Okay, I was hanging by a thread 2/3 through level 3.
@@williamzinedineh Real numbers are numbers that aren't imaginary, like the square root of -1. For any x, x^2 cannot be negative. Thus, the square root of any negative number is imaginary.
I really don't get why I enjoyed this so much, I would have watched hours and hours of this woman speaking and explaining math concepts. Thanks for the video!
Right?? She’s a dream to listen to, even for someone who knows nothing about maths! She is the type of teacher who can gift learning itself, not just knowledge
It's great to see Emily Riehl getting so much recognition, she's a great mathematician and educator. I've had the pleasure of meeting her in person and she's wonderful to be around.
I took Advanced Calculus in my freshman year of college and failed my first test because I couldn't understand these concepts. This woman just explained nearly everything that my professor sucked at explaining over our first 10 lessons in the span of 24 minutes. Great video.
I feel you. If only maths were as easy to manipulate and practice for me as they are to conceptualize. Conceptually I get it. I get lost in all the notation and coding though.
@@danielcohn6884same. The concepts make sense but applying the concepts and remembering formulas, steps and all the components are were I’m lost. Feels like a whole world of knowledge exists behind mathematical knowledge.
Maybe you've also matured in your reasoning ability...I wouldn't necessarily put ALL of the blame on your professor nor give all of the credit to this woman!
It's so nice to meet people in our lives who can explain 5 minutes of information using 5 years of your time, (soap opera on tv) and people who can pack 5 years of information down to an understandable video that is 20 minutes long. (Guess which one you would rather sit thru?)
I'm so impressed by how smart that little girl is! She not only understands the concepts pretty quickly, but also asks really pertinent questions to further the conversation
She is smart, has the ability to communicate, beautiful, who knows how many skills she has beside this. We need more women in the field of stem and she is inspiring to see reach this level of success. May she succeed in all her endeavors.
I do not understand anything about mathematics, physics or whatever is being discussed on this channel but what I find interesting is the fact that the conversation with PhD students becomes very technical and when you reach the expert level, the conversation becomes much more clear and philosophical.
It was so fun to watch her using a Socratic approach with the grad student. And the expert was luminous. I especially loved when she said that mathematics does not really explore a universal truth but is a human construct. Mathematics is an art indeed!
@@RT-ol4hh Maths as a human construct does not mean it's a social construct... It's a human construct because it relies on chosen axioms. It does have a connection to "reality" (a concept not that easy to define) as it is the main tool of physics and we see its practical consequences every day. So, we can build some very tangible objects relying on an massively incomplete human construct. And that's awesome!
IS an art and ISNT an art, as a language has a lot of fails(really big fails to be honest, incomplete, undecidable, sometimes inconsistant) and something like that shouldnt be art but at the same time chaos(and all non-chaos thing inside it) is often a type of art too ❤️😂 ok too much math logic
Experts explaining complex ideas in this manner, gradually increasing that complexity through 5 levels, is my new favourite process. I'll be binge watching all of these now.
The child level kid is so good the way she understands the concept and answers the question asked by the mathematician is so good.The basic intuition she had about infinity was great that's how you start your beginners class of mathematical analysis.
Oh my god, I know her book "category theory in context" which is among the best math books ever written! Love to see Emily Riehl here! She deserves all the love she gets! I love how the grad student explains projective lines and Emily Riehl explains category theory, where definitely both of them are very familiar with both of those concepts.
@@magiquemarker idk if you're having a dig at the video or not, but just in case you aren't: infinitesimally means to an extremely small degree, not to an infinite degree :)
@@tophmyster But I do believe the 'infinite' in that word is the very same infinite we are speaking about in this video (as in infinitely centisimal or something like that).
@@JesseTate The correct terminology is "infinitely better," not "infinitesimally better," and the infinitesimal quantities that you encounter in several disciplines of mathematics are not defined with the same "infinite" sense as the infinity concept discussed in this video.
I’m just amazed by her ability to teach ! She breaks down the topic so beautifully considering their age and level of studies without a single pause ! I wish I had a professor like you ! What an amazing person!
Would love to see more math videos!! applied areas like differential equations, Topology etc. would be great and there are many excellent doctorates and people in the field who can get people excited and would make a great video!!
I'm a high school Math teacher, and I got totally lost in the grad level of the video, as it got very technical. I saw some of it when I was getting my masters (I remember getting very confused with the sphere doubling itself), but not in depth, and I don't remember much. I thought the expert level would go waaaaaay over my head, and some of it did, of course. But I absolutely loved that the expert level was very much about philosophy! That's one of the things to love about Math - how it can get very complicated, yet often finds its way back to basics.
@@Ms19754 You could do it if you have an undergraduate degree in Math or extensive experience reading proofs but probably not otherwise, unfortunately. Lot of great resources on UA-cam, though!
I’ve seen every episode of this series and this is the first time I feel genuinely loss. And at no fault to the presenter who is absolutely brilliant and so freaking clear in her explanations. I just literally felt my brain explode by level 3 and had to pause, regroup, and return just to remotely keep up
A good sign you're understanding the concepts. If you're comfortable with your intuitions on infinity, you probably don't really understand the question. Infinity is deeply weird.
I think it was the single most mind blowing thing I have learned in University (2nd to that is the idea of proving something is unprovable using a simple device as a Turing Machine). I am still as confused as you. Now think about how many Rational numbers you can put between every two Reals you pick ;)
i have mild dyscalculia and have struggled with maths beyond the elementary level all my life, somehow graduating high school while also failing manths and physics, but i love hearing talking about maths this way. i don't necessary understand the examples they use in the video, but when they're just having a conversation about the topic is feels surprisingly easier to understand. i used to have SO many questions during maths class because my teachers wouldn't explain concepts further than it was useful to us to do our homework, so this is super refreshing to see.
It might interest you that while I was studying math in uni, we almost never used any numbers, and I can't help but wonder if higher level maths would be easier for you than the school stuff :)
@@Nezumior that's very interesting! (even though it sounds oddly suspicious lol) I used to ace the theory portion of my tests, so, probably? i don't want to give myself too much credit since i struggle with the simplest of arithmetics 😅 but i definitely understand more of math when it's in the written form, it's when numbers come in that my brain completely checks out, which is so frustrating because it drove my teachers mad! they couldn't understand why i would get the theory and could not for the life of me put it in practice.
I struggled through Algebra all 4 years of high school. I’d fail a semester, retake it, and barely pass. Rinse and repeat. It wasn’t until the last semester when we focused solely on word problems that it clicked with me and I got a B in the class. Then I failed college Algebra and decided to take a lower level class to meet my degree requirements.
Working with math in simple programming languages (like Python) where you can easily adjust variables and see the outcomes makes it easier to learn. People that think literally and pragmatically can struggle with math in an academic setting.
@@Nezumior it's possible - I am not necessarily great with the calculations etc in the head but by the end of college when we got into the theory something clicked and I went to grad school for it even
It’s interesting how a basic level of understanding is more applicable, like to the child it’s used for counting and as the understanding increases in levels, it’s it transforms into more philosophical applications.
I love that when we get to experts in these videos It's just 2 people having a conversation on equal footing about what their passionate about, there is no need to simplify or explain, just a fun conversation
As a final year Math student who just finished a functional analysis module (basically, the study of infinite dimensional spaces), this was really fun to watch!
I love infinite dimensional spaces. The only problem is most people think they are imaginary. It is extremely difficult for people to visualise a duocylinder in 4D. There is an object in 164,438 dimensions, which the 164,437-headed Brahma cannot visualise. The 10,000,000- headed Brahma cannot be imagined by anyone. You need to understand what I am talking about. It need not be gross physical spatial dimensions.
What I don't understand is that infinity must be destroying information. If you add a infinity to a random number, it's infinity. And if you subtract that same number again, then it's still infinity. But if you add Infinity to a number and then subtract Infinity right away, it's still gonna be Infinity. You can never return to that number, or any number again. So Infinity is like the event horizon of a black hole? I dont understand what to do with this information.
@@ralphwiggum1203 but that's what they did in the hotel experiment? But to be honest I couldn't really follow the last examples because I have no idea what the mathematical terms mean. Cardinal Principe, transfinite numbers, ordinal etc. And also I don't understand why it is important that there is a difference to be made between countable infinity and uncountable. Cool and all, but since it's infinite anyways, what use does that distinction have? I only studied chemistry which had two math classes, so my mathematical knowledge is pretty limited ^^
@@supu8599 the talk with the graduate student definetly a little bit I think, they both try to explain it in a way a normal person could understand. They both definetly know what the axiom of choice is in depht, but they explain it in a very general way.
@@codesuzakugeass Yeah round four lost me a bit. I knew the college stuff (took those classes in college myself). Then they went crazy for a bit(as in beyond my comprehension at the moment). Had me feeling like I do when people start talking about topology beyond 3 dimensions(well I'm not great at 3d either, but still). Then back to sense at the end lol.
I think the truest thing to come out of this was when they were having the ‘Expert to Expert’ conversation: “We are humans constructing meaning”. Powerful stuff.
Her explainations are invaluable. I study computer science and she pretty much covered most stuff about sets. She mentioned set builder, bijection, injection. Another is surjection. She also mentions isomorphism, cardinality and proof very useful stuff.
These series are fantastic! Would it be possible to create a set of videos that explain concepts related to social sciences and economics? It would be fascinating to watch.
Took discrete math first year of uni and learned a lot about cardinality, 1-1 funcitons, onto functions, and bijective functions. Highly recommend taking discrete math its very interesting.
What I like most about these videos is that because of the different levels I can get an idea of just where I am on understanding the different subjects.
It's interesting how once we get past the College student level, the concepts become more a debate about axioms and philosophy and less about the study of various infinities.
It's because as you become more educated in a subject, you will naturally become more and more familiar with the foundation behind the subject. Philosophy is the foundation of pure mathematics.
@@AlaiMacErc As the expert said in the video, mathematics aren't foreign to philosophy. It was obvious during the Ancient Greeks era and it's still the case. Especially when you choose your assumptions, axioms, to explore further.
PhD stands for doctorate of philosophy - in any field getting to that level is no longer about basic facts and mechanics and more about various ways to think about the field.
@@AlaiMacErc Pure math is built upon its axioms. But those axioms aren't some magical universal truth. The axioms are literally only "true" because a bunch of scholars decided they wanted them to be true. The reasons behind why we would accept some axioms or reject some others is based on philosphy
I’m a musician who operates mostly by ear, rather than theoretically-I don’t know if there’s anything especially inherently mathematical about my approach, but I think not, because I’ve always been extremely jealous of others that seem to have a “mathematical mind”. I struggled with algebra, I struggled with physics. I still do as an adult. I’ve always felt like so many of these concepts just weren’t “clicking” with me the way they seemed to with many others, yet I remain so fascinated by all of this and have learned a little bit more as an adult. I thought the comparison of mathematics and philosophy here was interesting, I had never considered the shared DNA between the two.
Infinity is a fun concept to teach to students of all levels. Once you start playing with it and discover some of the paradoxes, it expands your ability to think Math.
@@mouthpiece200 philosopher bachelor here, i want to say that your comment reminded me of Wittgenstein concept of linguistic limitation. I don't think the idea of paradox in itself is a lie or a weakness of understanding because we can understand the implications of both the consequence and the cause of it, but rather a limitation of what we are able to express in symbols to convey a perfect message. If we can identify a problem and we know how to replicate the problem, the only reason that problem has to continue existing as a problem is because we can't get past its blockade on the specific path we take to get to the final location we want or the conclusion we want. It is the same concept of a broken bridge between islands, if we have a boat or we take a plane we can cross between one to the next but if we repeatedly try to get through it by the bridge we will always fail. Such is the language limitations that makes paradoxes real and at the same time not damning us to stagnate understanding because we can take other means, in the case of math, other possible symbols and equations to actually get where we want to get. The paradox remain but our way of thinking develops.
This video is a great display of why I always tell my students that "math is a language" You can use math to describe something just like you do traditional words.
The college level shows there are different ways to make collections of numbers. Intuitively, someone would say the natural numbers (0, 1, 2, 3, 4, etc.) are a smaller collection than the collection that is called integers, which contains all the natural numbers AND their negative counterparts (etc., -4, -3, -2, -1, 0, 1, 2, 3, 4, etc.). Then there's another collection called rational numbers. These are all the previous numbers and now they're allowed to have commas (for example 1.25, like dollars and cents). What she explains at the college level, is how you can prove all three collections are infinite, but one infinity isn't bigger than the others, even though someone with no mathematical knowledge would intuitively state that the rational numbers are more infinite than the integers and the integers are a bigger collection than the collection of natural numbers. She proves this by assuming numbers are nothing but symbols we use to order a collection. So the numbers in the natural collection could have a value that is similar to the integers or the rational numbers, but because that's not how most humans think of numbers we don't naturally feel inclined to agree with this. I hope you will understand it better by this explanation. Let's talk money: everyone agrees 500 dollars on your account is more than 2 dollars, -200 dollars would imply you paid for something or have debt. Let's say we would write down all the numbers that appear on your bank account and order them. We can count them as the amount of transactions. We would start with 0, this is when you opened your bank account. Then we say 1, for example, your first paycheck. Then 2, you bought gas for your truck, which probably has a negative value. 3 is a gas bill, another negative value. 4 could be your friend paying back a pizza, so that's a positive value. And so on, and so on. As we progress, you will have a very large collection. If you were to live forever, or pass your bank account down to your children and they pass it on to your grandchildren, given enough time, the amount of transactions will become infinite. You may have noticed the numbers we used to rank the transactions are natural numbers. You probably also noticed the values, ie the amount of dollars that were exchanged during that transaction, are part of the rational numbers. Because the natural collection has become infinite over time, your rational collection has become infinite as well. Because we know there are equal amount of natural numbers as there are rational numbers within this bank account, we can agree the infinity of natural numbers is equal to the infinity of rational numbers.
The expert mathematician is so professional. She is able to make others understand her & her POV. She will be a good teacher. Her students must really enjoy her class.
I was hoping she'd mention the continuum hypothesis because it's one of those open problems that's accessibly understood but also so so difficult to even begin to tackle
This is my favorite subject to discuss with 3rd graders. I found that the most outgoing or more advanced students were the one who felt the most uncomfortable about infinity, and because of that they were not as quick to solve some of these questions. Many said it made them feel smaller or insignificant. Meanwhile, some students that historically struggled with math were the first to correctly answer more complex concepts/solutions (like the n+1 portion of the infinite hotel) because they were already used to feeling smaller in many ways compared to their higher-achieving counterparts. Infinity didn’t seem to scare them as much.
@@CharlsDiggens Yes - dumb, lazy students are better at dealing with infinity. Infinity + 1 … Infinity! 👋🏼😀 Infinity + 2 … Infinity! 👏🏼😀 Meanwhile, brainiac over there is thinking … This is BS. Something has to change; otherwise, what’s the frickin’ point … ?! Pretty sure the expert had a ‘What’s the frickin’ point?!’ moment, as seen on her face at 1:57, when for a moment she realised she was spending her life talking about how infinite jars full of infinite glitter wouldn’t fit into the room she’s in … as if that might help anyone do anything …
This was a really interesting episode in this series, and Emily Riehl is a really good choice of moderator for the discussions at all levels! I think that every interviewee made a good contribution to the discussion. Thank you, Wired, for hosting this!
I'm honestly so thankful for these incredibly brilliant people. They have so much to give back to society along with all else who do. They help us with an INFINITE amount of problems we face and come up with solutions. Love it.
It's interesting how she explain really well topics to each one for his level but with the PhD student she already starts just chatting and with the expert she just talk about what are they doing without telling something new
Oddly enough, I actually found level 5 way easier to follow than level 4! Level 4 sounded like cryptic mathematics; level 5 almost like epistemology or metaphysics. Fantastic video anyway! ✨
@@scotimages Philosophically, I tend to adhere to a constructivist conception of science, which inclines me to think that, past a certain point, all scientific thinking has to reach epistemological thinking: having to question the very principles your scientific thinking is based upon. But I know nearly nothing about mathematics past a high school level, so I wouldn’t be as bold as throwing around wild assertions about what mathematics are reductible to or terminate in.
@@AlexisDayon Yes that is true. Science is like a house where epistemology is the foundation, and if the foundation is true then the peak of the house will be true. I believed that for a long time until i seen with my own eyes, bell's inequality violated by nature. Now im not sure about anything anymore. But im not qualified to speak with any certainty on any of these subjects. I just self study for fun. Do you think reality is deterministic?
@@duckyoutube6318 I like Bertrand Russell’s take on determinism. It goes like this: reality being either deterministic or random "per se" is a metaphysical problem we will never be able to solve scientifically, since it applies to a fundamental principle of reality which is absolutely out of range of any possible empirical knowledge. Nevertheless, determinism is necessary as a methodological assumption for scientific research. Trying to understand phenomenons scientifically is essentially trying to find out by which causes and principles they are determined to happen. (Even thinking that reality is probabilistic like many tend to think in the field of quantum physics is another way to determine phenomenons.) Therefore, if there is a point in reality where determinism ends, all scientific effort ends with it. So, a thorough, sceptic and empirical answer to that question would be: we have no way to assert whether reality is metaphysically deterministic, but we can assert that science needs to be methodologically deterministic.
The way i wish I could have the advantage of getting taught by a teacher like her, who seems to make maths look so interesting & give other the opportunity to question & explore it. perhaps then the fear of doing math wouldn’t have stuck with me till now,even after graduating.
Hands-down, this was the best one of these things I've seen. Make a weekly "street math with Emily" and sign me up. I want to go on a math journey and I want her to lead the way.
as i was watching the child and teen section i was thinking about the infinite hotel problem and how infinity can run out and there she goes and uses it as an example! quite cool
Honestly I thought the undergrad underperformed. He had basically the same conceptual understanding as the high schooler and child with a little more ability to follow proofs. She was still spoon feeding him everything. The biggest jump in this ranking was undergrad to phd.
Of course I couldn't follow the grad student and the expert discussions, but for some reason... the expert discussions made more sense to me were i understood it. It was more philosophical and about the concept in itself - i really enjoyed that! I love this series.
The expert they are talking about foundation of mathematics while the grad they are talking about boundaries of infinities. Boundaries of infinity is a lot harder concept because it is counter intuitive. How can an infinity beyond infinity has a boundary and yet there are still things beyond that boundary. The fact that there are structures above infinity is pretty amazing.
I'm glad this video came up as soon as I began reading 'Beyond infinity' by Eugenia Chung, I'm only just getting deeper into maths as an A-level student. it is completely about how we think about infinity and the progress that mathematicians have made on it so far touching a lot on the types of topics brought up in the first two sections of this video and I'm expecting it to get more in depth later on. I'd absolutely recommend it for anyone interested in this topic whatever your mathematical knowledge!
A true expert can explain their field of expertise in simple words and I‘m really impressed how she does that. And I really like her style, what an interesting person!
It's not true if you attended any top tier universities, most professors teach horribly/Ok'ish but they do amazing research. Teaching and research both take a lot of time to be good.
@@Jack-op8bb Rephrase on the word "true expert" because as Einstein once said, "If you can't explain it to a six year old, then you don't understand it yourself".
@@firstlegend5105and Einstein himself is not a good teacher. I recommend you read Walter Isaacson's book on Einstein. “Einstein was never an inspired teacher, and his lectures tended to be regarded as disorganized.”
@@Jack-op8bb well, that could be justified. He wasnt a sole "teacher" but gave lectures at various universities. I'd say that he's leaning more into research as a job prospective if anything
As a software Engineer Infinity is sometimes apart of systems development in my field and I find this video very pleasing in its manner of explaining infinity in a manageable manner.
The concept of induction over the reals is mind blowing if that is actually possible. I’m still somewhat confused about how a set that is dense in another set like how the rationals are dense in the reals can have a different cardinality.
That’s such a great question! This is how I think about it: When we say that the rationals are dense in the reals, we’re saying that if we choose some sort of epsilon (a certain tolerance where we say “close enough”) and then pick a random irrational number, then we can build a rational number that gets within “close enough” to the irrational number we got. So say we pick pi. If we care about being at least 0.1 close to pi, we can just choose 31/10. If we want to be at least 0.01 close, you can choose 314/100. And so on But! No matter what epsilon you choose, there are infinitely smaller epsilons that you COULD choose. What ever digit you stop matching at, there are infinitely MORE digits you could be matching. You can’t ever build a rational number that matches digits to pi far out enough to not have infinitely more digits to go. That’s why the reals are just SO BIG. In this infinite pool of real numbers, each irrational number is, itself, also infinite Not the most rigorous explanation out there, but I hope it helps! Sincerely, an applied math masters student
@@the_piano_nerd4960 Thank you, my question is more about how that doesn't imply the rationals should have the same cardinality as the reals, although this is probably just me not knowing very much about cardinality. Sincerely, a high schooler who does maths olympiad.
I love how she is actually at no point using inaccurate mathematical statements. On the board she used subset and injection symbols without explaining them because it is not necessary to know. But she at no point said Q = ZxZ or that she constructed a bijection between Q and N, because it would not be mathematically correct.
@@Scary_Balthazar I mean, Q is certainly is not equal to N×Z*. And even the natural function f(p/q) = (p,q) is not bijective because for example (1,3) is the same element as (2,6). Also, Z* is not all of Z without 0, but actually just +1 and -1 if you look up what * is defined as. Sorry, I am being a little nerdy :D but these are exactly the little inaccuracies that she does not make at all in this video, without overexplaining unnecessary details. This is what impressed me.
A cs grad here ... The mathematician explain the functions so well here by providing a model and then deriving it . I remember in my older days when I have to prove those functions...we only took an example of 2 numbers and substitute and then prove it ...kind of reverse engineering one would say...but that was often hard for me to grasp onto.. this model explained it way clearer. How I wish she was my maths prof. 🥺
I'm a final year masters student of math & I easily understood upto grad student. I've heard( or studied a lil bit) of the concepts/technical language that the grad student & the expert used but was unable to understand what they were saying most of the time. It's like learning a new language, you know a bunch of words & concepts but that doesn't mean you're fluent in it.
Level 2 I’ve heard way too many times, yet somehow this explanation made me understand it better. Level 3 was fine two, but somehow the level 1 explanation (and reaction) was my favorite😄
I find the youngest child to be really impressive, she kept her cool when confronted with unexpected and complex thoughts, and had surprisingly accurate intuitions.
It's scripted and staged...
@@dark_sunset "if there's unlimited pieces of glitter we need unlimited pieces of jar"
lol what part of that sounds scripted to you?
she already could handle more than me… you can have more infinities than infinity?
is that like when i told my mom i loved her infinity plus 1 and thought i had her beat? lmao
you mean she could’ve said i love you times infinity plus another infinity…?
jeez maybe i did love her more lmao
@@ChunderThunder1 he’s just disappointed that a child is already thinking at a higher level than he is.
@@dark_sunset I love how people say that, when it is clearly easier to just have an expert tell something to a kid and have them react to it, than writing dialogue that sounds authentically kidlike and getting a childactor to remember it. Also if it were scripted wouldn't you write a kid character who is intuitively wrong more rather than less often?
i love how giddy and enthusiastic the expert is. she is clearly energized by the discussion. love to see people living their dreams.
Heck yeah, it's a super endearing quality you see a lot in mathematicians. I figure you only become a professional math doer if you really, really like doing math.
Being an expert is one thing, being a communicator is unrelated. Being both is really unusual. Brilliant.
She once got into an argument with a triangle and honorably conceded when she determined it was right.
Every mathematician is like this, almost down to a t in personality. It's just the effect really understanding math has on people, no matter who they might be
she's on molly
As a father I can tell you right now that child holds in her hands a jar of infinite glitter
Lmao 😂
Well said.
Glitter is forever.
😂😂 soo true.
The herpes of the craft world
You can tell how much a person understands a subject by the way they explain it to people. The fact that she can explain such a complex idea in simpler terms shows how much she understands about the subject.
Her phd in mathematics is another way to tell she understands the subject lol.
This is how circular reasoning works. If you explain how your lies is the truth
you have to do it that way otherwise people will laugh at you.
@@mauricemenard2243why are you copy pasting this comment
Ask the god Hazard .@@pinto_8261
If the god of atheist HAZARD is able to create life that no one is able to reproduce he should be able to respond to this simple question.@@pinto_8261
She communicates complex concepts so clearly with no filler sounds, clearly thinking at top speed the whole time. I'm no fan of mathematics, but somehow I'm feeling an interest through her passion
I thought it would take FOREVER to explain. ;)
Yeah, she's sharp.
Absolutely. There is a kind of magic that happens when you hear a person talk about something they are passionate about. You could be totally uninterested in the topic before, but suddenly something sparks when you hear them talk, and you want to know more. It's fascinating and beautiful.
@@AracneMusic we kinda are a good passion detector , don't it?
you just know that they put their lifetime into the subject
before making the every sentence to make us understand.
Definitely beyond fascinating ,i would say.
As a fan of maths, she communicates extremely complex concepts very precisely with no filler sounds. The points she made about the axiom of choice blew my mind. People talk about how counterintuitive it is but the examples she chose are so perfect yet not the natural go-tos that I've seen people mention. I've mainly heard the Banach-Tarski Paradox like the student mentions. Doing induction on the reals is so mind-boggling. It's like drawing infinitely many, infinitely small dots on a page to colour it in. And then the circle thing? Chef's kiss.
the youngest child's "can infinity get bigger" was a surprisingly deep question the brings you into some significantly more advanced topics, i wouldve been completely stumped trying to answer that in a way that would address the question and also make sense to the child
It's the equivalent of asking "can there be something that isn't a part of everything." Infinity isn't a description of a product, it's a description of a process that has no end.
To restate: it isn't simply that there are "infinite integers" it is that if you were to try and create the set of integers, that process would not end. And so, any description of infinity needs to include the interval of calculation to make comparisons between the processes (and an injected stopping point).
eg, if you were limited to calculating the numbers of "X" category between -10 and 10, 1 per second and had 10 seconds, you might come up with 0-9, or 1.1 1.2 1.3 ... 2.0 or -10, -9.9 ... -9 which we can prove are different % of the infinite sets of integers v. natural numbers v. real numbers. It depends on the process used.
There's the classic infinite hotel thought exercise.
@@nathanberrigan9839 my thoughts went to the diagonal argument and countable vs uncountable infinity
As a mathematician I often tell people that the (countable) infinity is not the biggest you can think of and sometimes they even get angry with me and tell me that if something never ends, there can't be anything bigger. Especially people on social networks quickly get aggressive. I think people should learn the concept of infinity as soon as possible. Maybe at the age of that first girl. I think at that age students would already understand the Cantor trick. At the same time the would learn the concept of proof by contradiction, which may be the most essential tool in maths.
@@skyscraperfan I still don't get how some invites are bigger than others.
Is it more so that you can get to a bigger number faster? But the "size" of each set is the same?
I saw someone draw a circle centred on a point x, and someone else drew a bugger circle centred on the same point. Then someone claimed that for every point in the small circle, there would be the same point on the big circle (injective). But then they tried to say that the bigger circle had more points and was therefore a bigger infinity. That's just false to me. For every point in the big circle, there is also a point in the small circle. If you couldnt always find another point, it wouldn't be infinity.
Both of those circles have the same "amount" of points to me.
Is a countable infinity perhaps something that is more tangibly infinite. Whereas uncountable infinite has so many "linking" points that you count it far enough. But each set is the same size
It cannot be omitted that this woman has a tremendously impressive ability to teach. She was able to walk a very wide range of people through the topic of infinity, adjusting the flow and terminology to the interlocutor, regardless of their age or degree. As a teacher myself, I can only offer my admiration and congratulations.
What's that white thing on the college student's head? At 6:40 ?
@@ishakHafiz12I would assume it’s a Yamakah.
@@masneomlock5344 wow
Though somebody already told me that it's something called kippah and then i googled and found out that yamakah and kippah both are kinda the same thing. Thanks though
"she's" trans you know? "she" has balls.
How do you know it’s a woman? What is a woman?
As a person with a bachelors degree in math, she is a much better professor than so many I had!
So true, I still regret not having a good professor my whole grad life, in high school, I was so much addicted to Maths and all the wonders it had in it, but later on I started hating Maths just because the professors sucked out all my interests....All I had to do was learn and mug up the questions and their solutions delivered by the professors because that was only what used to show up in exams, no creativity, nothing...
@@bumblebeeflies20 that's interesting, I'm curious what the highest courses you took were. I've found that my upper level math courses were all solely proof-based and required lots of creativity and critical thinking rather than just computation
she's simply a superb communicator
She?
Watch your language....
😂
@@arisgreek8697 keep yourself safe 😁😁
The youngest girl was incredibly bright and intuitive when answering and asking the questions. She understood concepts even some adults struggle with.
Who struggles with knowing what infinity is and where it's used?
@@marrycinati2604 wait until you meet an average adult.
@@marrycinati2604 Bro i know adults who cant do division. Trust me people are dumb
Such as ? What concepts did she understand that some Adults didn't ? Give examples .
@@philharmer198that more than what you can count doesn't necessarily mean it is infinity lol..
I love how every time we reach Experts stage, there's no teaching or educating done, just discussion and sharing of thoughts and ideas.
No? At the experts state, they were literally stating theorems and ideas previously used in mathematics to explain infinity, just like in the other stages
Once you stop counting you start thinking.
Because there are experts. They know a lot about the subject as much as the host themselves, so there can only be discussions and sharing of ideas, not actual teaching.
Cause they are at the same level
The way she spoke to the young girl was on point. She explained things in a way that could make anyone understand it, while building up the complexity at a rate she could keep up with. Granted, the kid seems to be a very intuitive individual with a good ability to connect the dots. Overall, a great video.
It's not a WOMAN. He is a biological male who identifies as a female mathematician.
@@emmapasqule2432 does it matter tho ?
@@midchib9236 Pronouns are important and HE should be afforded the correct pronoun based on what is in his pants. You may not care about science, but it's important. He is a mathematician with a johnson.
@@emmapasqule2432 lady, she has a female skull shape which is congruent to female ratios of shoulders to hips. its a biological woman wearing ugly clothing with a bad haircut. are you stupid?
@@emmapasqule2432 do you have any evidence of that
Fascinating that the expert conversation dives right into the realm of philosophy
I’m actually surprised it wasn’t
another “been trying to find patterns in the primes” kind of conversations because generally speaking if you study mathematics and physics for long enough you reach similar conclusions about the nature of mathematics itself and how…arbitrary it is. Usually mathematicians are oblivious to this fact and just focus on the particular interesting pattern they are looking into (like finding Waldo in the digits of pi), rather than identifying the nature of patterns themselves and why they exist.
Her ability to engage with such a wide variety of students on different level is incredible.
this was obviously scripted
True .
@@ladderlappen4585 why do you think so ? I'm saying true to the wide variety of students .
repent to God
@@Baggerz182Or what
I just love how all the explained in 5 levels of difficulty, starts of as a lecture and ends in a discussion
It's interesting that when she explains infinity to kids, it's about instincts. When she explains it to teenagers, college students, grad students, it becomes mathematics. And when she reaches the final level, it goes back to instincts.
Infinity is a sandwich
@@ATIARImusica tasty sandwich
Probably tastes like chicken
Brain development.
This is how circular reasoning works. If you explain how your lies is the truth
you have to do it that way otherwise people will laugh at you.
It was very interesting to watch Emily's eyes during each conversation. Eye contact was strong early on because everything that was said was so routine. Later on, they would drift away as they reached more difficult to explain ideas. Really fascinating.
My first instinct was that she has autism. The eye contact, the incredible depth into a single subject, and even her hands on her knees seemed like she wanted to stim, but couldn't. Whenever I explain something deeply (I wish to be on her level someday), I also avoid eye contact. When listening intently as well
thought the same thing bari
Emily is infinitely better than any math teacher I had in high school.
i see the pun in here and i hate it
but yeah agreed shes great
She riehly is
having studied math myself and partially crossed over courses with soon to be teachers, and i gotta say that's kind of unsurprising to me.
Math teachers aren't mathematicians. They are teachers. They stop in their pursuit of mathematical knowledge to be able to put time into getting better as an educator.
But at some point, you reach a level of mathematics, where the level that the teacher is familiar with is no longer sufficient such that the teacher has enough excess knowledge to be able to explain things very well.
Basically the higher level you go, the more important math will be and the lower (younger) you go, the more you need to be an educator and well versed in pedagogics.
High school is somewhere in the middle of that.
Yes, being a good educator is important, but if the teacher doesn't know their Taylor Series for example, then it gets a lot harder to explain what a derivative is, no matter how well he could explain it if he DID know their Taylor Series.
SO yeah ... not easy being a teacher if you wanna be a good one.
You must have never had me as a teacher.
@@tanmaygarg3885 I've lost it
I like how all of these levels can be summarized by a single question each.
Level 1: What's the difference between a really large finite number and infinity?
Level 2: What happens if you try to do basic maths with infinity?
Level 3: How can one kind of infinity be larger than another?
Level 4: What kinds of weird logical consequences are there to the fact that infinity exists?
Level 5: Why are we asking these questions about something we literally can't imagine?
I have never actually had to study maths beyond high school, and videos like this make me feel like I'm missing out. Maybe I'll pick it up for fun once I'm retired. It'll be super interesting to see what mathematicians will come up with until then.
Well, we can imagine infinity just fine. We make bad conclusions with it, but that is different than not being able to imagine it. Most people reach erroneous conclusions about most things they imagine.
There is no use in knowing what infinity means to real life. I think it's much better for retiring individuals to study aspects of medicine, so that they could understand their doctor appointments, and what they're conditions means. It is much more interesting to study what is already there to what is theoretically possible
Do it now! Like in the video most concepts are intuitive and you don't have to do the math. Then in most cases it is much more useful to know that a concept exists rather than knowing how to do it exactly.
You say "videos like this" so I assume you've seen other maths stuff on youtube, but Numberphile is a cool channel for anyone that hasn't seen it
@@josephdahdouh2725 Not everything you do has to be useful, and things that don't have real world applications can still be interesting. If you have the time and you find it interesting, why not learn about anything you want to?
I think the most interesting part of this was the conversation with the fellow expert. When speaking with the graduate student, the vocabulary and concepts reached a point one could no longer connect with. But speaking with a fellow expert made it human again, in how they became philosophical and how they were in fact vulnerable humans in a world of.... infinite possibilities, but with finite knowledge and capabilities. It brought it all back to earth as it were.
This kind of happens in businesses as well I guess. Interesting thought.
Beautifully articulated!
My favorite part was the technical discussion with the grad student, but then, I'm also a graduated grad student that imbibes all math and science I can on UA-cam, so I'm quite biased here lol
I guess this is what Terry Tao describes as “post-rigorous”. They both have such deep knowledge that they get each other without using rigorous language.
lol, the "expert" should have been right after the child.
The expert is obviously amazing, but at 14:50, the grad student shows how infinity is practically useful in his field very concisely, awesome
i loved seeing how she interacted with all the different people. never talked down to anyone and she really explained everything quite well.
Fact: She’s a leading expert in the field of category theory.
I've been reading her book 'Category Theory in Context', it's amazing.
@@thanics1 good to know, thanks!
Interesting. I wasn’t aware of this field when I studied but that was early 2000s and I suppose this wasn’t as advanced as it is today.
I would have confused ‘category theory’ with taxonomisation! 😂
@@icarusflying1814 😂
Instant infinite amount of respect for that ;D
Emily Riehl is amazing. Just discovered her through this video. She communicates so clearly, and 100% does not sound/feel like a 'nerd' at all. She almost tricked me into thinking I was smart enough to understand all the concepts in this video. Which... I fell apart at level 4. Okay, I was hanging by a thread 2/3 through level 3.
yeah they lost me at the "real" numbers. i have no idea what that means
I lost at level 4 and came back at level 5 lol
@@williamzinedineh Real numbers are numbers that aren't imaginary, like the square root of -1. For any x, x^2 cannot be negative. Thus, the square root of any negative number is imaginary.
@@knayvik yeah... that does NOT answer my unstated question
@@williamzinedineh real numbers are any number that isnt the square root of a negative number
I really don't get why I enjoyed this so much, I would have watched hours and hours of this woman speaking and explaining math concepts. Thanks for the video!
Right?? She’s a dream to listen to, even for someone who knows nothing about maths! She is the type of teacher who can gift learning itself, not just knowledge
It's great to see Emily Riehl getting so much recognition, she's a great mathematician and educator. I've had the pleasure of meeting her in person and she's wonderful to be around.
Stop lying
@@ohnah6261 Infinity yourself
@@Alic4444 ?
I have definitely met people who chose their field of math because she and her circles are so welcoming!
I read “meeting her in prison” 👯♀️👯♀️
I took Advanced Calculus in my freshman year of college and failed my first test because I couldn't understand these concepts. This woman just explained nearly everything that my professor sucked at explaining over our first 10 lessons in the span of 24 minutes. Great video.
I feel you. If only maths were as easy to manipulate and practice for me as they are to conceptualize. Conceptually I get it. I get lost in all the notation and coding though.
free bj for you when we next meet
@@danielcohn6884same. The concepts make sense but applying the concepts and remembering formulas, steps and all the components are were I’m lost. Feels like a whole world of knowledge exists behind mathematical knowledge.
Maybe you've also matured in your reasoning ability...I wouldn't necessarily put ALL of the blame on your professor nor give all of the credit to this woman!
It's so nice to meet people in our lives who can explain 5 minutes of information using 5 years of your time, (soap opera on tv) and people who can pack 5 years of information down to an understandable video that is 20 minutes long. (Guess which one you would rather sit thru?)
I'm so impressed by how smart that little girl is! She not only understands the concepts pretty quickly, but also asks really pertinent questions to further the conversation
She is smart, has the ability to communicate, beautiful, who knows how many skills she has beside this. We need more women in the field of stem and she is inspiring to see reach this level of success. May she succeed in all her endeavors.
I do not understand anything about mathematics, physics or whatever is being discussed on this channel but what I find interesting is the fact that the conversation with PhD students becomes very technical and when you reach the expert level, the conversation becomes much more clear and philosophical.
the little girl answered the questions better than i did in my head 😭
Exactlyyyy
You dumb?
Tough
literally 🤣🤣
It was so fun to watch her using a Socratic approach with the grad student. And the expert was luminous. I especially loved when she said that mathematics does not really explore a universal truth but is a human construct. Mathematics is an art indeed!
Yeah I got surprised when they give a solid answer to that (afaik) on-going debate about Mathematics whether it's a discovery or an invention.
@@RT-ol4hh Take your meds and breathe :)
@@RT-ol4hh Maths as a human construct does not mean it's a social construct... It's a human construct because it relies on chosen axioms. It does have a connection to "reality" (a concept not that easy to define) as it is the main tool of physics and we see its practical consequences every day. So, we can build some very tangible objects relying on an massively incomplete human construct. And that's awesome!
@@RT-ol4hhPeace and happiness upon you too. No need to be so harsh with yourself and the noodle; ramen can be delicious.
IS an art and ISNT an art, as a language has a lot of fails(really big fails to be honest, incomplete, undecidable, sometimes inconsistant) and something like that shouldnt be art but at the same time chaos(and all non-chaos thing inside it) is often a type of art too ❤️😂 ok too much math logic
Experts explaining complex ideas in this manner, gradually increasing that complexity through 5 levels, is my new favourite process. I'll be binge watching all of these now.
The child level kid is so good the way she understands the concept and answers the question asked by the mathematician is so good.The basic intuition she had about infinity was great that's how you start your beginners class of mathematical analysis.
The first girl is amazing in how she understands infinity and how she is able to verbalize her thoughts.
Oh my god, I know her book "category theory in context" which is among the best math books ever written! Love to see Emily Riehl here! She deserves all the love she gets! I love how the grad student explains projective lines and Emily Riehl explains category theory, where definitely both of them are very familiar with both of those concepts.
Downloading it now because of this recommendation ❤
I am 1st year undergrad and will save it now too for later, thanks for the recommendation.
I love how much positive , and intelligent conversation that First child was exposed to through their childhood.
This video is countably better than all the previous 5 Levels of Difficulty videos
I think it's infinitesimally better!
@@magiquemarker idk if you're having a dig at the video or not, but just in case you aren't: infinitesimally means to an extremely small degree, not to an infinite degree :)
@@tophmyster The smallest number thats after 0, which has infinite decimals at that
@@tophmyster But I do believe the 'infinite' in that word is the very same infinite we are speaking about in this video (as in infinitely centisimal or something like that).
@@JesseTate The correct terminology is "infinitely better," not "infinitesimally better," and the infinitesimal quantities that you encounter in several disciplines of mathematics are not defined with the same "infinite" sense as the infinity concept discussed in this video.
I’m just amazed by her ability to teach ! She breaks down the topic so beautifully considering their age and level of studies without a single pause ! I wish I had a professor like you ! What an amazing person!
Would love to see more math videos!! applied areas like differential equations, Topology etc. would be great and there are many excellent doctorates and people in the field who can get people excited and would make a great video!!
more math and please do at least one on chemistry! How can you do bio and physics but not do the central science of chemistry
@@saimaurice3652 maybe on organic chemistry
She once got into an argument with a triangle and honorably conceded when she determined it was right.
🙏🙏
I'm a high school Math teacher, and I got totally lost in the grad level of the video, as it got very technical. I saw some of it when I was getting my masters (I remember getting very confused with the sphere doubling itself), but not in depth, and I don't remember much. I thought the expert level would go waaaaaay over my head, and some of it did, of course. But I absolutely loved that the expert level was very much about philosophy! That's one of the things to love about Math - how it can get very complicated, yet often finds its way back to basics.
Riehl is a great category theorist, and her book category theory in context is superb
Is the book readable for people who don't study mathematics? Do you need a lot of formal knowledge to understand it?
@@Ms19754 You could do it if you have an undergraduate degree in Math or extensive experience reading proofs but probably not otherwise, unfortunately. Lot of great resources on UA-cam, though!
Maybe one day she'll invent the Riehl numbers...
@@Ms19754 not really, you need pretty much an undergrad in maths to follow it
@@UsernameXOXO well done 👍🏻
I’ve seen every episode of this series and this is the first time I feel genuinely loss. And at no fault to the presenter who is absolutely brilliant and so freaking clear in her explanations. I just literally felt my brain explode by level 3 and had to pause, regroup, and return just to remotely keep up
A good sign you're understanding the concepts. If you're comfortable with your intuitions on infinity, you probably don't really understand the question. Infinity is deeply weird.
I think it was the single most mind blowing thing I have learned in University (2nd to that is the idea of proving something is unprovable using a simple device as a Turing Machine). I am still as confused as you. Now think about how many Rational numbers you can put between every two Reals you pick ;)
@@heyman620 None if you pick the same two numbers 🤔
@@bartholomewhalliburton9854 Wouldn't it make it 1 real?
@@bartholomewhalliburton9854 Refer to it: en.wikipedia.org/wiki/Dense_set
i have mild dyscalculia and have struggled with maths beyond the elementary level all my life, somehow graduating high school while also failing manths and physics, but i love hearing talking about maths this way. i don't necessary understand the examples they use in the video, but when they're just having a conversation about the topic is feels surprisingly easier to understand. i used to have SO many questions during maths class because my teachers wouldn't explain concepts further than it was useful to us to do our homework, so this is super refreshing to see.
It might interest you that while I was studying math in uni, we almost never used any numbers, and I can't help but wonder if higher level maths would be easier for you than the school stuff :)
@@Nezumior that's very interesting! (even though it sounds oddly suspicious lol) I used to ace the theory portion of my tests, so, probably? i don't want to give myself too much credit since i struggle with the simplest of arithmetics 😅 but i definitely understand more of math when it's in the written form, it's when numbers come in that my brain completely checks out, which is so frustrating because it drove my teachers mad! they couldn't understand why i would get the theory and could not for the life of me put it in practice.
I struggled through Algebra all 4 years of high school. I’d fail a semester, retake it, and barely pass. Rinse and repeat. It wasn’t until the last semester when we focused solely on word problems that it clicked with me and I got a B in the class. Then I failed college Algebra and decided to take a lower level class to meet my degree requirements.
Working with math in simple programming languages (like Python) where you can easily adjust variables and see the outcomes makes it easier to learn. People that think literally and pragmatically can struggle with math in an academic setting.
@@Nezumior it's possible - I am not necessarily great with the calculations etc in the head but by the end of college when we got into the theory something clicked and I went to grad school for it even
It’s interesting how a basic level of understanding is more applicable, like to the child it’s used for counting and as the understanding increases in levels, it’s it transforms into more philosophical applications.
I love that when we get to experts in these videos It's just 2 people having a conversation on equal footing about what their passionate about, there is no need to simplify or explain, just a fun conversation
They're*
As a final year Math student who just finished a functional analysis module (basically, the study of infinite dimensional spaces), this was really fun to watch!
ok
I love infinite dimensional spaces. The only problem is most people think they are imaginary. It is extremely difficult for people to visualise a duocylinder in 4D. There is an object in 164,438 dimensions, which the 164,437-headed Brahma cannot visualise. The 10,000,000- headed Brahma cannot be imagined by anyone. You need to understand what I am talking about. It need not be gross physical spatial dimensions.
What I don't understand is that infinity must be destroying information. If you add a infinity to a random number, it's infinity. And if you subtract that same number again, then it's still infinity. But if you add Infinity to a number and then subtract Infinity right away, it's still gonna be Infinity. You can never return to that number, or any number again. So Infinity is like the event horizon of a black hole?
I dont understand what to do with this information.
@@maduude8809 you are still thinking of infinity as finite
you can not add a number to infinity because it is already a part of infinity
@@ralphwiggum1203 but that's what they did in the hotel experiment? But to be honest I couldn't really follow the last examples because I have no idea what the mathematical terms mean. Cardinal Principe, transfinite numbers, ordinal etc.
And also I don't understand why it is important that there is a difference to be made between countable infinity and uncountable. Cool and all, but since it's infinite anyways, what use does that distinction have? I only studied chemistry which had two math classes, so my mathematical knowledge is pretty limited ^^
The approach to explaining infinity to a child is brilliant. I am touched by the narrator's sincere desire to make the other person truly understand.
Warning at 14:18 they start speaking a different language and closed captions do not help....
I feel a little stupid because I needed to read three times to understand lol
Clever child. They almost came up with Hilbert's Hotel on their own.
If you are given the answers it helps
She is Indian after all
@@supu8599 It's also scripted and staged, like all videos in this series...
@@dark_sunset is it ? 🤨
@@supu8599 the talk with the graduate student definetly a little bit I think, they both try to explain it in a way a normal person could understand. They both definetly know what the axiom of choice is in depht, but they explain it in a very general way.
Experts that can communicate really well like Emily can are so beneficial to the world.
I found the Last explanation easier to understand than the fourth.
@@codesuzakugeass Yeah round four lost me a bit. I knew the college stuff (took those classes in college myself). Then they went crazy for a bit(as in beyond my comprehension at the moment). Had me feeling like I do when people start talking about topology beyond 3 dimensions(well I'm not great at 3d either, but still). Then back to sense at the end lol.
why is she dressed like a dude tho?
No.
I think the truest thing to come out of this was when they were having the ‘Expert to Expert’ conversation:
“We are humans constructing meaning”. Powerful stuff.
Her explainations are invaluable. I study computer science and she pretty much covered most stuff about sets. She mentioned set builder, bijection, injection. Another is surjection. She also mentions isomorphism, cardinality and proof very useful stuff.
That youngest girl has a better grasp on infinity than 99% of adults.
This is true.
99% of illiterate adults maybe
@@marcioamaral7511 fr
You gotta be one of them if you don’t understand how much 99% is
99% is an outrageous statement
These series are fantastic! Would it be possible to create a set of videos that explain concepts related to social sciences and economics? It would be fascinating to watch.
Took discrete math first year of uni and learned a lot about cardinality, 1-1 funcitons, onto functions, and bijective functions. Highly recommend taking discrete math its very interesting.
I love how she explains the concepts. She has the gift of clarity.
I love the number guesses of the child. But one can tell that the child is super intelligent and absolutely got the concept.
What I like most about these videos is that because of the different levels I can get an idea of just where I am on understanding the different subjects.
I agree, but the more I watch these videos the more I realise I only have the understanding of a small child and that my brain also hurts!
I'm on level .55555555555555555
She needs to start a podcast on education, math and science with the name "Keep it Riehl"
😂
Love the 9yo child. She’s so smart for her age. “How many jars do you need?”, instantly answers: “infinite amount of jars” 1:37 😮
im 18, and i would probably say something stupid like 10
I mean, it’s pretty obvious
I absolutely love how passionately she talks about mathematics, some people don't really have that much energy behind their jobs, but Emily does
It's interesting how once we get past the College student level, the concepts become more a debate about axioms and philosophy and less about the study of various infinities.
It's because as you become more educated in a subject, you will naturally become more and more familiar with the foundation behind the subject. Philosophy is the foundation of pure mathematics.
@@downsonjerome7905 I think we have an escaped philosopher in the room. No, pure maths is its own foundation -- that's kinda the entire point.
@@AlaiMacErc As the expert said in the video, mathematics aren't foreign to philosophy. It was obvious during the Ancient Greeks era and it's still the case. Especially when you choose your assumptions, axioms, to explore further.
PhD stands for doctorate of philosophy - in any field getting to that level is no longer about basic facts and mechanics and more about various ways to think about the field.
@@AlaiMacErc Pure math is built upon its axioms. But those axioms aren't some magical universal truth. The axioms are literally only "true" because a bunch of scholars decided they wanted them to be true. The reasons behind why we would accept some axioms or reject some others is based on philosphy
I’m a musician who operates mostly by ear, rather than theoretically-I don’t know if there’s anything especially inherently mathematical about my approach, but I think not, because I’ve always been extremely jealous of others that seem to have a “mathematical mind”. I struggled with algebra, I struggled with physics. I still do as an adult. I’ve always felt like so many of these concepts just weren’t “clicking” with me the way they seemed to with many others, yet I remain so fascinated by all of this and have learned a little bit more as an adult. I thought the comparison of mathematics and philosophy here was interesting, I had never considered the shared DNA between the two.
Infinity is a fun concept to teach to students of all levels. Once you start playing with it and discover some of the paradoxes, it expands your ability to think Math.
I don't believe in real paradoxes. Every "paradox" is simply revealing a weakness in understanding. Paradoxes don't exist except in the human mind.
@@mouthpiece200 philosopher bachelor here, i want to say that your comment reminded me of Wittgenstein concept of linguistic limitation. I don't think the idea of paradox in itself is a lie or a weakness of understanding because we can understand the implications of both the consequence and the cause of it, but rather a limitation of what we are able to express in symbols to convey a perfect message. If we can identify a problem and we know how to replicate the problem, the only reason that problem has to continue existing as a problem is because we can't get past its blockade on the specific path we take to get to the final location we want or the conclusion we want. It is the same concept of a broken bridge between islands, if we have a boat or we take a plane we can cross between one to the next but if we repeatedly try to get through it by the bridge we will always fail. Such is the language limitations that makes paradoxes real and at the same time not damning us to stagnate understanding because we can take other means, in the case of math, other possible symbols and equations to actually get where we want to get. The paradox remain but our way of thinking develops.
This video is a great display of why I always tell my students that "math is a language" You can use math to describe something just like you do traditional words.
She totally lost me at the college level, but its easy to see just how intelligent she is, as well as passionate about math.
The college level shows there are different ways to make collections of numbers.
Intuitively, someone would say the natural numbers (0, 1, 2, 3, 4, etc.) are a smaller collection than the collection that is called integers, which contains all the natural numbers AND their negative counterparts (etc., -4, -3, -2, -1, 0, 1, 2, 3, 4, etc.). Then there's another collection called rational numbers. These are all the previous numbers and now they're allowed to have commas (for example 1.25, like dollars and cents).
What she explains at the college level, is how you can prove all three collections are infinite, but one infinity isn't bigger than the others, even though someone with no mathematical knowledge would intuitively state that the rational numbers are more infinite than the integers and the integers are a bigger collection than the collection of natural numbers.
She proves this by assuming numbers are nothing but symbols we use to order a collection. So the numbers in the natural collection could have a value that is similar to the integers or the rational numbers, but because that's not how most humans think of numbers we don't naturally feel inclined to agree with this. I hope you will understand it better by this explanation.
Let's talk money: everyone agrees 500 dollars on your account is more than 2 dollars, -200 dollars would imply you paid for something or have debt.
Let's say we would write down all the numbers that appear on your bank account and order them. We can count them as the amount of transactions. We would start with 0, this is when you opened your bank account. Then we say 1, for example, your first paycheck. Then 2, you bought gas for your truck, which probably has a negative value. 3 is a gas bill, another negative value. 4 could be your friend paying back a pizza, so that's a positive value. And so on, and so on.
As we progress, you will have a very large collection. If you were to live forever, or pass your bank account down to your children and they pass it on to your grandchildren, given enough time, the amount of transactions will become infinite. You may have noticed the numbers we used to rank the transactions are natural numbers. You probably also noticed the values, ie the amount of dollars that were exchanged during that transaction, are part of the rational numbers. Because the natural collection has become infinite over time, your rational collection has become infinite as well. Because we know there are equal amount of natural numbers as there are rational numbers within this bank account, we can agree the infinity of natural numbers is equal to the infinity of rational numbers.
@@DarkAngelEU very well explained especially with the bank account thingy props to u
I gather infinity is a made-up concept that doesn't make any sense when analysed... thus going into philosophical.
@@ChillerBaby Thanks mate, makes my day :D
@@DarkAngelEU Thanks for taking the time to write this wonderful explanation! You are awesome :)
Her domain expansion must go crazy
0:24 Child
2:52 Teen
6:35 College Student
14:15 Grad Student
19:40 Expert
infinite amount of thanks
@@usernameisamyth Which size of infinity?
@@pooky3672 Uncountable, for sure!
@@sankang9425 which uncountability? Continuum?
Ngl, that black girl and i are the same age but she looks more mature than me. About 17 or 18
I deeply admire those who can explain such abstract concepts concisely
The expert mathematician is so professional.
She is able to make others understand her & her POV.
She will be a good teacher.
Her students must really enjoy her class.
I was hoping she'd mention the continuum hypothesis because it's one of those open problems that's accessibly understood but also so so difficult to even begin to tackle
She did mention it.
@@divinepraiseeric yep that's what I meant. I was hoping she would and was happy she did. poor wording on my part
This is my favorite subject to discuss with 3rd graders. I found that the most outgoing or more advanced students were the one who felt the most uncomfortable about infinity, and because of that they were not as quick to solve some of these questions. Many said it made them feel smaller or insignificant. Meanwhile, some students that historically struggled with math were the first to correctly answer more complex concepts/solutions (like the n+1 portion of the infinite hotel) because they were already used to feeling smaller in many ways compared to their higher-achieving counterparts. Infinity didn’t seem to scare them as much.
And it's looking at maths a different why compared to how it's usually taught.
So you say your dumber students are not afraid of infinity, cause they dont get the greater picture of it and what it means?
Nice
@@CharlsDiggens I don't think that's what they were saying
Learn what a strawman argument is and you will understand what you just did there @@CharlsDiggens
@@CharlsDiggens Yes - dumb, lazy students are better at dealing with infinity.
Infinity + 1 … Infinity! 👋🏼😀
Infinity + 2 … Infinity! 👏🏼😀
Meanwhile, brainiac over there is thinking … This is BS. Something has to change; otherwise, what’s the frickin’ point … ?!
Pretty sure the expert had a ‘What’s the frickin’ point?!’ moment, as seen on her face at 1:57, when for a moment she realised she was spending her life talking about how infinite jars full of infinite glitter wouldn’t fit into the room she’s in … as if that might help anyone do anything …
This was a really interesting episode in this series, and Emily Riehl is a really good choice of moderator for the discussions at all levels! I think that every interviewee made a good contribution to the discussion. Thank you, Wired, for hosting this!
I'm honestly so thankful for these incredibly brilliant people. They have so much to give back to society along with all else who do. They help us with an INFINITE amount of problems we face and come up with solutions. Love it.
The pure joy and excitement on the experts face when she talked about the philosophy of math 🥹
It's interesting how she explain really well topics to each one for his level but with the PhD student she already starts just chatting and with the expert she just talk about what are they doing without telling something new
Her book on category theory is amazing! Easily one of the best authors/communicators of higher level maths.
Oddly enough, I actually found level 5 way easier to follow than level 4! Level 4 sounded like cryptic mathematics; level 5 almost like epistemology or metaphysics.
Fantastic video anyway! ✨
Is the implication of what you are saying is that mathematics is reducible and terminates (in logical terms) with the epistemology of definition ?
@@scotimages Philosophically, I tend to adhere to a constructivist conception of science, which inclines me to think that, past a certain point, all scientific thinking has to reach epistemological thinking: having to question the very principles your scientific thinking is based upon.
But I know nearly nothing about mathematics past a high school level, so I wouldn’t be as bold as throwing around wild assertions about what mathematics are reductible to or terminate in.
@@AlexisDayon Yes that is true. Science is like a house where epistemology is the foundation, and if the foundation is true then the peak of the house will be true.
I believed that for a long time until i seen with my own eyes, bell's inequality violated by nature. Now im not sure about anything anymore.
But im not qualified to speak with any certainty on any of these subjects. I just self study for fun.
Do you think reality is deterministic?
@@duckyoutube6318 I like Bertrand Russell’s take on determinism.
It goes like this: reality being either deterministic or random "per se" is a metaphysical problem we will never be able to solve scientifically, since it applies to a fundamental principle of reality which is absolutely out of range of any possible empirical knowledge.
Nevertheless, determinism is necessary as a methodological assumption for scientific research. Trying to understand phenomenons scientifically is essentially trying to find out by which causes and principles they are determined to happen.
(Even thinking that reality is probabilistic like many tend to think in the field of quantum physics is another way to determine phenomenons.)
Therefore, if there is a point in reality where determinism ends, all scientific effort ends with it.
So, a thorough, sceptic and empirical answer to that question would be: we have no way to assert whether reality is metaphysically deterministic, but we can assert that science needs to be methodologically deterministic.
grad students are usually just REALLY deep in the sauce to be honest
I love that as she moved into higher levels of explanation they have to engage in conversations that become more abstract and philosophical
The way i wish I could have the advantage of getting taught by a teacher like her, who seems to make maths look so interesting & give other the opportunity to question & explore it. perhaps then the fear of doing math wouldn’t have stuck with me till now,even after graduating.
Hands-down, this was the best one of these things I've seen. Make a weekly "street math with Emily" and sign me up. I want to go on a math journey and I want her to lead the way.
yes, i NMEED THIS
Had to stop it at 1:50 to come here and say that’s no ordinary child. She does not speak for the general population of children her age.
She is fatter than normal 😂😂😂😂😂😂😂🤓🤓🤓😝😝😝
LOL stopped at the same time for the same reason 🤣video is definitely SCRIPTED!!!
Wtf me 1:49😂😂😂 she's above average for sure
Bro, same. She is smart!
Not necessarily @@Deison-Srz
Math is so awesome and she is really good at communicating it.
She riehly is
The kid in the beginning is brilliant. I hope she gets the proper guidance and education to pursue her passion.
Dr. Riehl is amazing, and she explains complex processes so well. Thank you for featuring her!
That means she’s an actual expert not a fake
@@thienthetyga3462 It doesn't mean that, no. She _is_ an actual expert, but it has nothing to do with what you said.
he is not a she, their pronounces respect them
as i was watching the child and teen section i was thinking about the infinite hotel problem and how infinity can run out and there she goes and uses it as an example! quite cool
Loved this, and that undergrad is incredibly sharp! Emily's explanation of those proofs were very clear.
More 5 levels of math please
I know the undergrad student and he’s the best!!
@@KBin727
Hello Yoni's alt account 👋
Honestly I thought the undergrad underperformed. He had basically the same conceptual understanding as the high schooler and child with a little more ability to follow proofs. She was still spoon feeding him everything. The biggest jump in this ranking was undergrad to phd.
I would love an explamation of dividing by 0
@@appa609 Almost like he's not a math major...
She once got into an argument with a triangle and honorably conceded when she determined it was right.
😆 math jokes
Of course I couldn't follow the grad student and the expert discussions, but for some reason... the expert discussions made more sense to me were i understood it. It was more philosophical and about the concept in itself - i really enjoyed that! I love this series.
The expert they are talking about foundation of mathematics while the grad they are talking about boundaries of infinities. Boundaries of infinity is a lot harder concept because it is counter intuitive. How can an infinity beyond infinity has a boundary and yet there are still things beyond that boundary. The fact that there are structures above infinity is pretty amazing.
I'm glad this video came up as soon as I began reading 'Beyond infinity' by Eugenia Chung, I'm only just getting deeper into maths as an A-level student. it is completely about how we think about infinity and the progress that mathematicians have made on it so far touching a lot on the types of topics brought up in the first two sections of this video and I'm expecting it to get more in depth later on. I'd absolutely recommend it for anyone interested in this topic whatever your mathematical knowledge!
A true expert can explain their field of expertise in simple words and I‘m really impressed how she does that. And I really like her style, what an interesting person!
It's not true if you attended any top tier universities, most professors teach horribly/Ok'ish but they do amazing research. Teaching and research both take a lot of time to be good.
@@Jack-op8bb Agreed. Being an expert does not equate to being an educator.
@@Jack-op8bb Rephrase on the word "true expert" because as Einstein once said, "If you can't explain it to a six year old, then you don't understand it yourself".
@@firstlegend5105and Einstein himself is not a good teacher.
I recommend you read Walter Isaacson's book on Einstein.
“Einstein was never an inspired teacher, and his lectures tended to be regarded as disorganized.”
@@Jack-op8bb well, that could be justified. He wasnt a sole "teacher" but gave lectures at various universities. I'd say that he's leaning more into research as a job prospective if anything
This was inexplicably, infinitely fun to watch explained at various levels of understanding. Well done.
As a software Engineer Infinity is sometimes apart of systems development in my field and I find this video very pleasing in its manner of explaining infinity in a manageable manner.
well in computer science you only deal with countable infinity at most anyway
The concept of induction over the reals is mind blowing if that is actually possible. I’m still somewhat confused about how a set that is dense in another set like how the rationals are dense in the reals can have a different cardinality.
That’s such a great question! This is how I think about it:
When we say that the rationals are dense in the reals, we’re saying that if we choose some sort of epsilon (a certain tolerance where we say “close enough”) and then pick a random irrational number, then we can build a rational number that gets within “close enough” to the irrational number we got. So say we pick pi. If we care about being at least 0.1 close to pi, we can just choose 31/10. If we want to be at least 0.01 close, you can choose 314/100. And so on
But! No matter what epsilon you choose, there are infinitely smaller epsilons that you COULD choose. What ever digit you stop matching at, there are infinitely MORE digits you could be matching. You can’t ever build a rational number that matches digits to pi far out enough to not have infinitely more digits to go. That’s why the reals are just SO BIG. In this infinite pool of real numbers, each irrational number is, itself, also infinite
Not the most rigorous explanation out there, but I hope it helps!
Sincerely, an applied math masters student
@@the_piano_nerd4960 Thank you, my question is more about how that doesn't imply the rationals should have the same cardinality as the reals, although this is probably just me not knowing very much about cardinality. Sincerely, a high schooler who does maths olympiad.
I love how wired brings diverse smart appealing people to their videos. The presenter here is a perfect example. Please keep up these great videos!
I love how she is actually at no point using inaccurate mathematical statements. On the board she used subset and injection symbols without explaining them because it is not necessary to know. But she at no point said Q = ZxZ or that she constructed a bijection between Q and N, because it would not be mathematically correct.
I mean we can say Q is N×Z*
@@Scary_Balthazar I mean, Q is certainly is not equal to N×Z*. And even the natural function f(p/q) = (p,q) is not bijective because for example (1,3) is the same element as (2,6). Also, Z* is not all of Z without 0, but actually just +1 and -1 if you look up what * is defined as. Sorry, I am being a little nerdy :D but these are exactly the little inaccuracies that she does not make at all in this video, without overexplaining unnecessary details. This is what impressed me.
A cs grad here ... The mathematician explain the functions so well here by providing a model and then deriving it . I remember in my older days when I have to prove those functions...we only took an example of 2 numbers and substitute and then prove it ...kind of reverse engineering one would say...but that was often hard for me to grasp onto.. this model explained it way clearer. How I wish she was my maths prof. 🥺
One of my favorite infinity concepts is that an infinite shape has no boundaries and therefore no center.
I feel glad that I could follow at each level of this. Turns out the math degree wasn't a total waste of time.
It'd be embarassing if you couldn't follow any of this with a math degree
I'm a final year masters student of math & I easily understood upto grad student. I've heard( or studied a lil bit) of the concepts/technical language that the grad student & the expert used but was unable to understand what they were saying most of the time. It's like learning a new language, you know a bunch of words & concepts but that doesn't mean you're fluent in it.
Level 2 I’ve heard way too many times, yet somehow this explanation made me understand it better. Level 3 was fine two, but somehow the level 1 explanation (and reaction) was my favorite😄
my wife says I’m infinitely stupid, so I must be at level 0