Ah you're right! I don't know how that error got through, thank you for spotting that mistake; I've added it as a correction in the caption of the video.
Yes, sorry that it came out like this, I'm quite new to editing and making videos. I'll be reuploading a version of this in the future that is hopefully way better produced not only in the music but the voice-over as well!
I've heard of it at 16 I think. The thing is yt is good at popularizing math topics in recent years so that's why current young adults might be able to say such thing
It's also possible he meant that topological problems interesting him, but without the formal language. Wondering how different shapes relate to each other and can be turned into each other is something even a young child is capable of thinking about. Things like "how many holes in a straw" and "how many holes are in a coffee cup" aren't unreasonable.
Thinking about it, it kinda feels like Lie Algebras have more in common with a metric than they do with a vector space. The point of disanalogy really is that distances can be negative, and the triangle inequality becomes the Jacobi identity. I mean, given that the determinant - which tracks oriented area/volume/metric scaling under a linear transformation - is a Lie algebra, doesn't that make some sense?
3:50 nit pick: iirc the circle is the boundary, so distance equal to the radius. A disc contains all the points less than the radius. Same with sphere (boundary) vs ball (solid).
@@pi_squared2 It isn't a small error. You go on to talk about the interior of a sphere, but under your definition, there is no interior. The point/ball that you said was in the interior of the sphere was a part of your sphere. Good that you fixed it. I guess you could fix it by using a slightly different definition of "interior". English is easier to manipulate than math.
Sorry about that, I'm new to editing and making these videos. I'll be making more with hopefully better audio levels, and plan to re-upload this with better editing in the future as well as my other current vids.
Nice video, I would recommend trying to explain something like this without using your university level understanding of sets and set operations. Sets and subsets can be described simply in the beginning so it makes it easier for the viewer to understand. Anyways cant wait for your chanel to grow
@@solidpython4964 Tf u mean doing topology? its a youtube video, also who said I dont understand what sets are, I meant for others watching because the set theory was heavy. Also understanding intresting topology topics is possible without set knowledge, sets are just a good language for describing relations that occur in sets
I thought sets and subsets were high school topics in most places... And everybody interested in topology will eventually need this kind of knowledge anyway. Cool animations can only go so far.
Music makes the video presentation better not sure why everyone else is complaining. I enjoyed it and I could hear you perfectly fine. I look forward to more of your videos thank you.
Honestly, if I'm gonna be real with you, this was not what I expected when I clicked on this. It was certainly intriguing for what it's worth, but for some reason I wanted more of the way that distance is measured? I might be tripping, I'm quite tired honestly, but did anyone else have a different expectation from this title? It's probably closer to the idea of (a metric) space rather than distance, but eh it's kinda the same.
Ah I'm sorry to hear that. For me, it was that mathematicians abstracted distance by creating the definition of a metric space with the distance function relying on 4 key properties outlined in the video. Those 4 properties are what abstract distance, but I apologize if the video was in anyway misleading.
@@pi_squared2 Don't stress over it, worst case scenario one could call it "clickbait", which honestly isn't that bad nowadays anyway. As long as the content is there, it's mostly okay!
Sorry if it felt misleading, the abstraction of distance are the 4 properties listed under the distance function of a metric space. These 4 properties allow you to construct various different possible distance functions, some of them quite different to how we ordinarily measure distance in daily life.
You have earned a new subscriber! Very cool video! !! I'd only suggest to turn down the music a little more when you are speaking because it makes it a little hard to listen to your voice :)
hey brother can you please make a vid explaining basics of sets cuz it is hard to understand what the notation says and i have not yet learnt sets so its hard to understand the vids too
Beautiful music, unfortunately, it tends to drown out your voice. Maybe try re-releasing the video with the music only 25% as loud. Thanks, I appreciate the content
Now these open balls could have a metric though, so how exactly did we abstracted/generalized metrics spaces? Just by avoiding speaking of Pythagoras theorem?
The open balls are really just a part within our metric space. The cool idea behind abstraction was the idea that you could describe distance with those 4 properties, which seem quite obvious but are actually enough to help you define objects like open balls and distinguish them from closed balls.
Before this video, I had never heard of topology (in maths), that is very interesting (even if I only understood some of the proofs and notation). Amazing graphics and animations, and explanations. Question: What level of study do you learn about topologies in maths/education? I’m starting my 2nd year of college, so I assume it’s University level content? Also, could you explain more about the needs/ applications of topology?
Really glad you enjoyed the video, topology is a super interesting area of math! Typically from what I understand it'll be first introduced in the 2nd year of a pure math degree, but you can also get exposed to some of its ideas in a course called Real Analysis in 1st year. In terms of the needs/applications, there's a massive variety of subjects within topology, some which seem incredibly abstract and not applicable, while others are quite helpful. For example, topology is used to develop the theory of knots which has been used in quantum mechanics. Topology also plays a role in differential geometry, which is used quite famously to describe General Relativity in physics. As for the more abstract/inapplicable subjects within topology though, there's a rich history in mathematics of things which at first seem totally inapplicable becoming massively applicable/important in the future; so that also might just be the case with a lot of topology!
Ah ok, so I probably won’t come across it in my studies (I plan on studying a Mechanical Engineering degree), but it’s always something I can just research/look up about. and wow, interesting applications.
Studying mechanical engineering probably wouldnt lead you to topology, topology is an advanced "pure" math subject, generally you'll see it after a course in real analysis, there is also a pure math subject. So it goes to Differential and Integral Calculus -> Real Analysis -> Maybe Analysis on Rn -> Maybe Metric Spaces -> Topology. Of course all of these you'll learn in an undergraduate or graduate course in math.
Where did you pull out closure? Also, in my opinion, what you're talking about it's not beauty of topology. What you're talking about is basic rules of proofs where you use definitions + axioms + theorems to lead to proof. Beauty of topology is generalization of what we call "distance" to other things which you no longer call "distance" in some cases. For all *metric spaces* you can use term distance. But there are also non-metric spaces with their own "topology", this is where beauty lies. Actually, properties of metric spaces which work for *all metric spaces* is also beautiful but in my opinion non-metric spaces more "mysterious" and thus, more beautiful.
I do agree that there's a lot more beauty outside of the concepts in this video, this is definitely just an introductory subject in the world of topology. I do think it is still quite cool for a beginner to see the abstract proofs used in this beginner subject. I remember when I first learned about it thought it was really cool and beautiful, even though there is definitely an infinity of more advanced and beautiful topics!
In general, can it be said that all concepts other than distance itself need to be reified via this foundational magnitude? This would imply that there are no units other than length, but only abstractions built on top of this core notion. This line of thinking echoes in dimensional analysis and geometrization of physics, where all physical quantities are fundamentally reducible to a foundational unit-typically length. In this interpretation, many other physical concepts (such as angles, areas, volumes, and even time) can be seen as derived abstractions built on top of the core concept of distance or length.
Really interesting idea, I'm not quite sure on whether we need to redefine other concepts, but I can tell you it definitely gets more abstract than this. In the video I just talk about metric spaces, but they fall under a more abstract space which is the topological space, which is defined even without a distance function! If you're curious about these abstractions I also recommend looking into measure theory, where the notion of volume is abstracted.
@@pi_squared2 When Set Theory is our foundation then everything must be abstracted! And yes.... Rubber-sheet geometry was once regarded as Higher Geometry but has substituted its class so that it can turn donuts into mugs..... now mythomagics and mathrobatics are center stage. YIKES!
@@rielblakcori971 As I showed, from d(x,x)=0, d(x,y)=d(y,x), and the triangle ineqality, I demonstrated that d(x,y)>=0. From that, we get your condition.
2 elements a and b of the set A are at distance 1 if the distance function outputs the real number 1 when aplied to a and b. There is no single function of distance, you can use any function from A to the real numbers that verifies those properties. The set A can be as weird as you want it to be, and different distance functions applied to a and b can output different values.
@@Xmodg4m3 Agree. There are infinitely many distance functions but the point I'm making is that author *escapes* from bringing any real example of such a function. We can pick 2 arbitrary points A, B on a plain and declare distance of 1 unit. But what property constites distance of 2 between B and C? The qq is how to build such a function.
@@kcg26876 A nice thing about math is that most of the time you are not working with concrete things, but with abstractions and generalizations instead. This is a neat thing as it replaces calculation with thinking. You are now for example proving things not just about the real space R^3, but about any abstract vector space. This makes so that when you are working with a really abstract and complicated object, but that can be seen as a vector space, you automatically unlock all the vast knowledge humanity has about vector spaces and can apply it right away, whithout needing years of research just for your example. He is not fleeing the need to define a concrete distance, as there is no such need to do so. It's just working the way up from a general concept to reach truths that can be applied whenever you try to solve any kind of real life problem involving any kind of notion of distance.
it would be so much easier if humans ditched mathematical notation and we just used simple programming. 3 lines of JavaScript could convey some of these concepts more intuitively. great video, not trying to bash. it's funny when i finally realize what the set notation is saying and how trivial of a statement it really is when expressed in object.method(parameter) "notation" haha
Great video! Your definition of Q is incorrect though. 1/3 = 0.33333… is rational
Ah you're right! I don't know how that error got through, thank you for spotting that mistake; I've added it as a correction in the caption of the video.
@@pi_squared2 You could've also said finite or repeating infinite.
@@pi_squared2Don't let it frustrate you too much. Making simple mistakes is an essential part of doing something complicated. Keep up the good work.
@@pi_squared2you also missed that in a metric space d(x,y)=0 if an only if y=x
@@marigold2257no need for that cuz he showed d(x,x)=0
Maybe reupload with the music turned down or off
Yes, sorry that it came out like this, I'm quite new to editing and making videos. I'll be reuploading a version of this in the future that is hopefully way better produced not only in the music but the voice-over as well!
Why is the music so loud
WHAT?
WHY IS THE MUSIC SO LOUD?!
@@turtle926Are you reta***d?
@@turtle926Hello, u r not okay
@@Joshs8707 I CAN'T HEAR YOU
“As a child I was interested in topology”
What child knows what topology is lmao I heard the term for the first time as an adult
Children with highly educated parents
@@axelnils I have highly educated parents, they never talked to me about math topics
I've heard of it at 16 I think. The thing is yt is good at popularizing math topics in recent years so that's why current young adults might be able to say such thing
It's also possible he meant that topological problems interesting him, but without the formal language. Wondering how different shapes relate to each other and can be turned into each other is something even a young child is capable of thinking about. Things like "how many holes in a straw" and "how many holes are in a coffee cup" aren't unreasonable.
@@axelsanchez5849 Well, highly educated is a very broad term. I mean if they're biologists, they won't ever talk about topology.
New incredible math channel is cooking...
Thinking about it, it kinda feels like Lie Algebras have more in common with a metric than they do with a vector space. The point of disanalogy really is that distances can be negative, and the triangle inequality becomes the Jacobi identity. I mean, given that the determinant - which tracks oriented area/volume/metric scaling under a linear transformation - is a Lie algebra, doesn't that make some sense?
3:50 nit pick: iirc the circle is the boundary, so distance equal to the radius. A disc contains all the points less than the radius. Same with sphere (boundary) vs ball (solid).
You're right, that's a good catch! I'll add that to the description of the video, sorry about the small error and thanks for pointing it out.
@@pi_squared2 It isn't a small error. You go on to talk about the interior of a sphere, but under your definition, there is no interior. The point/ball that you said was in the interior of the sphere was a part of your sphere. Good that you fixed it. I guess you could fix it by using a slightly different definition of "interior". English is easier to manipulate than math.
So, I guess you cleared that up a bit later when you defined interior mathematically...
That music ruined everything
Sorry about that, I'm new to editing and making these videos. I'll be making more with hopefully better audio levels, and plan to re-upload this with better editing in the future as well as my other current vids.
@@pi_squared2keep cooking dude, you got this
@@pi_squared2you can reupload again. But great job 👍
Super interesting learning about math concepts that I have yet to reach. Love the video, beautifully explained. 🙏
Glad you enjoyed it!
@@pi_squared2What is the name of the song in the video
Background noise not loud enough, as I could decipher some of the narration, thank you.
just the music makes the voice a little hard to hear clearly. but great content
great video with good visual explanation. keep going :)
Got recommended the Quadratic Formula vid just before, and now you post a new vid 👀
(also no bots can say first now)
Nice video, I would recommend trying to explain something like this without using your university level understanding of sets and set operations. Sets and subsets can be described simply in the beginning so it makes it easier for the viewer to understand. Anyways cant wait for your chanel to grow
Bound a unit and extend it with scalars
Why are you even doing topology if you don’t know what a set is
@@solidpython4964 Tf u mean doing topology? its a youtube video, also who said I dont understand what sets are, I meant for others watching because the set theory was heavy. Also understanding intresting topology topics is possible without set knowledge, sets are just a good language for describing relations that occur in sets
I thought sets and subsets were high school topics in most places... And everybody interested in topology will eventually need this kind of knowledge anyway. Cool animations can only go so far.
I got confused during the proof largely due to the subset symbol being used, which was the ambiguous ⊂. By analogy with
You should turn down the volume of the music, or at least setup the side-chain compression.
nice explanation
you got a subscriber
Music makes the video presentation better not sure why everyone else is complaining. I enjoyed it and I could hear you perfectly fine. I look forward to more of your videos thank you.
Glad you enjoyed it!
Honestly, if I'm gonna be real with you, this was not what I expected when I clicked on this. It was certainly intriguing for what it's worth, but for some reason I wanted more of the way that distance is measured? I might be tripping, I'm quite tired honestly, but did anyone else have a different expectation from this title? It's probably closer to the idea of (a metric) space rather than distance, but eh it's kinda the same.
Ah I'm sorry to hear that. For me, it was that mathematicians abstracted distance by creating the definition of a metric space with the distance function relying on 4 key properties outlined in the video. Those 4 properties are what abstract distance, but I apologize if the video was in anyway misleading.
@@pi_squared2 Don't stress over it, worst case scenario one could call it "clickbait", which honestly isn't that bad nowadays anyway. As long as the content is there, it's mostly okay!
Wow! What a great presentation!
Great video and topology (especially algebraic topology) is awesome.
great vid brother as always :D
¿Where do you actually explain what the title promised? I don't see it anywhere.
Sorry if it felt misleading, the abstraction of distance are the 4 properties listed under the distance function of a metric space. These 4 properties allow you to construct various different possible distance functions, some of them quite different to how we ordinarily measure distance in daily life.
You have earned a new subscriber! Very cool video! !! I'd only suggest to turn down the music a little more when you are speaking because it makes it a little hard to listen to your voice :)
I forgot I left speakers on max volume and the music just blasted lol
A distance function that only returns 0 is very interesting
hey brother can you please make a vid explaining basics of sets cuz it is hard to understand what the notation says
and i have not yet learnt sets so its hard to understand the vids too
What is a higher dimension shape? Please use a ruler and a square to demonstrate.
Thank You
Beautiful music, unfortunately, it tends to drown out your voice.
Maybe try re-releasing the video with the music only 25% as loud.
Thanks, I appreciate the content
Great video.
with the music I can barely hear you talk, awesome video other than that
The music was dope though
Great content
How do I turn off the background music on Math TH-cam? Every video seems to have it...
Besides that, great video!
I really like your music
Now these open balls could have a metric though, so how exactly did we abstracted/generalized metrics spaces? Just by avoiding speaking of Pythagoras theorem?
The open balls are really just a part within our metric space. The cool idea behind abstraction was the idea that you could describe distance with those 4 properties, which seem quite obvious but are actually enough to help you define objects like open balls and distinguish them from closed balls.
Before this video, I had never heard of topology (in maths), that is very interesting (even if I only understood some of the proofs and notation). Amazing graphics and animations, and explanations.
Question: What level of study do you learn about topologies in maths/education? I’m starting my 2nd year of college, so I assume it’s University level content?
Also, could you explain more about the needs/ applications of topology?
Really glad you enjoyed the video, topology is a super interesting area of math! Typically from what I understand it'll be first introduced in the 2nd year of a pure math degree, but you can also get exposed to some of its ideas in a course called Real Analysis in 1st year.
In terms of the needs/applications, there's a massive variety of subjects within topology, some which seem incredibly abstract and not applicable, while others are quite helpful. For example, topology is used to develop the theory of knots which has been used in quantum mechanics. Topology also plays a role in differential geometry, which is used quite famously to describe General Relativity in physics. As for the more abstract/inapplicable subjects within topology though, there's a rich history in mathematics of things which at first seem totally inapplicable becoming massively applicable/important in the future; so that also might just be the case with a lot of topology!
Ah ok, so I probably won’t come across it in my studies (I plan on studying a Mechanical Engineering degree), but it’s always something I can just research/look up about.
and wow, interesting applications.
Studying mechanical engineering probably wouldnt lead you to topology, topology is an advanced "pure" math subject, generally you'll see it after a course in real analysis, there is also a pure math subject. So it goes to Differential and Integral Calculus -> Real Analysis -> Maybe Analysis on Rn -> Maybe Metric Spaces -> Topology. Of course all of these you'll learn in an undergraduate or graduate course in math.
music is too distracting...
Where did you pull out closure? Also, in my opinion, what you're talking about it's not beauty of topology. What you're talking about is basic rules of proofs where you use definitions + axioms + theorems to lead to proof. Beauty of topology is generalization of what we call "distance" to other things which you no longer call "distance" in some cases. For all *metric spaces* you can use term distance. But there are also non-metric spaces with their own "topology", this is where beauty lies. Actually, properties of metric spaces which work for *all metric spaces* is also beautiful but in my opinion non-metric spaces more "mysterious" and thus, more beautiful.
I do agree that there's a lot more beauty outside of the concepts in this video, this is definitely just an introductory subject in the world of topology. I do think it is still quite cool for a beginner to see the abstract proofs used in this beginner subject. I remember when I first learned about it thought it was really cool and beautiful, even though there is definitely an infinity of more advanced and beautiful topics!
In general, can it be said that all concepts other than distance itself need to be reified via this foundational magnitude?
This would imply that there are no units other than length, but only abstractions built on top of this core notion.
This line of thinking echoes in dimensional analysis and geometrization of physics, where all physical quantities are fundamentally reducible to a foundational unit-typically length.
In this interpretation, many other physical concepts (such as angles, areas, volumes, and even time) can be seen as derived abstractions built on top of the core concept of distance or length.
Really interesting idea, I'm not quite sure on whether we need to redefine other concepts, but I can tell you it definitely gets more abstract than this. In the video I just talk about metric spaces, but they fall under a more abstract space which is the topological space, which is defined even without a distance function! If you're curious about these abstractions I also recommend looking into measure theory, where the notion of volume is abstracted.
@@pi_squared2 When Set Theory is our foundation then everything must be abstracted!
And yes.... Rubber-sheet geometry was once regarded as Higher Geometry but has substituted its class so that it can turn donuts into mugs..... now mythomagics and mathrobatics are center stage.
YIKES!
the music it is too loud but great vidfeo
Nice video👌👍
Thank you so much!
Whats your background?
Slight bug bear of mine, property 3 is unnecessary as it follows from 1,2, and 4 in the following way:
0=d(x,x)=0.
also he should have included the property 1 as follows: d(x,y) = 0 x=y
Otherwise we have no metric but a pseudo metric
@@rielblakcori971 I believe that was encoded as d(x,x)=0.
@@mathunt1130 those are not the same since you can only imply right hand side to left hand side by his definition but no equivalence
@@rielblakcori971 As I showed, from d(x,x)=0, d(x,y)=d(y,x), and the triangle ineqality, I demonstrated that d(x,y)>=0. From that, we get your condition.
@@mathunt1130 yes but you still didn't show why you get from d(x,y)=0 that this implies x=y, that's the whole point
You provide properties of distance as a reduce function but don't define it. How do we rank distances? How do we define a distance of 1 unit?
2 elements a and b of the set A are at distance 1 if the distance function outputs the real number 1 when aplied to a and b.
There is no single function of distance, you can use any function from A to the real numbers that verifies those properties. The set A can be as weird as you want it to be, and different distance functions applied to a and b can output different values.
@@Xmodg4m3 Agree. There are infinitely many distance functions but the point I'm making is that author *escapes* from bringing any real example of such a function. We can pick 2 arbitrary points A, B on a plain and declare distance of 1 unit. But what property constites distance of 2 between B and C? The qq is how to build such a function.
@@kcg26876 A nice thing about math is that most of the time you are not working with concrete things, but with abstractions and generalizations instead. This is a neat thing as it replaces calculation with thinking.
You are now for example proving things not just about the real space R^3, but about any abstract vector space. This makes so that when you are working with a really abstract and complicated object, but that can be seen as a vector space, you automatically unlock all the vast knowledge humanity has about vector spaces and can apply it right away, whithout needing years of research just for your example.
He is not fleeing the need to define a concrete distance, as there is no such need to do so. It's just working the way up from a general concept to reach truths that can be applied whenever you try to solve any kind of real life problem involving any kind of notion of distance.
Can you share video code?
Yes, will be dropping video code for my vids soon in the future!
it would be so much easier if humans ditched mathematical notation and we just used simple programming. 3 lines of JavaScript could convey some of these concepts more intuitively. great video, not trying to bash. it's funny when i finally realize what the set notation is saying and how trivial of a statement it really is when expressed in object.method(parameter) "notation" haha