Point about Points - Numberphile
Vložit
- čas přidán 31. 07. 2013
- Part 1: • Number Line - Numberphile
An extra bit after this: • Numberline (little ext...
More links & stuff in full description below ↓↓↓
This video features Simon Pampena, Australia's numeracy ambassador.
Ilustration by Pete McPartlan
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Patreon: / numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9
Numberphile T-Shirts: teespring.com/stores/numberphile
Other merchandise: store.dftba.com/collections/n... - Věda a technologie
"Without my feelings I can't do math because it is all very emotional" I love this line
"Humans... and mathematicians... we are really analogizing"
Mathematicians are not humans confirmed
I noticed that too :P
+Hoo Dini
Mathematicians are aliens.
It's a step up from humans
@@rewrose2838debateable :D
@@oz_jones Yes. I’d say: ”Quite a few steps.” :D.
1:00 "Why is this a problem??...Why is this a problem?!?..Why is this a problem!?!"
Math.exe has stopped working
"Are we starting to come towards philosophy here?"
You've been there the whole time, yo.
Right or wrong, I must admit that he has a point.
Shamak XD I GET IT
illuminati confirmed
+Shamak
You do have a point there.
But if he has a point...
is it really worth anything?
69 likes.
3:18 Did a well respected mathematician just say "thirty-twoth"
+BlueUmbrella I paused the video and came down here to post the same thing, only I was going to spell it "thirty tooth" for fun.
"dirty tooth"?
"Suddenly something make sense in maths.
[BUZZER NOISE]
It doesn't make sense!"
NICE!
Simon Pampena is a wonderful addition to the Numberphile regulars! I love his enthusiasm and demeanor. His enthusiasm is infectious...
"... this is already an analogy"
That's brilliant.
i love this guy
I love him too !
I love him too too!
I love him 3. And 3 is a special number
the nth person loves him n.
I hate this guy
"without my feeilings I can't do maths". Wonderful! I love maths and I love the way this guy does maths and I like Brady's insightful questions. They are always to the point, so to speak....
I first watched this video when it came out and I was still in secondary school and it fascinated me. Now I'm studying for my PhD in maths and measure theory plays a heavy role in my work!
"We know numbers, they're never gonna change." - Romans
Well they haven't.
@@AloisMahdal yes they did
@@brennonstevens467 Numbers didn't change. They way they're represented changed. Numerals are not numbers.
@@Jivvi nerd
@@Jivvi The numbers themselves also changed. We added zero. We added imaginary numbers. We added several dimensions
Interesting to think about. If points have 0 length, and a line is made up of a succession of points, then how does a line have a length greater than zero. Seems a little paradoxical.
its because there's infinite points. infinite is weird like that
Makes no sense to me... a line is not made out of points. A line is a dimension, whereas a point is just a representation of a location within that dimension. An infinite amount of locations isn't the same thing as a dimension.
en.wikipedia.org/wiki/Lebesgue_measure
en.wikipedia.org/wiki/Lebesgue%27s_density_theorem
I personally think this is because there is a countable infinte amount of rational numbers but an uncountable infinte amount of irrational numbers(did the proof during lectures, i'm sure you'll find it online somewhere).
So I think there "infintely more" irrational numbers than rational numbers thus they don''t really matter lengthwise.
Not sure if this deduction is correct though xD
A single point has a measure zero. (It can be seen that given any positive real number, the measure of a single point is less than that number, because the point is contained in an interval of that length; because in real numbers there are no non-zero infinitesimals, it follows that the measure must be zero.) Now let's see what is the measure of countably many points, such as of the set of rational numbers. Because the set is countable, it means that its members can be put in a sequence. So let's cover the first point with an interval of length 1, the second with an interval of length 1/2, the third with an interval of length 1/4, the next with length 1/8, the next 1/16, and so on; the total length of this infinite sequence of intervals is at most 2. (I say "at most", because the intervals can - and do - overlap. But here's the trick: I have started with an interval of length 1. I could have instead started with an interval with an arbitrarily small length ε, and the infinite sequence would still cover all rational numbers, with a total length of at most 2*ε. It follows that the measure of rational numbers, and indeed of any countable set, is zero.
Guys, just because it is non-intuitive does not mean it is incorrect. Infinite sets and countability are well-established mathematical concepts. Maybe what is preventing you from being intuitively satisfied with this notion is that you are constantly trying to relate these concepts to "real" or malleable things. When you manipulate different sizes of infinity strange things happen. And COUNTABLE infinities (infinite sets that can be enumerated by the natural numbers) are quite different then a "dense" infinity. These concepts very real in mathematics, but they are not anything that you can physically pick apart or experiment on. This does not make it untrue.
It seems to me that Simon is the one who did what you're suggesting "guys" did. It was him who started talking about "removing" points, which IMO does only make sense in physical world...
I don't see how he's talking about anything related to the physical world. He's talking about the number line as the set that contains all the real numbers and talking about how we define the size of subsets of real numbers. His point is that if you take any segment [a,b] from the number line which is the set of all real numbers between a and b, and then define a set that contains all points from this segment except from a countable amount of them then this subset has the same size as the segment [a,b] which is b-a. You can look at the wikipedia article on Lebesgue measure but i really don't understand why everyone thinks it has anything to do with the physical world or pure philosophy when it is 100 years old well established math.
Yeah unfortunately this video started abruptly with no ground work from a previous video. For example we are suddenly talkin about countable infinity and uncountable which is really all this boils countable infinity and uncountable infinities, which is really what this boils down to. We just need a Starter video to introduce this topic, or else stay on the topic of points and continuous space or lines, and talk about integers rationals and density in another one, or else make this video over 30 minutes long.
"....No, Im just weirded about you removing my feelings" LAWL
There is no spoon.
I don't now why you guys have a problem with this. I trust this guy. He looks like math.
It is indeed very, very fascinating!
If we look on numbers as points on the number line, every such point is not only infinitesimally small, but so infinitesimally small that it _can't get any smaller_, and if it can't get any smaller it has to be nothing... which is also why there between two such points, no matter how close together they are, always exists an infinite amount of other points, because the distance between two points can't be smaller than a point, and if you can fit _one_ point between them, you can also fit an infinite amount of points.
So wouldn't this make the number line infinitely _short_ instead of infinitely long, stretching no more than the infinitesimal distance between the two points "minus infinity" and "infinity", with all other points fitting in-between?
The problem here, as I see it, is that these two points "don't exist" because there is no such thing as "The Biggest Number" ("infinity"), which is why the ends of the number line race outwards in pursuit of that unreachable "biggest number", making the number line infinitely long.
When we count, "1, 2, 3, 4, ..." and so on, we traverse through an infinite amount of numbers each time, which is a tremendous progress to say the least, and yet we are the exact same distance away from "infinity", so at the same time we haven't moved at all.
This fellow makes me cry per beauty. He talks with a very huge passion and power !
Would removing a point from a number line be kind of like cutting the number line?
+Circuitrinos Nope, points have no length.
rango3526 If you have a piece of paper with a number line on it, and you cut it in half with scissors, did you remove something with length? No, that would require two cuts. In a way, removing a point is like removing the connection between the number line. If you cut it at five, each of the two parts of the number line still have a 5, but they are no longer connected, so you have effectively removed a point with no length.
Circuitrinos Oh ok, I misunderstood your question. I suppose it would be like that, but I'm not sure how much of an effect that interpretation would have versus the "the point is just gone" interpretation.
I don't know if that way of thinking has any use, I was just throwing it out there.
Circuitrinos Ok, cool.
Moreover, if I have three ducks and I destroy the word three from the language, how many ducks do I have? THREE!
This series is one of my very favorite of all Numberphile videos.
So many people in comments think they know better, when in reality they don't know what they are talknig about
Алексей Салихов so?
Pawel X Well, the problem is, either they have the audacity to think that they know math better than the lecturer does. Or they think that math is a subset of common sence. Which it is not.
Алексей Салихов Talking about things I don't understand is the only way I know how to learn.
+ShonkyLegs There's a difference between talking about things you don't understand and what most of the people in this comments section say. It's okay to talk about things you don't understand as long as you qualify that you don't understand. For example, you could be like, "I don't understand why he said [this]." Or, "I thought it should be like [this] because of [this], but he says differently. Why is that?"
Most of the people in the comments section are like, "Wow, he's totally wrong because [this]" even though they're perpetuating a misconception or complete nonsense.
+Алексей Салихов I liked this because you have an anime style avatar. Here, I'll give you a russian style smiley )
James "bradey you broke math. Stop that"
lol, which video is that from?
Zero Factorial. Just shy of the 3 minute mark :)
If a point has no lenght, removing it from a line doesn't leave a hole. To say that a line is an infinite number of colinear points is to imply that every point has an infinitesimal lenght (1 - 0.999...). I think the core of the problem here is that we're treating a point like a spacial thing, while it is not. It is a concept, an abstract tool that we use to locate things. Or did I miss the point?
Simon's "why is that a problem???" sequence at 1:03 is probably the most intense thing I've ever heard in a Numberphile video.
Bro was perplexed when he said "if i take away your feelings" like "why would you do this?!?!?"
Zero added to itself finitly many times is zero, but zero added to itself infinitly many times is infinity*zero, which is undefined (as a limit it could be any number, just like 0/0). Don't lie to me!
it sounds like your argument is
"3*0 =0 thus 0/0=3 but 0/0 is undefined therefore 3*0=/=0"
0 add 0 infinately many times is still 0 just like how any number multiplied by 0 is still zero
The geometric definition for a point is that it has no size and determines a position in a system, that system being the number line. Therefore, I don't like to think of numbers as objects that can be manipulated but more as destinations or positions. You don't have 3, you are at 3.
but how do you define 3 ? I guess it would be the point at 3-0 excactly but I meant to suggest that 3 is undefined it has only one significant number I mean it could also be 3.1; 3.4; 3.49... if you wrote these numbers with only one significant number. And defining 3.000000... is impossible. That was the contradiction I was wanting to convey.
+damien olivier I guess my question was where are you when you say you are at 3?
There's another numberphile video in which someone says we define a number by all the space to its right and left on a number line
I'm beginning a course in linear functional analysis and I just started looking at measure theory, so watching this guy talk about it and seeing how excited he gets really makes me happy! Man.... I wish I could just sit and talk with him about this stuff.
Please never stop producing these!! Im blown away every time.
I don't think i've ever seen a numberphile video with quite so many people straight up disagreeing in the comments.
***** Then you haven't seen the -1/12 videos ;)
Penny Lane the most ridiculous summation series result in Mathematics - and yet used in String Theory and other areas of Physics
He looks like the young version of the old professor on periodic videos :P
"It's an analogy." I like that. Especially in regards to number lines.
"But what's the truth?"
[Explains truth]
"Can't I have an analogy?"
"Think of numbers as points on a line . . . "
when I draw a line on paper, I draw a representation of a line. Same thing if I draw or print a tree, in actuality it is colored paper, but still Id say "thats a tree", for it models reality or rather a sense of reality most humans share. As the guy who was filming pointed out, numbers (and math) are not objects, though we like to think of them as such, since our thinking is very object oriented. (this is why I like thinking in classical programming languages while descibing systems)
I wish he'd go a bit deeper. I already thought of the number line like this. When I learned about points they were defined as that which as position, but no size. A line is that which has length but no breadth. Of course the number line has no "next number:" no line has a "next point." Of course taking out a point, even an infinite number of points, does nothing to the length of the line: points have no size to remove; they only reference position. Simon here was really excited about something and I've gotta think that it has to do with the further implications that he glossed over/sped through at the end there, because this whole business of removing points seems just painfully obvious. I just wish he spent more time going into what came out of this revelation that "almost broke maths" and how that happened and less on elaborating on what is essentially a direct result of definitions.
Look at Measure Theory and unmeasurable sets. While you're at it, when he mentioned 'almost broke maths' I think he's referring to the Banach-Tarski paradox
Maybe a point is infinitely small?
That sigh at the end is the icing on the cake for this whole video because it just goes to show how incomprehensible numbers really are
Thought it'd interest you guys to know that I'm holding a LOT of points in my hand right now.
Ironically, this was actually one of my favorite videos on Numberphile. In my eyes, maths is not some magnificent force conducting the ways of the Universe and guiding us all in some sort of grand scheme, it is something that mankind has create to better understand the world around them. In short, there was no maths before man, but there was man before maths. I really suggest you watch the follow up to this video as that one seems to drive a point home unlike this one.
Anyone who is interested in this should look up cardinal numbers and ordinal numbers because that is what he is really talking about.
He is actually talking about measure theory and Lebesgue measure in particular.
It's weird how something most of us gave for granted is actually a LOT more complicated (to the point of being very difficult to really grasp) than what we thought it was. From the comments I see a lot of people don't like to be told "nah, what you thought you knew it's actually quite imprecise and not really all that correct". Seems like a lot of people don't like to have their beliefs challenged ;)
Great video!
This guy is a math fanatic! Pretty cool to see someone so into what they do.
Fundamentally here though, it seems that those asking this question are treating numbers as though they're discrete, when they're continuous.
This is the mind set of those before asking the question. Can you explain why 1 has any relation to 2 or 0? Why is it we think numbers are connected in any way at all? The our main connection is cardinality which is the position compared to another number. You must change how we think about numbers in 0-dimensional space before thinking about the 1-dimensional space of a line and how we get to the other.
that can neither be proven or disproven with our current knowledge
9:21 Three is a magic number. Yeah it is: It's a magic number.
This discussion is absurd for similar reasons that analysis of infinity does. With infinity we say it is a process without end but then go on to treat it as if it does end, it is viable mathematics because it doesn't lead to contradiction - but that doesn't mean it is a viable 'reality'. Here we have the idea of a point as a place in a continuous space ... which clearly can never be removed by definition.
+Hythloday71 There is nothing absurd about it and understanding how to work with infinite sums is essential to understanding how the universe works. The video on why 1+2+3+4+... = -1/12 is a prime example of that.
The ones that proclaim the loudest that mathematics and reality aren't the same tend to be the ones that need to shut up more.
It took me the longest to understand what this man was talking about, but I'm glad that I finally do. I erased my earlier comment because I couldn't understand seeing my ignorance.
These videos are about how set theory was implemented in the 1800's as a way to frame mathematics. We made very many assumptions up until that point that couldn't be proven, such as the width of a point on a number line. To fix this, we had to redefine numbers and what they meant, so starting with the null set, we defined numbers by value as as sets that contained the sets of the previous set, the null set being 0 and the set containing the null set to be 1, etc.
All-in-all, it's not a bad way to structure mathematics. As of recently though, I've been supporting Type Theory which, to a potential degree, could replace set theory as the foundation for mathematics one of these days, or at least a variation of Type Theory.
is this why limits work? There's really nothing there at (0,1) so we can say lim(x→0) of sin(x)/x = 1.
The idea of limits is separate from this, I know because limits are used in measure theory which is the stuff which this guy is talking about, limits in essence don't depend on measure theory for that reason.
I really like this guy--he's charismatic, but what he's saying in these videos is simply either going over my head, or it's nothing but philosophical nonsense. I'm open to the idea that I'm simply not intelligent or read up enough about this, but much like a child doubting their parents about the existence of Father Christmas, I either have to blindly swallow this as truth, or simply not accept it. The way I see it, as the person out of the shot was saying, the notion of measuring an abstract...thing doesn't make sense to begin with. It's like asking "how heavy is the average dream?"
GerikDT yeah, and the guy recording got it, but what he (Simon) was saying is that he made an analogy to represent the analogy of numbers.
Energy has a very subtle weight. So to determine the weight of a dream, you have to measure the amount of energy the neural activity is generating.
Watched most of this video before even caring to come to the comments, then decided I was going to express how much love I have for him. Annnnnd of course you beat me to it, so I guess I Love You too brother!
6:09 "The problem is that humans... and mathematicians..." xD I knew it!
Yup, totally not the same and never were
When he talks about numbers, he can't give you an analog in real life but those feelings he has, there is only concepts that you can understand but you can't express it in another way.
There is no spoon
If I threw a dart on the real number line between 0 and 1 what is the odds that it will land on say 0.7? Is it ε or exactly 0?
Zero. In fact, if u define the line as having infinitely small width, u d not be able to even reach the line...
+Roberto Vilela Obviously it is not an experiment we can do irl since there's physics making everything discrete rather than continuous. The point is that if we could do it it will definitely hit some number, so it is a paradox that we can have an event with an exactly 0% chance of happening actually happen.
There can't be multiple answers, it's obviously 0. ε for you is a limit. And you know that this limit is 0..!
Roberto Vilela Why that ? oO : your dart as a true width ; and your line as a position in space so your dart can find not only one but multiple position to hit the line. You are confusing with the geometry of the dart. But even if we defined it as none, physically, just a point, mathematically, we could make it find another point also no problem in there.
Let's say you throw a dart on a cube and we measure the distance from the left edge of the cube to the dart. In an event in which all results are equally possible the possibility of one result is
1/(number of possible results).
There is an unlistably infinite number of possible results therefore the possibility of the distance being 0.7 is 1/infinity. It is an infinitesimal, not zero.
I love the passion in his voice. Thank you for inspiring me to become a mathematician.
You're right. Set theory wasn't the single turning point where maths went from a collection of arithmetic and drawings to having deeper meaning. It's an anecdote used by the professor, to show that maths is bigger than intuition.
That shouldn't be very surprising, but apparently many people don't get this. Just look at all the comments redefining a "point" using only their intuition, or describing what they think "infinity" means.
i just removed a point...
nothing happenned
This is a silly question. A number line isn't a real thing which you could remove lengths from. It is a pleasant but nonetheless arbitrary representation of the language system we use to quantitatively describe the world in a way which would be useful to us. Since we have already defined what a "point" is, it is crazy to start asking "well what is it really? What if it was actually something that it wasn't? Wouldn't that blow your mind?"
3 isn't a strange number. 3 = "here is a sheep, here is another sheep, and here is yet another sheep." I could show that on a number line which counted the number of sheep. I could even let it be fractional if I had no moral qualms about dicing up sheep into little bits, but it still wouldn't make sense to ask "well how much of a sheep would be lost if I removed the point 3.5 from this number line?" Because that point doesn't in and of itself quantify an amount of sheep, it simply marks how much we count up to from zero before we stop counting.
What if a ruler then? Isn’t it a number line for measurement?
Videos like this, are the reason for CZcams. Loving it
"What happens when we take away a point? Well nothing, there is no next point, there is no hole per se" says Simon. Consider a point P as the centre of a small circle drawn on the surface of a large sphere. Now imagine that circle expanding, but P remains as its centre. The circle expands and moves over the surface of the sphere until it becomes a great circle then as it continues, diminishes again to zero at a position exactly opposite to P. As soon as it does so P vanishes, since the surface of a sphere can have no centre, so effectively you've taken away the point. Just before that happens when the circle is very tiny again but still in existence, we have a little hole which is the counterpoint of P and whose closure annihilates P , a kind of negative entanglement.
That was the most confusing talk on this channel. And what do you mean by "getting emotional" when he asks you a legitimate question???
There isn't a problem with any of this in my view. The numberline is simply infinitely dense. He seems to be neglecting the defined and undefined operations with infinity. (+∞) - (+∞) is undefined, so an infinite number of points cannot be taken away from the set of infinite points. He said 0 added to itself an infinite number of times was zero, however, +∞ * 0 is undefined. A point is simply infinitesimal, and +∞ - 0 = +∞.
Infinity is not a number, and you can't just apply it to things and say it's undefined. 0+0+0+0+...=/=0*infinity because infinity is not something you can multiply something by.
I noticed that part too when adding infinitely many zeros. Infinity*0 is indeterminate.
If one were to "take" a point, like 3.5, out of a number line, wouldn't it be akin to just breaking the number line like it was a ruler at that specific point, but placing it back together?
yeah you are taking nothing out of the ruler besides the idea that the point exists
"Without my feelings, I can't do maths." Hahaha.
this guy is bonkers
What if you just say the real line contains points but is not made of them? Then you could say if we removed a point it would be exactly one point smaller even if it remained the same length
He's passionate mate. People like this help us move forward.
'Without my feelings I can't do maths' - I love these videos so much :D
He explained this wrong. 0 + 0 + 0 infinitely many times doesn't necessarily equal 0. Length of each number between 0 and 1 is 0. Length of all these numbers is 1. Not 0.
Also, the strangeness of immeasurable sets comes from Axiom of Choice, not numbers themselves. Axiom of Choice is balls-tripping weird thing, you use it at any branch of mathematics you want, and you get something that everyone will tell you is obviously wrong. You get sets without measure, as described here. You can split a sphere into 4 parts, rotate them a bit, and put them together to form 2 new spheres identical to the original(Banach-Tarski). You can prove that x = f(x) + g(x), where both f and g are periodic, that is, f(t) = f(t+p) for every t, where p is period length of f. You can get events that have no probability.
Measure is countably additive but (0,1) is uncountable
this is pointless
+SmileyMPV
Point taken.
This takes me back to my "measure theory" course during the final year of studying maths at uni. There was a lot of weird stuff going on in that!
I agree. I wish James and Simon would make another video together trying to answer some of our questions. But then, some of our questions are redundant. Like Brady's. I think they owe us some attempt at resolution. It's pissing me off just as much as the next guy but I do recognize that there are some polite people here trying to help us understand.
Thank you.
One thing I have noted in numberphile videos, they always treat physical space and time like numbers and make a problem out of it. Ofcourse you can't. it's pointless because one is physical and the other is abstract. We cannot draw a parallel between them at infinitely small scales. Infact, this is what exactly what plancks unit is for. To tell us where to stop. Normally I enjoy numberphile videos, but this one is a total letdown.
3.5 is not a point its a length that ends in a point.
5:05 - What he is saying is so important. We have to question every assumption we are making. It is true, a number is infinite and infinite isn't a number. As such, "removing" points from the number line does not make sense with what a number line is suppose to help the mathematician do. It helps to analyze a thought process in a physical sense and then ask ourselves questions which seem incompatible with what the number line represents. Although it looks like we create "dots" and draw "lines", these are simplistic representations for the sake of understanding mathematical relationships. In other words, we must find ways to challenge the way we think. So, if you agree that there is a paradox here then you would be right. I would definitely recommend reading a book titled, "How Mathematicians Think" by William Byers. This would shed light on why this is a problem. 5:25 - "If i remove your feelings, your size doesn't change..." that is correct and that is the point. Likewise, i agree that you can't do maths without feelings. "It's all very emotional."
I was glad to hear Gödel mentioned. I can't believe there are no numberphile videos about Gödel.
It seems to me that Brady has a better understanding of the whole issue than this guy. Numbers are not points! How can you "remove a number"? Maybe I just totally don't get something, but such mental tricks remind me of scholasticism and apologetics in religion, which rely on twisting abstract concepts to "prove" things.
Its not. Basically a pony is a 0 dimensional object. Like how a true line has no width.
This video is not about trying to find a rigorous definition of a point or a length, those problems have been solved with the formulation of set theory.
The prof is trying to explain that this is the turning point where maths became truly fascinating. It became apparent how little mathematical objects are related to (/bound by) physical reality or human intuition. Instead of a tool, maths became the philosophy that explores the fundamental logical structure supporting reality.
I like the enthusiasm in this guy. He is full on.... :)
Brown paper and sharpies is how this channel works. It wouldn't be numberphile if it wasn't brown paper.
Yes. This video conveys that he is passionate about something, but not precisely what his passion is.
Stuff like this is why I picked engineering instead of maths, where things can be measured, numbers have units, "close enough" is good enough, and everything infinite can be truncated to something finite
1:01 as it was for me.
*Simon talking about doomsday and how it's horrible*
Braddy: "Why is this a problem?"
Simon: "Why is this a problem?! WHY IS THIS A PROBLEM?!"
He’s trying to explain Measure Theory which is probably a math major’s worst nightmare. Even after studying mathematics for like 4 years, measure theory is real difficult to understand. It’s not pointless, it’s just difficult.
True. However, it is not bad as you make it out to be. I'm a Stats major with a maths minor and I have taken a course in measure theory. I have however, read Terry Tao's book on measure theory and I do feel like he does over-complicate some very rudimentary concepts which can be easily explained. I think it's the way that it's taught, not the actual material.
김현빈 Then how can I, someone who completed sixth grade last week, understand this perfectly and find it highly intuitive?
I probably don't even remember watching videos like this that made me understand maths better
I laughed so hard at that part! It sounds like a Tumblr night-post.
The initial supposition is faulty. No matter how you want to obliterate a number, it still exists. You CANT remove any number from the line and erase it from existence.
He's not claiming it's removed from existence, he's saying how does it change the length.
If you ask what is the price of 3 objects that you have, and then you remove the object and ask the price of what you have left, you don't object by saying 'yeh, but the object STILL EXISTS!'
This is actually the most interesting video on the channel, and it is the most disliked. Why? I don't understand.
Part 2: Maybe the reason I so very much like this video is because it introduces us to the more abstract side of maths whereas, like Simon repeatedly brings up, we are for the longest time stuck in this preconceived notion that the mathematics is the most concrete academia there is. This video reminds us that that is most certainly not the case and, though Simon and most mathematicians will likely disagree, maths may only be as true as we allow it to be.
The number line is utilized like a tool, much as any fixed tool, but paradoxically it's an abstraction that, for all of its simplicity, has yielded conundrums that have led to revolutionary discoveries in math. This modestly approaches what Simon Pampena is trying to convey, I believe. It's like the question of, if the ground's holding us up, what's holding up the ground?- a query which can destroy the foundation underpinning knowledge itself, or, by logic, open up a new understanding of reality
very interesting! thanks numberphile. we love you :)
I don't think that's what he meant. This does have real life applications, but that is advanced mathematics, and Numberphile doesn't teach maths, it points out quaint trivialities that are of interest to lay people. The questions presented in this video led directly to measure theory and Lebesgue integration, which have produced advancements in pretty much all areas of mathematics involving modelling anything in the real world. I'm guessing that's a lot more than anything philosophy has to offer
Discrete maths is definitely something maths enthusiasts should look into. It's brilliant.
I don't remember the kids in my high school math class being all that upset, puzzled, etc when the teacher said that the dot of chalk on the blackboard wasn't actually a *point*, since points are infinitely small.
Did he say "thirty-tooth"? xD love it!
My brain exploded into fraction bits after watching this
"Oh suddenly math makes sense"
BOOOOONG
*buzzer*
he just gave me existential crisis
Because it lead to people to ask the question "What can be computed?" It turns out that only very boring things can actually be computed (at least in my opinion). To answer the question they had to formulate exactly how computation is done, and in doing so, they made it concrete enough that we can build machines to compute for us.
Not sure if you read the comment you just replied to. He did acknowledge at the end that he "did not explicitly state that integers are within the set of real numbers". In other words, integers are a sub-set of the real numbers. In simpler terms, integers are real numbers, but real numbers are not necessarily integers.
You can have a number line with countably infinite points removed because number lines represent unaccountably infinite values.