Every Math Paradox Explained - FULL Video

[@francomiranda706]

Kinda useless to talk about the hairy ball theorem without bringing up the motivation for it (flow fields over spherical spaces). The "hairy ball" thing is just a funny name, the actual meat of it is in, for instance, explaining why earths winds can never be in a non turbulent state.

[@CutleryChips]

Ahem, I know the real motivation is to find the optimal way to cool those jewels

[@willhenry2567]

Thanks for explaining the actual meat of the hairy ball theorem. I thought it was something else ya know

[@francomiranda706]

@@willhenry2567 yeah! It doesn't stop there, theres plenty of interesting things that come about by knowing that flow fields over a spherical space must contain these "tufts" or "spirals". Interestingly enough, this isnt true for flat or donut shaped spaces, just spheres!

[@anuragneelam8527]

What does non turbulent state mean?

[@ryuzaki5491]

It isn't even a paradox

[@jadeeliss]

the elevator paradox doesn’t seem like a paradox, it just sounds like logic and common sense. clearly if i’m on the top floor the elevator will be coming from below, there is only below to come from

[@sinnfuly]

did you pay attention

[@jadeeliss]

@@sinnfuly i was trying my hardest to understand what makes it a paradox

[@patrickmanasco5905]

@@jadeelissIm in same boat

[@juanguerrero2692]

The paradox suggests that in a building with only 1 elevator, the elevator will usually already be on its way to you. Expected behavior where it wouldn't already be going to you may be in a case where you're on the top floor and you queue the elevator, but someone on the bottom floor (unbeknownst to you) queued the elevator before you, making it go down the building first to the bottom floor, then the top. That is possible and very obviously reasonable. But what the paradox states is that in a real building with various floors calling that single elevator, it's very rarely the case that you'll be the one who makes the elevator change direction because the dang thing spends so much time going in one direction bc as it goes up (or down) more floors queue to go up (or down), so the elevator just keeps going in that same direction out of convenience.

[@Velereonics]

​@@sinnfulyI have nearly the same question as him. I think part of the confusion comes from one that the picture he shows.Is a person on the very top and the very bottom, vut then, with his words, he's just near to the top or near to the bottom and that kind of matterst And then he's talking about like. What are the elevators doing? As you're approaching it, as opposed to like at an instant and also like I don't think it even makes sense, because you assumed that the elevator had started at the bottom in the first place like that is its rust state, therefore, it lets people out when it's coming down, but that's i think supposed to be an arbitrary starting condition for the problem, and he stated it like this is how it is, and it's like.Well, no there could be a whole variety of ways that elevators are made, but he doesn't state that explicitly and so my brain was like bogged down and all this stuff and so I don't get it either and to me, it sounds wrong

[@JNDeaux]

The problem with the Monty Hall problem is that it presupposes I would not like to win a goat.

[@zakvargo9147]

Underrated comment 😂

[@1grapefruit]

The Dichotomy paradox isn't even a paradox, it's false. Limits in math exist for this exact reason. (lim)

[@julianthomas6362]

Is it just me or is their example for the Simpson paradox actually mathematically impossible. Because I don’t think 30% and 40% can combine to make 60% no matter how weird the data is

[@ieatgarbage8771]

Right. In order to have this paradox work, the treated chance for one group has to be lower than the untreated chance of the other.

[@ktushy4727]

Yeah, it can only be between 30 and 40

[@danielyuan9862]

Yeah Simpson's paradox only works under certain conditions, and the example shown in the video don't satisfy them.

[@IkiDimit]

I think the presented image is better for intuition than given explanation. If you consider two groups of dots separately and connect both of the groups individually to create two linear functions, both functions will show Y rising as X rises. Now, if you connect two groups with a single line, the function will show Y falling as X rises.

[@degle1923]

Yeah, if we assume that each of the 4 groups (treated/untreated M and treated/untreated F) is made up of 100 people for simplicity, the combined total number of treated people is 100 + 100 = 200 while the ones that survived is 70 + 80 = 150, which is 75% of 200. The math ain't mathing in my opinion.

[@simpli_A]

Erm actually, the lamp would be off, since turning it on and off infinite times would most definitely cause the light bulb to burn out 🤓👆

[@ArchoDarko]

it's a physics defying special paradox lamp that never gets worn out.

[@SirNobleIZH]

​@ArchoDarko but the circuit breaker isn't physics-proof

[@sweetypuss]

​@@SirNobleIZHthe circuit-breaker is also physics-proof.

[@Agent.036]

In the dichotomy paradox, by the second second the cheetah will have passed the snail.

[@sejin1]

yea are we that fing dumb. after the first sec they are only 1 m apart.

[@Verksento]

But the question still, how is that possible if the tiger need to catch the previous snail's position infinite times?

[@sejin1]

@@Verksento then the question needs to be changed to the tiger’s speed is 90% of speeds distance always

[@sensacoesinefaveis]

​@@sejin1 so if every single time the total tiger speed is 90% of the total distance then he can never catch up still

[@sensacoesinefaveis]

​@@sejin1the correct answer of the question is "for every second the speed of the cheetah grows in 1 factor of velocity thus moving itself in 10 in the number line of the distance and the distance of the snail grows in 1m, so in sec 1 they will be both on 10 and second 2 the snail will be on 11 and the cheetah will be on 20

[@huskadog7748]

I hate how "paradox" has lost all meaning in this video.

[@mikey-hm7dt]

You are taking one extremely particular definition and insisting everyone use only that one. Grow up, petulant child

[@sinnfuly]

paradox means something different in math the same way a "theory" in science isn't the same as a regular theory

[@huskadog7748]

@@sinnfuly what does it mean something vaugely cool? Or something vaugely different than what you may expect?

[@sinnfuly]

@@huskadog7748 "a statement or situation that appears logically sound but leads to a seemingly contradictory or impossible conclusion"

[@Cptn8993]

@@sinnfuly so basically, this video assumes that most peoples logic is so bad that people actually expect something else to happen?

[@QuikVidGuy]

A lot of these aren't paradoxes. Like "No, the area is still there, the pieces just overlap." Like... yeah, the pieces overlap, nothing is being contradicted.

[@rancidblock5615]

"A paradox is a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true."

[@MynameisnotGraey]

May I recommend jan Misali’s video “the 6 types of paradoxes”

[@TheAcidicMolotov]

@@MynameisnotGraeythere is one type. Something that is self contradictory

[@MynameisnotGraey]

@@TheAcidicMolotov and yet, in common language, the term is used to describe more things. But I’m not going to reiterate every point from that video, go watch it yourself.

[@OpossumYT]

that’s the type of paradox that isn’t really a paradox, just one guy getting confused and writing it down

[@Null2-irkutsk]

Picking this base colour for a "hairy ball" is devious work 😵😵😵

[@VLITMER]

0:00 the WHAT!

[@Idont_eatcrocs]

Hairy ball paradox

[@jacksonlee21]

0:47 wait…motion IS possible? in this economy?

[@lydon1767]

Zenos had the method, tap in

[@SentryFather]

Ah yes the hairy ball theorem, classic mathematics.

[@heatheretaithaha]

i don't think balls being hairy is a theory

[@jirayusangjampa1948]

if the snail starts 9 meters ahead wouldnt the chettah catch up in 1 second?

[@himehomemi5642]

Maybe that one wasn't explained correctly? After 2 seconds, the cheetah would be 20 meters from the start and the snail would only be at 11 meters? I think what he meant to say was that the cheetah can run half the distance of the snail each second. Therefore the cheetah would always seemingly be behind the snail because it is always running a fraction of the distance to the snail. I think on a graph this would look like an asymptotic curve. Where a line curves towards zero but never actually touches the zero point.

[@katana2k]

Im pretty sure he butchered this. A cheetah that runs 10 m/s could never catch a snail that goes 1 m/s and is 9m ahead? Obviously this is stupid

[@tylerduncan5908]

26:34 this "paradox" is a great example as to why n×∞ - ∞ is undefined, because even though it seems like you would get n-1×∞ it really always depends on the context.

[@LowellMorgan]

Guinea pigs are pretty round and I’ve been able to comb them pretty smooth

[@334nickg]

8:34 do I need to teach you kindergarten statistics? BOOOOOOONE!?

[@alika.dudu.tshela]

Oh captain holt and kevin🤭

[@duckner]

15:52 it's parallelogram. not pallelogram

[@gingersperg]

The first one makes me remember trying to lick a perfect hemisphere on an ice cream in a cone

[@TaylorGolmon]

Respectfully, the presenter might be mistaken.

[@jackkalver4644]

A constant flow of hair may be impossible, but I’ve seen a smooth flow online.

[@OtakuUnitedStudio]

What you called the Dichotomy Paradox is actually the Paradox of Achilles and the Tortoise. I don't know why it was explained with a cheetah and snail instead of the usual figures, but that's less important. It was also "disproven" by Diogenes of Cynic who simply pointed somewhere and walked to it to show how ludicrous it is. There's also a practical solution to it - since the next halfway point is half the distance to the destination, so to is the amount of time needed to reach it half that needed prior. Both follow the same log limit function until they reach 0. Some people argue that even with a mathematical solution, it's "the metaphysical argument he was really making for it that we still need to contend with." That sounds like psuedophilosophical nonsense to me.

[@Match348]

Hilbert’s hotel “paradox”: infinite rooms are… infinite! And the number of guests they can accommodate is… infinite! Mind blown 🤯

[@kamataros5172]

The paradox isn't simply that it accommodate an infinite amount of guests, it's the fact that it can do so despite already being fully occupied.

[@Match348]

@@kamataros5172maybe that’s because definitionally, an infinite hotel can never be fully occupied?

[@kamataros5172]

@Match348 also not true. Thats the point of the whole thing, it's a paradox, because "infinity" is a little wonky in general. Of course "occupuied" is not properly defined in any of these videos, but in this case it means "if you pick any natural number and check the room with that number, it will be occupied". That's also the reason why we can't just tell a new guest "hey, go walk this hallway and check the rooms, if you find a free one, you can have it" because by definition "the whole hotel is occupied", there is no such room. We have to make one vacant, and specifically we need to determine which room to vacate. So we say "alright, number 1 will be free for you, everyone has to increase their room number by 1". In any normal hotel, you'd throw out the guest with the highest room number, but hilberts hotel doesn't have a highest room number, so the chain of someone knocking on the next door saying "a new guest came in, we have to change rooms again" never ends. TL:DR if a hotel is completely occupied and you want to serve a new guest, you have to vacate a room. In a normal hotel, that means you throw someone out, in hilberts hotel, you don't need to.

[@COWHATE]

9:52 bro that's just called being an alcoholic

[@cam5816]

One day I’m going to make my own paradox 😌

[@davidhopkins6946]

The paradox of the future: Does the future exist? If yes, then how can we have any free will? If no, then why do we plan for and worry about tomorrow? If the future doesn't exist, but will exist, then that's only saying that the future exists in the future, but then you run into circular reasoning, like trying to define a word by using that word in your definition. Also, whether the past exists or not does not form a paradox like trying to debate whether the future exists or not, because the present is the sum of the past and cannot exist without the past, but the present CAN exist without the future. (i.e. in theory it could happen that one nanosecond after now the whole universe will suddenly cease to exist, that is if there is even such a thing as a "nanosecond after now" at all)

[@zacknattack]

​@@davidhopkins6946at the present moment, the idea of the future, or of possible futures, exist. the fact of the future does not.

[@cam5816]

@@davidhopkins6946 And here Hopkins’ Future Confuzzlement has led me to reject my dreams for not yet having been already achieved. I will never be like the rest. Nothing is for certain except that they already have their paradoxes which means that their paradoxes are real. I am too late... I am but a small dumpling in an infinite and untamable universe 😔

[@gosie-z]

No you don't! In your dreams!

[@DiscipleGames]

Anyone who watches this video should definitely watch the Jan Misali video immediately after

[@piotrborowski2716]

Elevator - not going up or down, but APPROACHING to YOUR FLOOR from above or below. If you come to an elevator in random moments, there's 50/50 it goes up or down, regardless the floor (skipping beeing idle). But, if you want to enter an elevator and ask yourself, if it'll stop on your floor coming from above or below, the chances varies depending on your floor.

[@ASAP_Kenny]

The cheetah/snail one makes no sense. If car A is driving 100 mph and car B is driving 10 mph with a 100 mile head start, Car A will catch up in just over an hour

[@josephinehendricks]

Its not really about if it catches up, its that its impossible to find the exact moment that it did "catch up". Its pretty weird to think about

[@alexandrecravo8618]

​@@josephinehendricks Ah yes because a thing that never happened in the history of everything is a animal cathing another

[@LagartoChicloso]

@@alexandrecravo8618u lack the exact thinking, it isnt logic, its exactitude, and we cant know

[@dwaynekeenum1916]

@@alexandrecravo8618u slow

[@damdevin1]

That is probably the greatest way to start a video btw

[@cheetosnour.scratch-learn]

the cheetah will catch the snail after 1 second

[@drewpocernich2540]

Yeah that's not a paradox of whether motion is possible, but whether we can be certain of the cheetah catching up to the snail at a specified point in time (which really asks if our measurements are as ideal as they seem to be as if we knew that the cheetah was going 10m/s and the snail was going 1m/s then the cheetah would catch up to the snail in one second no questions asked, but life does not work as intuitively as a math problem. If you were to design this thought experiment like a science experiment then you need a very clearly definable definition for the term "catch up" as well as the other measurements. This really gets at the fact that Zeno's paradox in some manner is necessarily intertwined with Plato's theory of forms in the concept that "forms/ideals (in the example of Zeno's paradox: the time at which the cheetah catches up to the snail) are not real. In mathematical terms/axioms: we can not be certain if our measurements are accurate, or if it really took the cheetah 0.999999999999999999999999 seconds to catch up to the snail. This is our best guess. It's a broader statement about science and objectivity that this gets at (that's what I love about philosophy).

[@lydiamourningstar2028]

Not if the snail has Infinity or is actually the Green Baby

[@pineapplejuice1109]

@@lydiamourningstar2028that reference will not go unnoticed

[@freddieperkins6953]

Xeno's paradoxes work because each time you measure by smaller and smaller distances people will just not think about the fact that the time that passed is also getting shorter and shorter

[@amysworld5521]

By inceasing the speed we can catch snail by 1 sec​@@drewpocernich2540

[@creslo]

I never understood the monty hall problem until i saw the 100 door version. The host reveals their hand like a FOOL :0

[@giuliavieira2905]

For hash functions it is not that it should be infiesible to find a collision necessarily, as the collision may exist you can theoretically brute-force yourself into finding it, which is basically what the birthday-paradox was used for. The thing about hash functions is actually that it should be an easy function to compute a -> b but hard function to compute b -> a. Making it more secure for the birthday-paradox make it more likely you need to brute-force it, but not that you can't

[@liamg1748]

It was all going great until he said centimeter like a robot

[@willcerne]

The cheetah and snail running "paradox" can actually be used to demonstrate the geometric series of constant a and rate r converges to a/(1-r)

[@Quinn2win]

Unfortunately for Hilbert, it takes infinite work to open up that new vacant room.

[@AtticusDragon]

A little known extension of the hairy ball theorem is the Gandalf's beard taint theorem. As wisdom increases, the ability to tickle behind one's knee approaches TRUE.

[@brandonm8901]

Many of these I don't see as true "paradoxes", even veridical paradoxes. The Birthday paradox especially seems so intuitive. What sort of numbers are people expecting?

[@freddiesimmons1394]

Like 100. Weird, right?

[@guillaumelagueyte1019]

Most people expect it to be higher than 23, that's for sure. I've read a post once from a maths teacher who would always try it with their students, in classes of 30, and win most of the time. I tried to see what people who didn't know about it answered, couldn't find an experiment like that to confirm, but I expect most people would say something between 50 and 100. It's definitely not intuitive for most people.

[@user-cz3sl5gr3n]

Bold way to start the video

[@capncook2006]

You can actually graph the cheetah and the snail and see the cheetahs line will overlap and take over the snail after 2 turns Because the snail starts at 10m, goes up 1 The cheetah is at 0 goes up 10 The snail goes up to 12 The cheetah goes up to 20

[@locallyhated364]

The narrator does a bad job of describing this paradox. It's not about if the cheetah reaches or catches the snail, but when it happens. The example is meant to show that time can be infinitely divided (there is infinite points in time between each second) and not that things happen instantly from one second to the next(cheetah does not teleport on top of snail). The "answer" to the paradox is that the cheetah reaches the snail at .999999999999... Repeating seconds, which is basically just 1. Each step is one smaller magnitude of a second.

[@dunsparce4prez560]

I think the elevator paradox is just common sense.

[@jadeeliss]

agreed. i even had to google the definition of paradox cause i feel like i’m going crazy 1. : a statement that seems to go against common sense but may still be true. 2. : a false statement that at first seems true. 3. : a person or thing having qualities that seem to be opposites. maybe it’s a paradox considering the last definition but not really, the qualities of the elevator aren’t different or changing, it’s the qualities of the observer. it’s just simple relativity

[@djackklingler3]

This may sound stupid but it really confuses me. If you have a finite volume say in paint. and it fills up the horn completely, since you have painted the inside with paint, wouldn't that insinuate you have enough paint to paint the outside of the horn?

[@canaDavid1]

The thickness of the paint gets thinner and thinner on the inside. The statement implicitly assumes that you try to cover the outside with an even coat of paint.

[@djackklingler3]

@@canaDavid1 Are you not still covering the same amount of area? If you imagining the difference in separation between the outer edge and the inner edge is a single plank unit and the thickness is also a single plank length on the inside and the outside so three planks long. It is applied by machine so there is no variance in the thickness. You also have extra paint that is contained in the part of the volume that does not touch the inner surface as small an amount as that may be. Unless you're saying somehow that the outside continues at some point where the inside stops because that is hard to understand.

[@aykarain]

1:35 the cheetah will catch the snail, as they probably both has sizes... if they were just infinitely small points somehow then i guess the cheetah wont

[@mikey-hm7dt]

Wrong

[@aykarain]

@@mikey-hm7dt can you explain pls

[@k.m.2874]

Your second part of this comment explained it. There are infinitely smaller distances that the cheetah must reach first. Obviously after the first distance the cheetah can just lean down and get the snail. lol

[@solvated_photon]

If you came here to hear a mathematician explain hairy balls: 👇

[@kidsamurai7662]

1:37 I like the fraction version more because of how terrible fractions are defined allowing the paradox to actually be paradoxical (the distance halved is how fast the cheetah is while the snail is a traveling a tenth of the distance)

[@fullfungo]

I’m getting a deja vu. Is this a reupload? I distinctly remember correcting you on your pronunciation of parallelogram before.

[@CFGalt]

Yeah I swear I’ve watched this before… Edit: I’ve just realised that this is a compilation of previous videos, so that’d explain why it’s so familiar XD

[@fullfungo]

@@CFGalt Oh, right! Makes sense

[@drasiella]

Wtf is wrong with me, Im starting to understand calculus

[@iminyourwallsjeremy]

I had a thought similar to the Dichotomy Paradox, so basically Take the number 0 Add 1 Add 0.5 Add 0.25 Add 0.125 Add 0.0625 And so on, halfing the amount added each time. If this goes on for infinite time, would it reach 2?

[@ijabbott63]

6:55 Shirley some mistake?

[@thezucclord-3738]

The cheetah and the snail was originally known to me as the ant on a line.

[@MrSockez]

The second example isnt an example of the dichotomy paradox, its an example of the achilles paradox. The Achilles paradox is designed to prove that the slower mover will never be passed by the swifter in a race. The dichotomy paradox is designed to prove that an object can never reach a stationary goal

[@jeffjo8732]

The Achilles Paradox ignores that both measures - time and distance - are bounded by a limit and so don't represent what can happen when that limit is surpassed. Specifically, that one can be surpassed, so the other can be as well.

[@jeffjo8732]

Example, in a reply since it is a separate point. What is presented as The Sleeping Beauty Paradox (9:19) is not the original problem, or even the problem presented in the paper that originated the paradox. (1) The original lasted a trillion days, and had the subject being wakened on a random day in that period, or on every day, based on an (unspecified) coin flip. She is asked for the probability that this is the only awakening. (2) The first public representation changed that to two days, specified that Heads would mean the one (random) awakening, and asked for the probability of Heads. (3) The solution used in the first public representation assumed the “random” day was Monday, since it can’t matter which day it was. (4) But the solution described here, which is the one that is usually quoted, was not the solution presented there. It “proved” that each waking was equally likely; the video asserts it. And the issue is that the problem is presented as one of Sleeping Beauty’s awareness. This treats each day she potentially sleeps through as if it does not, and cannot, happen. Literally, the other days are erased from the calendar. At the end of the experiment, if the coin landed against her, Sleeping Beauty thinks that only one day has passed even though she was told otherwise. In Mathematical fact, there are 2N (N=10^9, or N=2) day and coin combinations that can happen during the experiment. Being awakened is a random selection from N+1 pre-defined combinations. They are independent samples due to the sleep-amnesia drug. The probability, given that she is awake, that she is wakened once (or the coin landed on Heads) is 1/(2N+1). The "new information" is that it is not one of the N-1 combinations where she would not be wakened.

[@mannypardo8900]

Mathematical paradoxes are mostly just overthinking simple solutions.

[@giga-ratsey1420]

That is a CRAZY opening

[@V3racious3]

I have two proofs for the hairy bal theorem.

[@romypotash7114]

0:54 ok but what is wrong with Achilles and a turtle

[@davidhopkins6946]

You forgot Newcomb's Paradox.

[@rch5395]

Where did the 2nd ball go?

[@asdfasdf71865]

i have a bad feeling about the proof by contradiction as the Gödel said on every syntactically defined system there are statements which cannot be true nor false. that diagonal argument feels like those contractions and what happens after it is a matter of definition, an expansion to the original syntactical system and not a logical consequence. is ot really so that you can not make similar arguments other way around and come up to some other system than cardinality of infinities?

[@spencerriding2904]

the basic idea behind the proof is that if the list truly contained every real number between 0 and 1 then it would be impossible for there to be a number in that range that was not listed. the diagonalization is just a method to create a number that is not in the list. since that created number is then not in the list, the list must have been flawed from the beginning and so the original assumption (that you can list all real numbers between 0 and 1) is impossible

[@saldetii]

Why is 23 getting multiplied by 22 in the birthday paradox? Where does 22 come from

[@ItsJustMe0585]

"Birthday attack" sounds like what will happen to me when i turn 40 next year. :'(

[@tokimotois]

Isn't Ross-littlewood paradox the same as the ant on a stretching rope? You add A to the pot/rope and remove a from the pot/rope, assuming A>a.

[@giu3077]

Thw only way the cheetah paradoz works is if the cheetah stops every time it gets to where the snail was, and even then..

[@vegassmb1]

This is what you experienced when you get hit by infinite void

[@Velereonics]

You can demonstrate the um three door problem to yourself just by having an object and having three cups, I guess you could just have one object and then two of another obje, and then just put them down in different configurations and just make a choice for the first part and then make a choice for the second part and then the result. then get all the results together.There's not that many. but it's like it's something where like you can explain it. But unless you do it, it doesn't seem real, like you have to just do it, watch it happen, and then you're also proving it at the same time, and then it is believable

[@joshuavogel861]

For the ant problem on the rope the problem becomes at some length of the rope The ant will never make it because it will starve to death first

[@rejectiomundi]

Where is the contradiction?

[@wva6809]

The date chosen during the birthday paradox. Is coincidentally, my birthdate.

[@christiankettlewell]

In (23*22)/2, where does the 22 come from? What is that representing?

[@Aera223]

23 people compare with 22 other people, since the person comparing can't compare with themself

[@darionbalkaran790]

Banach Tarski - Does the two object have the same surface area?

[@ParkerShimoda]

you should do every chess gambit. There are 619 of them. I'm pretty sure that some of your videos are either outdated (or inaccurate in a way that seems outdated, like "actually this thing was proved to be unsolvable, therefore it's solved") or don't contain everything despite claiming to (like in the cases were less than ten examples are presented. comparing to other videos, you could do more). In the second case, just change the title to reflect that (don't say "every"). In the first case, just add an apostrophe to the title (no need to address it in the thumbnail, or at all).

[@mawns]

Does "explained" to you mean "told without explanation"? Because you did not explain how these are solved.

[@eastlondonhustler]

15:26 *Specifically

[@SLAV326]

Finally, someone says it!

[@collie1610]

I actually am getting annoyed by these "paradoxes" which start off with an incorrect premise to make the "problem". The dichotomy paradox assumes for some reason that the cheetah must catch up to the snail's previous location before pursuing the next point, which makes no sense. If you want to catch something, you need to be faster than it. The cheetah can run past the snail or grab it. The elevator "paradox" has nothing paradoxical. Its just common sense. If youre at a location it spends 90% of the time going up to reach and 10% of the time going down to reach, you will notice you get on as its going up more often. Banach-Tarski are amazed that items can be hollow. Holy shit. Monty Hall problem does not add the chance to any specific door. Its still 50/50. If you hadnt stood in front of any door and he opened all but two, you are in the same position, the doors are 50/50. This is gambler's mindset. Hilbert's Hotel has infinite rooms. How the fuck could it be "fully occupied". That places a limit on infinity. I am not finishing this video bruh

[@E.T.S.]

Thank tou for this video, very interesting. I never agreed with Hilbert's Hotel being a paradox, I think it is cheating. Initially *all* rooms are already occupied, there is no empty room left. By moving all existing guests 1 room up you'd basically add 1 to infinity, as if infinity were a number, which it is not. It's similar to letting the guests stay in their room and then dig a cellar floor for an extra room, below the surface. Theoretically you could lift the entire hotel including the cellar room up one floor, and get the same infinity, but there is no space left above the top of the hotel to do that. There is no "above the hotel." Imagine an infinite spiral as seen from above, it looks like a circle. At every 90 degrees there is a box, representing a room plus all rooms below it. If all rooms in the infinite column are occupied the box is black, otherwise it is not. What Hilbert Paradox suggests is that when you rotate all occupants 90 degrees the first box becomes white/available, since the guests of the "last" room of infinity does not rotate into the first one since there is no last room, and guests in an hypothetical last room would not move to the first room, but there is no need to. All rooms are occupied. Guests could move like forever, but that suggests that the first room is free because some guests infintely far away are moving stuff in the hallway, not occupying a room. For Hilbert's Hotel to work there should be an end to infinity, so you can add 1 floor from the bottom, but there is no space. Another example, an infinite deep hole cannot be made deeper, not even if the top entrance were walled in for an extra level. The topology chances, while in Hilbert's Hotel only the guests are moving and the hotel construction stays the same. It's like adding a new positive integer to the list of all positive integers, which is impossible. One can add 0, or -1 if floor 0 is the lobby, but 0 and -1 depict other domains. Sorry for rant, it seems too obvious.

[@E.T.S.]

Cantor's magnitudes of infinitiy are not a paradox. Natural numbers and rational numbers are different domains. One could use a number line to find both natural numbers as rational numbers, but this doesn't work well when comparing the both sets. All natural numbers can be depicted as an x-axis with only integers, while all rational numbers with 1 digit can be represented as an x-y axis form, where x and y only represent integers. All rational numbers with 2 digits can be depicted on an x-y-z axis system, and so on. So, all natural numbers is just an infinite x-axis with integers, all rational numbers is a system with an infinite number of integer axises since an infinite number of digits is possible. Then it becomes clear that the domain of rational numbers is like "natural infinite (times) natural infinite". Irrational numbers would be an axis (or infinite set of axises?) that are 90 degrees to the already existing "infinite (times) infinite" axises and each other. It feels like the axis that make irrational numbers are rational number axises, not natural. Sorry for bad English, not my native language. Math is just an interest.

[@E.T.S.]

The outcome of the lamp paradox seems to depend on the initial state. Simplified, the lamp is either OFF-on-off-on-... / 0.10101010101010101..., or ON-off-on-off-... / 1.0101010101010101... Intuitively we'd say the outcome is undetermined since anywhere you stop it is either 1/on or 0/off, but as a whole everything is taken in account. Eventually the difference with the intial state is infinitely small and diverges to 0. However, when the initial condition is 'off' then the end condition is being 0.1 'on'. There might be a 10.101010101010...% chance that the lamp is on, while the initial state of 'on' may result in a 1.01010101...% chances of being off. Since turning on and off a lamp takes time the lamp soon lacks enough electricity to give light. It will be off, no matter the initial state. After point zero statistics kick in. You could run the experiment countless times and get different outcomes. Sometimes it is on, sometimes it is off. It wouldn't be a reliable way to check the last digit of infinity since there is no last digit, it's pure probability.

[@E.T.S.]

That dart board paradox is not really a paradox. Chances for any infinitely small exact spot to be hit are next to 0, compared to infinite other possibilities, but as the magnitude grows so does the relative area that gets hit. Infinites are in balance. Simplified, say a dart point is 1/100,000th of the area of a dart board, in imaginary pixels, then (at random) there is 1/100,000 chance it hits a certain pixel. With van Barneveld and Stompee throwing the chances are 100%. Checkmate, dart board.

[@prototypeinheritance515]

You're right Hilbert's Hotel isn't a paradox. That's because adding one guest doesn't make the size bigger. I'll explain in more detail: Imagine that the Hotel staff keep a register in order to keep track of all booked rooms and guests. The register is list of entries of the form (Room, Guest) where on the left side is the number of the room and on the right is the name of the guest. The Hotel is considered to be full if the following Conditions are met: - Every room is booked i.e. there is an entry where the left side matches the room number. - The rooms aren't overbooked, so there no two different guests booked the same room. For instance (5, Alice) and (5, Bob) would violate that. The same goes for the guests: - Every guest has a room - Each guest can book only one room, so (9, Andy) (10, Andy) is disallowed. This is what is called a one-to-one relation and is the essence of the problem. In mathematics we call two collections the same size if there is a one-to-one relation between them. We can for instance prove that Hotel with two rooms can fit Alice and Bob because of the booking (1, Alice), (2, Bob) is a one-to-one relation. The booking (1, Bob), (2, Alice) is equally valid. We can even prove that there is no way to fill a hotel with two rooms with one guest. Let's assume every room is booked. Since there is only one guest (we'll name him x) every entry must have x on the right side. So since every room is booked we have the entries (1, x) and (2, x) in the register. But x can only book one room, so there is no valid booking. This is all fine but this way of comparing sizes really shines when you consider infinite collections. So we return to Hilbert's Hotel. The Hotel is already fully booked with a infinite number of guests called g_1, g_2 and so on to infinity. But then a new guest called x comes along and wants a room in the Hotel. Now how we actually book everyone doesn't matter since this is a thought experiment. You could imagine everyone waits in the Square in front of the Hotel until the new booking is decided and the moves into their newly assigned rooms. Again it does not matter. But afterwards the register will look like this: We have one entry (1, x). And for every natural number n there is a entry (n+1, g_n). You can easily check that all four conditions hold: - room one is obviously booked, all rooms n > 1 are booked by g_(n-1). For instance (2, g_1), (3, g_2). - by the same logic every room has exactly one occupant. - obviously every guest has a room. - every guest also has a unique room since every original guest is in a room n > 1, meaning only x is in room one, but if a and b are in the room n, that means a = g_(n-1) = b. That means they are the same guest and room n is occupied by only one guest. Now, if you are convinced you can stop reading. But you might still have questions. One valid concern might be that maybe there are ways to make one-to-one relations between collections of different sizes. To elaborate I will introduce a notation: If A and B are any arbitrary collections |A| = |B| means there is a one-to-one relation between them, or we can say they are the same size. Now to reformulate our concern, maybe there are arbitrary collections A, B, C where |A| = |B| and |A| = |C| but |B| ≠ |C|. That is in other terms there are one-to-one mappings between A , B and B, C but there is no such mapping between B, C. Such a thing is not possible, because our method has three properties: i) |A| = |A| ii) if |A| = |B| then the converse |B| = |A| is also true. iii) if both |A| = |B| and |B| = |C| is true the also |A| = |C| These things are fairly easy to prove, but i'm not going into detail here. But using these properties we can solve our earlier problem. To restate the problem, we are given |A| = |B| and |A| = |C|. Our goal is disproving |B| ≠ |C|. Because of property (i) we can conclude |B| = |A|. Now using property (iii) it follows easily that |B| = |A| & |A| = |C| => |B| = |C| which was our goal. Furthermore it is possible to show that because of the three rules, all collections can be divided into groups where every collection has the same size, which is to say our definition of size behaves nicely. We have shown until now that one-to-one relations can compare sizes of arbitrary collections and behave nicely. But you might still have one complaint. What if there are different ways to measure size? The answer to that is yes there are, the one I explained is simply the one mathematicians commonly use. But I can show that it is a very useful and natural one. If we go back to the definition of one-to-one mappings and relax the conditions a bit, we can make a new operation. By getting rid of the requirement that a map between A and B has to include an entry with a matching right side for everything in B, we get get a so called injection. In other word there are things left over in B that have no corresponding partner in A. Using that new definition we can say that |A|

[@S_Tone_Rock]

xeno never considered size and center of gravity according to you.

[@alexandrecravo8618]

If a cheetah moves 10m/s and the snail 1m/s in 2 second the cheetah would have move 20m and the snail would need 20 second to move those 20m. Why this is a paradox

[@MikeHawkJuices]

How does the cheetah chase the snail for infinity? That makes no sense. If the Cheetah was nine meters behind then, -9 meters + 10 meters = 1 meter ahead of the snails starting point. The snails starting point is 0 meters + 1 meter = 1 meter ahead of its starting point. They both have traveled 1 meter from the snails starting point. The cheetah catches the snail after 1 second. If this isn't true explain how

[@mark-gg7kj]

1:51 lol, not me sitting here being born on feb 2.

[@shortycareface9678]

Elevator paradox is just basic probability? I don't see the paradox in it, but then again, I'm not a math bro

[@abderrahim552]

1:41 actually no the cheetah reach him immediately as the snail start at 9 and move 1m/s so when 1 sec pass he's at 10 which is the same position where the cheetah would be after 1 sec

[@netheritecraftondrugs5126]

It has been mathematically proven otherwise

[@abderrahim552]

@@netheritecraftondrugs5126I guess but I really have a hard time seeing why it work like the movement speed should be enough

[@dominicjenkinz3722]

This theory makes no sense

[@JacksonPfeiffer]

Zino was an idiot who confused himself by the fact that the finite amount of time it takes for the cheetah to catch the snail can be sliced into an infinite number of pieces. If the guy ever learned about integration, he’d probably have an aneurysm.

[@behnzo1988]

Simmons paradox doesn’t make since. Wouldn’t the combined percentage for treated be 75% survival rate and for untreated 35% survival rate? I don’t understand how the math adds to 60% survival rate untreated and 50% survival rate treated, assuming the male and female groups have the same amount of subjects for each sub group.

[@ilikechairsz]

just going straight into the action ig

[@seanstuchbery]

why did the voice change for simpson’s paradox wtf

[@narwhalboss5226]

that overlapping area paradox sucks. “oh the missing area is just hidden lmaooooooo”

[@realwaterenderman]

all of us are only here because we saw hairy ball on the thumbnail

[@PatsliTV]

The dichotomy paradox doesn’t even make any sense

[@SSCoversSP]

Damn , feb 2 is my actual birthday

[@dreaarune]

me too! my jaw dropped LOL

[@QuikVidGuy]

The Monty Hall problem still makes no sense. Removing options... removes options. Nobody ever explains how the probability is absorbed into the unchosen door, they just say that this is how it is. You make one choice out of 3 options, you are then given a different choice out of 2 options. It's like asking "If I roll a die and it landed on 5, what are the odds it will land on 5 on the next roll?" The answer is still 1 in 6.

[@jeffjo8732]

"Nobody ever explains how the probability is absorbed into the unchosen door" They can't, because it isn't "absorbed." Those that explain it that way do more harm than good. It's a simple matter of conditional probability, not like your dice example. But it is almost never explain correctly. Here's a simple example of conditional probability. Say a bag contains 10 red marbles; 3 are striped, and 7 are solid red. It also contains 10 green marbles; 7 are striped, and 3 are solid green. If I draw a marble out at random, the chances that it is red, or green, or striped, or solid are all 50%. BUT, if I tell you that it is striped, then the chances that it is red drop to 30%, and that it is green jump to 70%. This isn't a matter of some of the red probability being "absorbed" into green. Half of the possibilities are eliminated, and you need to _recalculate_ the probabilities based on the possibilities that are not eliminated. In the Monty Hall Problem, those who get the wrong answer ("the remaining doors each have a 50% probability") are correctly using conditional probability, but make a mistake in how they do so. Many of those who get the right answer ("switching improves you chances from 1/3 to 2/3") are doing two things wrong, but it is a rare case where two wrongs do make it right. Say you originally pick door #3. 1) There is a 1/3 chance that the car is behind door #1. The host must open door #2. 2) There is a 1/3 chance that the car is behind door #2. The host must open door #1. 3) If the car is behind door #3, the host has a choice. 3A) There is a 1/6 chance the car is behind door #3, and the host opens door #2. 3B) There is a 1/6 chance the car is behind door #3, and the host opens door #1. Let's say, since it doesn't change anything, that you see the host open door #1. Those who get the wrong answer are using conditional probability. But they don't recognize that case 3 has to be divided into cases 3A and 3B. So they eliminate all of case 1, and "keep" all of cases 2 and 3. Since both case 2 and 3 started equal to each other, they stay that way and change to 1/2 each. Most of those who get the right answer don't use conditional probability. They ignore that case 1 needs to be eliminated, and "absorb" its probability into case 2. Both steps are wrong - and as you point out, the second is very wrong. The correct solution is that all of the cases where door #2 is opened need to be eliminated. That's case 1, and case 3A. Case 2 is twice as likely as case 3B, and they stay that way. So case 2 has a 2/3 chance, and case 3B has a 1/3 chance.

[@scritoph3368]

In one of 3 possible scenarios you pick the door with the prize first, so switching is wrong. In two of three possible scenarios you pick one of the doors with a goat, and the host reveals the other goat, so switching is correct. Therefore, with no knowledge of what’s behind the doors except for the host revealing one of the doors you didn’t pick to be a goat, switching will be the correct option two out of three times.

[@jeffjo8732]

@@scritoph3368 Nobody ever seems to want to read the parts that show they are wrong. If you have to decide whether to switch *_BEFORE_* a door is opened, then you solution is correct. Because it ignores which door that is. The problem is that you are asked *_AFTER_* a door is opened. And guess what - YOU SEE WHICH DOOR. That makes your solution wrong. The answer is right, under the best assumption that "In [the] one of 3 possible scenarios [where] you pick the door with the prize first" the host chooses randomly between the other two doors But it is not the correct answer if you don't make that assumption. Since your solution requires an assumption that is neither explicit nor accounted for, as a solution it is wrong. Want to try another problem, to see why this is important? Mr. Smith tells you that he has two children, and that at least one is a boy. What are the chances that he has a boy and a girl? Hint: the solution methodology that says switching can't matter says this answer is 2/3. Any solution methodology that says switching wins 2/3 of the time says this answer is 1/2.

[@peterondrus7065]

@@jeffjo8732 Congratulations to Jeff who just won a brand new goat.

[@spencerriding2904]

@@jeffjo8732your second problem is different in that you are not making a choice before he tells you. not a comparable problem. the monty hall problem relies on the fact that monty will always pick a door that has a goat behind it. he has knowledge you don't. the example with a greater number of doors makes the benefit of that much clearer. the only scenario in which you lose by switching is when you choose the correct door on the first guess, whose probability is 1/number of doors. this will always be less than the probability you were incorrect on the first guess, which is 1-(1/number of doors).

[@williamwinstrop3918]

The birthday paradox never seems to explain if its month and day or year month and day? 23 kids in the same class would probably easy share a year/month/day 23 people random picked from the teachers and students from an elementary, middle, and high school would probably not fair the same. I think this paradox needs some explanation.

[@kittenbouquet]

It's specifically month and day. No year involved.

[@bassett_green]

Many of these aren't paradoxes

[@_d_a_n_c_e_]

01:35 The snail and cheetah That's literally just stupid If the cheetah moves 10m/s and the snail 1m/s In 1s the snail moved 1m and the cheetah 10 So it takes the cheetah an extra ⅒ of a s to catch the snail???

[@noahhirons8223]

But in that 1/10 the snail continues to move and the Chetah must again make up that distance.

[@Zerco526]

Lemme start off by saying, very good vid. I enjoyed it. Subbed. But watching this made me realize that mathematicians can sometimes lose the plot of reality. The Bertrands Paradox really got me. If you change the way that you come to the two points (The randomness) of course it changes the likelihood of outcome. The three methods are mathematically different. So would it not be abundantly obvious before even running the numbers that it would change the chances of desired outcome? The analogy that went through my mind when I heard this was "I'm going to test the likelihood of myself getting knocked unconscious when I strike my head. For the first test I will strike my head with a fluffy pillow, the second test a block of wood, and the third test a solid steel garter. Oh heavens! The likelihood changed on each test!" Like, you are fundamentally changing the test. It seems pretty apparent that the outcome of the test would change as well. I think that mathematicians sometimes lose the reality of this when they look purely at the abstractions. "Well it's just finding random points (I'm just striking my head) if the outcome changes, how could it NOT be a paradox?" Little background, former CS nerd here, so I may have internalized a different way of looking at randomness as not one thing but a procedure, but I still think that the astonishment holds up. Will certainly be proposing this to others to gauge the reaction.

[@jeffjo8732]

“Mathematicians can sometimes lose the plot of reality.” Well, not quite. We try to prove the consequences found in different versions of reality, whether or not these versions match “actual” reality. That isn’t as otherworldly as it sounds. The problem here is that people can disagree about what “reality” means. So they end up with different sets of consequences. And that is what is illustrated in the video. A “paradox” means contradictory, but valid, conclusions. Mathematicians use them to prove that there is something wrong with a particular version of reality. But non-mathematicians think that they mean there is something wrong with Mathematics. A good example is the one you used, Bertrand’s Paradox: “If you change the way that you come to the two points (The randomness) of course it changes the likelihood of outcome. The three methods are mathematically different.” And that was the point. That you need some criteria to define what “random” means, and it goes further than the parameters being random. This particular “paradox” was actually solved by Professor Edwin Jaynes, who taught my Statistical Thermodynamics class in college. A so-called “random chord” of the circle should also be a “random chord” of smaller circle with the same center (if it passes thru the smaller one, that is). Said another way, the density of these chords should not depend on how far away from the center they are. Method #1 has a higher density near the center, and Method #3 has a higher density away from the center. Only Method #2 has a density that is invariant. Each conclusion is valid, with its definition of "random," but only one can apply in a "reality" where "random" is implied, but not defined. But I’m not trying to convince you of that. I want to point out that these issues survive as “paradoxes” because of the internet-based environment. Armchair mathematicians arrive at a conclusion they like, and then try to massage Mathematics to fit that conclusion. True Mathematicians need detailed explanations to debunk such logic, but get criticized (and ignored) for that detail. I can explain why each of these examples is not really a paradox, but it will lead to endless debate with an adversary who blindly ignores Mathematics.

[@Zerco526]

@@jeffjo8732 That was going so well until the weird little jab at the end there. Those kinds of things are typically why people are so standoffish with the self proclaimed high value minds. I think the real point of my original statement was that this categorization and way of thinking fundamentally strays from the common sense and "reality" way of thinking. That's not necessarily a bad thing, and has lead to objectively helpful innovations in the past. I do think however, that its a dangerous path because it can infect the overall thought pattern and corrupt it towards paths of non-usefulness. If you pair that with an already disproportionately large ego and general discontent for social cooperation, as seen in the apparent need to end a peaceful reply with an unnecessary insult, you get to a point where using critical thought for the betterment of society is simply lost. The most complex and intricate invention can still be worthless if it's creator had no concept of value. Brilliant? Maybe. Useless? Certainly. I'd also like to point out that the seeming contention held for "Armchair mathematicians" and educational internet groups, is precisely the cause for the general social outcasting we see societally that drives the wedge between thinkers and non thinkers further and further down. Essentially, congratulations you just proved both that you missed the point and are the problem. Have a wonderful day

[@ghostmantagshome-er6pb]

Xeno needed Napier

[@cam5816]

The ant one puzzles me. How does infinity have to eventually exceed 1 when you can have infinite numbers/fractions between 1 and 0?

[@Aera223]

Apparently, the ant goes faster relative to the ground since the string expands both before and after it, iirc from Wikipedia.

[@aidenchoe4971]

Even if they are all composed of fractions smaller than 1, there are series that can eventually become greater than 1. A great is example is 1/2 + 1/3 + 1/4 +…. If you try inputting it in your calculator after every term, you will see that it becomes greater than one by the 3rd term.

[@codysmith3366]

There is an issue with the logic of the math done in the problem. The fractions added are based upon the previous length of the rope. In reality the ant walks 1% of the rope, then the rope grows causing it to be 0.5% across, then the ant walks 0.5% of the new rope. This continues and the ant never makes it past 1% of the rope.

[@LagartoChicloso]

Simpson paradox is just a failure in stadistical detailing, i just read the whole wikipedia article and its just that

[@ptrckhodges]

I think that's the wrong Zeno

[@robotrudy....]

this video has a script written by an AI

[@mrcleanisin]

Only one that makes sense is the Monty Hall puzzle.

[@logand488]

that one was one of the only ones i didn’t understand haha

[@mrcleanisin]

But do you understand it now?

[@logand488]

@@mrcleanisin not really, i don’t understand how it’s not 50/50 if there are only two available choices once the middle door is essentially eliminated

[@mrcleanisin]

The easiest way to understand is to use a deck of playing cards and mix only one black card in with all the red cards then choose one and throw out all but 2 cards, so do you think you most likely have a red card or a black card you choose? You most likely choose a red card, so the correct answer is to always switch.

[@logand488]

@@mrcleanisin that does make more sense! it’s like you’re selecting from a pool with a higher chance of getting the black card vs. the first card being the one that you chose from the entire deck. i see!! thank you

[@gingersperg]

The monty hall one is bs. Once one door is revealed to not contain what I value, the odds become 1/2. This is because not making the choice to change my initial choice, is still a choice. From the point the goat is revealed, I begin a brand new choice between 2 doors. The question is framed as if the 33.3% odds removed by the reveal of the goat, is entirely placed on the door I didn't choose. However, my choice is irrelevant. I still have no idea which remaining door could possibly contain the other goat. My initial decision bore no impact on the geographical location of my prize or on the entirely new decision I have to make. Imagine the other door was wiped from existence. There are now 2 doors to pick from (well, there were anyway). How are you going to tell me that my completely arbitrary stab in the dark previously constitutes a higher chance of the other of these 2 unopened doors being the correct one?!

[@logand488]

i’m glad i’m not the only one thinking this

[@TomatoBreadOrgasm]

There are two door pickers. It's not arbitrary from the other door picker's perspective, because Monty knows what's behind each door. If you picked a door with a goat, you have constrained his choices, because he has to open a door, and it can't be the one with the car. If you picked the door with the car, he has two options. Considering the whole decision tree, and the fact that the scenario is asymmetrical, will help you understand the problem.

[@spencerriding2904]

your initial decision affects which door monty can open. there is a 1/3 chance you were right originally, in which case monty can open either door and switching loses. the remaining options are that either door 2 or 3 have the prize, and given the restriction that monty can only open doors with no prize mean that if you pick door 1 in this scenario then switch you are guaranteed to win. both those scenarios are equally likely, so you have a 2/3 of winning by switching

[@bud-yo]

Hairy world