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.

[@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?

[@VLITMER]

0:00 the WHAT!

[@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

[@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.

[@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.

[@cheetosnour.scratch-learn]

the cheetah will catch the snail after 1 second

[@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."

[@jackkalver4644]

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

[@Agent.036]

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

[@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.

[@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!

[@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

[@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?

[@ijabbott63]

6:55 Shirley some mistake?

[@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

[@davidhopkins6946]

You forgot Newcomb's Paradox.

[@jirayusangjampa1948]

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

[@TaylorGolmon]

Respectfully, the presenter might be mistaken.

[@rch5395]

Where did the 2nd ball go?

[@PatsliTV]

The dichotomy paradox doesn’t even make any sense

[@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.

[@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

[@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|

[@rejectiomundi]

Where is the contradiction?

[@mrcleanisin]

Only one that makes sense is the Monty Hall puzzle.

[@abderrahim552]

9:20 euhhhhh what

[@baruffaparsley4710]

How is Zeno treated seriously? Do we lack context?

[@dogedev1337]

Calculus

[@baruffaparsley4710]

@@dogedev1337 ok it's a limit, but Zeno examples is terrible. Achilles/the cat reaches his prey in a second and tha's it

[@Boundless_Border]

​@baruffaparsley4710 I think the issue lies in the idea that real infinities can't be traversed. If you look at space and time as continuous, you also have to accept the reality of infinity and that it is possible to traverse infinity.

[@roryburch861]

Yeah since we live in a post calculus universe, we know that a process with infinite steps can complete in a finite amount of time. But in Ancient Greece that probably seemed like a logical paradox. It says a lot more about Greek perspectives than math imo.

[@fasthowto]

It's called a paRalellogram.. #fail

[@stuffthings1417]

the monty hall thing is a simple problem.

[@mrcleanisin]

It is simply, but did you solve it before you saw the answer?

[@stuffthings1417]

@@mrcleanisin sort of. it was a post on the fact people were taking different sides of the topic. i realized a diagram can prove it. but, i tried showing to some random co-workers and some were still confused. it was frustrating. so i saw the confusion it made. i believe the post mentioned "scholars" debating it. i can't believe that.

[@mrcleanisin]

I asked if you solved it without looking at the answer. I have not found anyone who did it on their own.