Combinatorics scares me, the immense size of seemingly trivial things.
For example: If you take a simple 52 card poker deck, shuffle it well, some combination of 4-5 riffles and 4-5 cuts, it is basically 100% certain that the order of all the cards has never been seen before and will never been seen again unless you intentionally order them like that.
52 factorial is an unimaginable number, the amount of unique combinations is so immense it really freaks me out. And all from a simple deck of playing cards.
Chess is another example. Assuming you aren’t deliberately trying to copy a specific game, and assuming the game goes longer than around a dozen moves, you will never play the same game ever again, and nobody else for the rest of our civilization ever will either. The amount of possible unique chess games with 40 moves is far far larger than the number of stars in the entire observable universe.
You could play 100 complete chess games with around 40 moves every single second for the rest of your life and you would never replay a game and no other people on earth would ever replay any of your games, they all would be unique.
One last freaky one: There are different sizes of infinity, like literally, there are entire categories of infinities that are larger than other ones.
I won’t get into the math here, you can find lots of great vids online explaining it. But here is the freaky fact: There are infinitely more numbers between 1 and 2 than the entire infinite set of natural numbers 1, 2, 3…
In fact, there are infinitely more numbers between any fraction of natural numbers, than the entire infinite natural numbers, no matter how small you make the fraction…
It’s called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let’s assume for a moment that the numbers between 1 and 2 are the same “size” of infinity as the natural numbers. If that were true, you’d be able to map every number between 1 and 2 to a natural number. but here’s the thing, say you map some number “a” to 22 and another number “b” to 23. Now take the average of these two numbers, (a + b)/2 = c the number “c” is still between 1 and 2, but it hasn’t been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers
It’s like when you say something is full. Double full doesn’t mean anything, but there’s still a difference between full of marbles and full of sand depending what you’re trying to deduce. There’s functional applications for this comparison. We could theoretically say there’s twice as much sand than marbles in “full” if were interested in “counting”.
The same way we have this idea of full, we have the idea of infinity which can affect certain mathematics. Full doesn’t tell you the size of the container, it’s a concept. A bucket twice as large is still full, so there are different kinds of full like we have different kinds of infinity.
Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.
Yeah, OP seems to be assuming a continuous mapping. It still works if you don’t, but the standard way to prove it is the more abstract “diagonal argument”.
It’s weird but the amount of natural numbers is “countable” if you had infinite time and patience, you could count “1,2,3…” to infinity.
It is the countable infinity.
The amount of numbers between 1 and 2 is not countable. No matter what strategies you use, there will always be numbers that you miss. It’s like counting the numbers of points in a line, you can always find more even at infinity.
It is the uncountable infinity.
I greatly recommand you the hilbert’s infinite hotel problem, you can find videos about it on youtube, it covers this question.
Basically, if two quantities are the same, you can pair them off. It’s possible to prove you cannot pair off all real numbers with all integers. (It works for integers and all rational numbers, though)
How many infinities you accept as meaningful is a matter of preference, really. You don’t even have to accept basic infinity or normal really big numbers as real, if you don’t want to. Accepting “all of them” tends to lead to contradictions; not accepting, like, 3 is just weird and obtuse.
We’re talking about increasingly smaller fractions here. It’s more like saying if you ground up all the rocks on earth into sand you would have more individual pieces of sand than individual rocks.
In one of Vsauce’s videos he suggested a good visualisation of the number of unique shuffles of a deck of cards that was originally suggested by Scott Czepiel.
Imagine you have a friend that is shuffling a deck of cards and ordering the deck uniquely every second. Also imagine that every action you take is completed instantaneously.
You stand on the equator. Wait a billion years. Then take a step. Wait another billion years. Then take another step. Continue this until you have got back to where you started.
Then take 0.02ml from the Pacific Ocean. Wait another billion years. Then take a step. Continue until you get back to where you started and take another drop out of the Pacific Ocean.
Repeat this process until the entire Pacific Ocean is empty. Then place a sheet of paper on the ground at sea level.
Refill the ocean and repeat - wait a billion years between steps as you walk around the equator, take a drop of water out of the Pacific Ocean every time you get back to where you started and place a piece of paper on the ground in a tower before refilling the Pacific Ocean and repeating.
When the tower of paper reaches the sun do you think that your friend has managed to produce each, unique ordering of the cards?
Nope! Not even close…
If you were to repeat all of the above 3000 times, then he’d be pretty much done.
Combinatorics scares me, the immense size of seemingly trivial things.
For example: If you take a simple 52 card poker deck, shuffle it well, some combination of 4-5 riffles and 4-5 cuts, it is basically 100% certain that the order of all the cards has never been seen before and will never been seen again unless you intentionally order them like that.
52 factorial is an unimaginable number, the amount of unique combinations is so immense it really freaks me out. And all from a simple deck of playing cards.
Chess is another example. Assuming you aren’t deliberately trying to copy a specific game, and assuming the game goes longer than around a dozen moves, you will never play the same game ever again, and nobody else for the rest of our civilization ever will either. The amount of possible unique chess games with 40 moves is far far larger than the number of stars in the entire observable universe.
You could play 100 complete chess games with around 40 moves every single second for the rest of your life and you would never replay a game and no other people on earth would ever replay any of your games, they all would be unique.
One last freaky one: There are different sizes of infinity, like literally, there are entire categories of infinities that are larger than other ones.
I won’t get into the math here, you can find lots of great vids online explaining it. But here is the freaky fact: There are infinitely more numbers between 1 and 2 than the entire infinite set of natural numbers 1, 2, 3…
In fact, there are infinitely more numbers between any fraction of natural numbers, than the entire infinite natural numbers, no matter how small you make the fraction…
Natural numbers being infinite, how it be possible for the values between 1 and 2 to be “more infinite” ?
It’s called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let’s assume for a moment that the numbers between 1 and 2 are the same “size” of infinity as the natural numbers. If that were true, you’d be able to map every number between 1 and 2 to a natural number. but here’s the thing, say you map some number “a” to 22 and another number “b” to 23. Now take the average of these two numbers, (a + b)/2 = c the number “c” is still between 1 and 2, but it hasn’t been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers
Great explanation by the way.
I get that, but it’s kinda the same as saying “I dare you!” ; “I dare you to infinity!” ; “nuh uh, I dare you to double infinity!”
Sure it’s more theoretically, but not really functionally more.
When talking about infinity, basically everything is theoretical
It’s like when you say something is full. Double full doesn’t mean anything, but there’s still a difference between full of marbles and full of sand depending what you’re trying to deduce. There’s functional applications for this comparison. We could theoretically say there’s twice as much sand than marbles in “full” if were interested in “counting”.
The same way we have this idea of full, we have the idea of infinity which can affect certain mathematics. Full doesn’t tell you the size of the container, it’s a concept. A bucket twice as large is still full, so there are different kinds of full like we have different kinds of infinity.
Please show me a functional infinity
Your explanation is wrong. There is no reason to believe that “c” has no mapping.
Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.
Yeah, OP seems to be assuming a continuous mapping. It still works if you don’t, but the standard way to prove it is the more abstract “diagonal argument”.
It’s weird but the amount of natural numbers is “countable” if you had infinite time and patience, you could count “1,2,3…” to infinity. It is the countable infinity.
The amount of numbers between 1 and 2 is not countable. No matter what strategies you use, there will always be numbers that you miss. It’s like counting the numbers of points in a line, you can always find more even at infinity. It is the uncountable infinity.
I greatly recommand you the hilbert’s infinite hotel problem, you can find videos about it on youtube, it covers this question.
Because the second one is bounded ?
Basically, if two quantities are the same, you can pair them off. It’s possible to prove you cannot pair off all real numbers with all integers. (It works for integers and all rational numbers, though)
How many infinities you accept as meaningful is a matter of preference, really. You don’t even have to accept basic infinity or normal really big numbers as real, if you don’t want to. Accepting “all of them” tends to lead to contradictions; not accepting, like, 3 is just weird and obtuse.
I thought the same but there is a good explanation for it which I can’t remember
I’m confused as well. Isn’t that like saying that there is more sand in a sandbox than on every veach on the planet?
We’re talking about increasingly smaller fractions here. It’s more like saying if you ground up all the rocks on earth into sand you would have more individual pieces of sand than individual rocks.
In one of Vsauce’s videos he suggested a good visualisation of the number of unique shuffles of a deck of cards that was originally suggested by Scott Czepiel.
Imagine you have a friend that is shuffling a deck of cards and ordering the deck uniquely every second. Also imagine that every action you take is completed instantaneously.
You stand on the equator. Wait a billion years. Then take a step. Wait another billion years. Then take another step. Continue this until you have got back to where you started.
Then take 0.02ml from the Pacific Ocean. Wait another billion years. Then take a step. Continue until you get back to where you started and take another drop out of the Pacific Ocean.
Repeat this process until the entire Pacific Ocean is empty. Then place a sheet of paper on the ground at sea level.
Refill the ocean and repeat - wait a billion years between steps as you walk around the equator, take a drop of water out of the Pacific Ocean every time you get back to where you started and place a piece of paper on the ground in a tower before refilling the Pacific Ocean and repeating.
When the tower of paper reaches the sun do you think that your friend has managed to produce each, unique ordering of the cards?
Nope! Not even close…
If you were to repeat all of the above 3000 times, then he’d be pretty much done.
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