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.
This reminds me of a one of Zeno’s Paradoxes of Motion. The following is from the Stanford Encyclopaedia of Philosophy:
Suppose a very fast runner—such as mythical Atalanta—needs to run for the bus. Clearly before she reaches the bus stop she must run half-way, as Aristotle says. There’s no problem there; supposing a constant motion it will take her 1/2 the time to run half-way there and 1/2 the time to run the rest of the way. Now she must also run half-way to the half-way point—i.e., a 1/4 of the total distance—before she reaches the half-way point, but again she is left with a finite number of finite lengths to run, and plenty of time to do it. And before she reaches 1/4 of the way she must reach 1/2 of 1/4=1/8 of the way; and before that a 1/16; and so on. There is no problem at any finite point in this series, but what if the halving is carried out infinitely many times? The resulting series contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. However it does contain a final distance, namely 1/2 of the way; and a penultimate distance, 1/4 of the way; and a third to last distance, 1/8 of the way; and so on. Thus the series of distances that Atalanta is required to run is: …, then 1/16 of the way, then 1/8 of the way, then 1/4 of the way, and finally 1/2 of the way (for now we are not suggesting that she stops at the end of each segment and then starts running at the beginning of the next—we are thinking of her continuous run being composed of such parts). And now there is a problem, for this description of her run has her travelling an infinite number of finite distances, which, Zeno would have us conclude, must take an infinite time, which is to say it is never completed. And since the argument does not depend on the distance or who or what the mover is, it follows that no finite distance can ever be traveled, which is to say that all motion is impossible. (Note that the paradox could easily be generated in the other direction so that Atalanta must first run half way, then half the remaining way, then half of that and so on, so that she must run the following endless sequence of fractions of the total distance: 1/2, then 1/4, then 1/8, then ….)
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”.
Yeah, that was actually an awkward wording, sorry. What I meant is that given a non-continuous map from the natural numbers to the reals (or any other two sets with infinite but non-matching cardinality), there’s a way to prove it’s not bijective - often the diagonal argument.
For anyone reading and curious, you take advantage of the fact you can choose an independent modification to the output value of the mapping for each input value. In this case, a common choice is the nth decimal digit of the real number corresponding to the input natural number n. By choosing the unused value for each digit - that is, making a new number that’s different from all the used numbers in that one place, at least - you construct a value that must be unused in the set of possible outputs, which is a contradiction (bijective means it’s a one-to-one pairing between the two ends).
Actually, you can go even stronger, and do this for surjective functions. All bijective maps are surjective functions, but surjective functions are allowed to map two or more inputs to the same output as long as every input and output is still used. At that point, you literally just define “A is a smaller set than B” as meaning that you can’t surject A into B. It’s a definition that works for all finite quantities, so why not?
because I assumed continuous mapping the number c is between a and b it means if it has to be mapped to a natural number the natural number has to be between 22 and 23 but there is no natural number between 22 and 23 , it means c is not mapped to anything
Then you did not prove that there is no discontiguous mapping which maps [1, 2] to the natural numbers. You must show that no mapping exists, continugous or otherwise.
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.
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.
When talking about infinity, basically everything is theoretical
Please show me a functional infinity
Right, an asymptote I guess, in use, but not a number.
It’s been quite some time since I did pre-calc, but I remember there being equations where it was relevant that one infinity was bigger than another.
This reminds me of a one of Zeno’s Paradoxes of Motion. The following is from the Stanford Encyclopaedia of Philosophy:
Your explanation is wrong. There is no reason to believe that “c” has no mapping.
Edit: for instance, it could map to 29, or -7.
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”.
But then a simple comeback would be, “well perhaps there is a non-continuous mapping.” (There isn’t one, of course.)
“It still works if you don’t” – how does red’s argument work if you don’t? Red is not using cantor’s diagonal proof.
Yeah, that was actually an awkward wording, sorry. What I meant is that given a non-continuous map from the natural numbers to the reals (or any other two sets with infinite but non-matching cardinality), there’s a way to prove it’s not bijective - often the diagonal argument.
For anyone reading and curious, you take advantage of the fact you can choose an independent modification to the output value of the mapping for each input value. In this case, a common choice is the nth decimal digit of the real number corresponding to the input natural number n. By choosing the unused value for each digit - that is, making a new number that’s different from all the used numbers in that one place, at least - you construct a value that must be unused in the set of possible outputs, which is a contradiction (bijective means it’s a one-to-one pairing between the two ends).
Actually, you can go even stronger, and do this for surjective functions. All bijective maps are surjective functions, but surjective functions are allowed to map two or more inputs to the same output as long as every input and output is still used. At that point, you literally just define “A is a smaller set than B” as meaning that you can’t surject A into B. It’s a definition that works for all finite quantities, so why not?
because I assumed continuous mapping the number c is between a and b it means if it has to be mapped to a natural number the natural number has to be between 22 and 23 but there is no natural number between 22 and 23 , it means c is not mapped to anything
Then you did not prove that there is no discontiguous mapping which maps [1, 2] to the natural numbers. You must show that no mapping exists, continugous or otherwise.
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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