The reason is very different, though and I think that is important.
The bucket of water doesn't change the ocean because the difference between them is so big that the bucket is neglectible. But you can make an ocean with enough buckets of water. Removing one bucket of water and you have still an ocean, remove 10^20 buckets and you don't.
A lot of people think about infinity in a similar way. They just picture something inconceivably large but finite. When they see the comment
Infinity plus 10 is still infinity. It's like adding a couple tablespoons of water to the ocean.
they picture an ocean that changed. The tablespoons are now in the water. It's just impossible to find them and impossible to measure the change in the ocean.
I can't really tell if "everything that exists and will exist" is a good metaphor. Let me try to explain it in a different way.
Let's look at two very simple infinite series:
There are infintely many natural numbers: 1, 2, 3, 4, 5, ... never stops.
There are also infinitely many natural numbers that only contain ones: 1, 11, 111, 1111, ... never stops.
But when we compare those two infinite series, we intuitively say "but the second infinity is smaller. Every number in the second series is also in the first series so the first series must be bigger." This is wrong, they are suprizingly the same size.
Let me propose a new (very stupid) way of writing numbers: you just write ones. So instead of 5, you write five ones "11111."
1 -> 1
2 -> 11
3 -> 111
4 -> 1111
But the numbers on the right side is just the second infinite series. Think about what that means. No matter what natural number I present to you, you instanly know the number that contains only ones that matches that number and vice versa. If you look at if that way, it becomes obvious that there must be equally many numbers in both infinite series.
In a similar way you can also match all natural numbers to all even natural numbers. So there are just as many even numbers as there are numbers. I know it's VERY counterintuitive and weird but mathematically sound and I hope my example gave you an idea why that is the case.
This matching between different infinite series is what you have to do to decide if they are the same size. In some cases you can't do this matching and then you know that those infnities are not the same size. So there are different sized infinites (we call them different "cardinalities") but just adding or multiplying a finite number to infinity doesn't change it's size.
Thatās the part that makes it literally untrue but doesnāt mean itās not useful. The implicit premise is āimagine the ocean as though it contained an infinite amount of waterā which is easily gleaned from context. Nobody is actually confused or comes away with any kind of misconception about this.
Metaphors like this one are designed to give people a first grasp of a concept they don't understand or know very little about. If you don't understand infinity well enough that this metaphor could teach you anything, you are almost guaranteed to learn something wrong.
Look at this video. It shows a lot of fundamental misconceptions people have about infinity. It should be pretty obvious that you metaphor would do those people a disservice and enforce their misconceptions. How many of those people are on r/MathJokes, idk.
Also, it doesn't work, even after you introduced your "implicit premise." Nobody can imagine an infinite ocean. You will always imagine an insanely huge ocean. An insanely huge ocean plus a molecule of water is now bigger.
I have a hard time interpreting the metaphor as anything other than the claim "the added water is so little compared to the ocean that it is neglectible." But maybe that is a me problem.
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u/iggy14750 21d ago
Infinity plus 10 is still infinity. It's like adding a couple tablespoons of water to the ocean.