Oh, I thought it might be a pun. But nothing about the surrounding description sounded like a poset (weights are totally ordered, right?), so I figured it was a typo :P
I think the answer is, "because we want to teach what a bijection is," but readers might be confused why we're doing this. Maybe some of the flavor text about the Count should say that he's not actually good at counting? :P (Though if we did that, I'd be worried about starting to be too long-winded.)
Or maybe there should just be a parenthetical saying there's a reason for not counting explicitly, which we'll come back later. And then we'd need to come back to that when we introduce "bijection" further down the page.
I might try to introduce these terms one at a time, and a bit more slowly -- the paragraph up to this point reads like Simple English (good!), and then in the last two sentences I've got two terms thrown at me.
I think if a reader doesn't yet know what an isomorphism is, it would be helpful to spend more time building the intuition that there's something the same about both boxes, maybe like this:
If there aren't any left over, then you know there were the same number of items in each box. So to Count von Count, who only cares about counting things, the two boxes are basically equivalent, and might as well be the same box. Whenever two objects are the same from a certain perspective, we say that they are isomorphic.
In this example, the way in which the boxes were the same is that you could pair up each item in one box with an item in the other (which you wouldn't have been able to do if the boxes had different numbers of elements). Whenever you can match each item in one set with exactly one item in another set, we say that the sets are bijective and the way you paired them is a bijection.[1] A bijection is a kind of isomorphism.
Note that two sets have to have the same number of elements to be bijective, but that's not enough — you also need some way to say which item in one should be paired with which item in the other. In the case above, we paired the items up using the order in which they were removed from their boxes.
I'd like to add some pictures to this page at some point, but due to current circumstances I can't for now. If anyone wants to add pics (say different station maps with the same connections, two 'boxes' with random items) please feel welcome.
I also think I'll change the names of the stations from a, b etc. to funny made up station names.
The majority of this page will probably end up in the least technical lens.
This is a great page! I think the intro/summary could be made a little more accessible though? The use case I'm thinking of is a person who wants a brief overview in relatively non-technical language, which is valuable for the popups from links to here.
Eric Rogstad But... but... poset office was a pun, not a typo.
I support the creation of a poset-office, but it's gotta be about posets!
Oh, I thought it might be a pun. But nothing about the surrounding description sounded like a poset (weights are totally ordered, right?), so I figured it was a typo :P
Eric Rogstad Elmo comes to visit. Does that seem fine you think?
Why not just count explicitly?
I think the answer is, "because we want to teach what a bijection is," but readers might be confused why we're doing this. Maybe some of the flavor text about the Count should say that he's not actually good at counting? :P (Though if we did that, I'd be worried about starting to be too long-winded.)
Or maybe there should just be a parenthetical saying there's a reason for not counting explicitly, which we'll come back later. And then we'd need to come back to that when we introduce "bijection" further down the page.
Patrick Stevens I agree completely. Along with some other pictures. However, due tomy current circumstances I can't make any pictures at thr moment.
If someone else is willing to, I would be very grateful. Otherwise I could probably do it in about a month? Month and a half?
I think this probably wants a diagram of the two graphs, being differently laid out in the plane but isomorphic.
Eric Rogstad The post has been updated with an isomorphic version of what you suggested. Thanks!
Joke stolen shamelessly from the latest post on slatestarcodex.com
I might try to introduce these terms one at a time, and a bit more slowly -- the paragraph up to this point reads like Simple English (good!), and then in the last two sentences I've got two terms thrown at me.
I think if a reader doesn't yet know what an isomorphism is, it would be helpful to spend more time building the intuition that there's something the same about both boxes, maybe like this:
What do you think?
Note that two sets have to have the same number of elements to be bijective, but that's not enough — you also need some way to say which item in one should be paired with which item in the other. In the case above, we paired the items up using the order in which they were removed from their boxes.
I'd like to add some pictures to this page at some point, but due to current circumstances I can't for now. If anyone wants to add pics (say different station maps with the same connections, two 'boxes' with random items) please feel welcome.
I also think I'll change the names of the stations from a, b etc. to funny made up station names.
The majority of this page will probably end up in the least technical lens.
Eric Bruylant Thank you very much! just to be clear, are you talking about the 'clickbait', the intro paragraph in the text itself, or both?
Feel free to suggest / make your own changes if you have anything specific in mind by the way.
The intro paragraph, the clickbait seems fine.
This is a great page! I think the intro/summary could be made a little more accessible though? The use case I'm thinking of is a person who wants a brief overview in relatively non-technical language, which is valuable for the popups from links to here.
Patrick Stevens Yeah I've been wondering about the convention of things like this. I've been calling my pages things like category_mathematics.
"identity" is probably not a sufficiently specific link; I'd go for math_identity, probably.