I probably would say that that is because your two sets A and B do not carve reality at its joints. What I think army1987 intended to talk about is “real” sets, where a “real” set is defined as one that carves reality at its joints in one form or another.
Er, no, I was just mistaken. (And forgot to retract the great-grandparent—done now.) For a pair of sets who do carve reality at (one of) its joints but still is like that, try A = {(10, 0), (30, 0), (50, 0), (70, 0)} and B = {(40, 1), (40, 1), (40, 1), (40, 1)}.
(What I was thinking were cases were A = {10, 20, 30, 40} and B = {11, 21, 31, 41}, where it is the case that “two random members of A are more alike than a random member of A and a random member of B”, and my point was that “Two random men are more alike than a random man and a random woman” doesn’t rule out {men} and {women} being like that.)
What I think army1987 intended to talk about is “real” sets
There will be some real sets that are similar to Nominull’s (well, natural numbers are a subset of reals, eh?), however army1987 did emphasize the any, so Nominull’s correction was well warranted.
I believe what Manon meant is that the difference in this case between two random members of the same class exceeds the difference between the average members of each class.
I don’t see where I assumed that the groups were disjoint. My point was that “Two random men are more alike than a random man and a random woman”, while technically true, isn’t particularly informative about men and women.
Ah, my mistake. I thought you were saying that given your proposition is (asserted to be true), the idea that two random men are more alike than a random man and woman must be meaningfully true.
Not necessarily—for example, if all the members of both groups are on a one-dimensional space, both groups have the same mean, and Group B had much smaller variance than Group A… But still.
For any two groups A and B, two random members of A are more alike than a random member of A and a random member of B, aren’t they?
No. A is [1,3,5,7], B is [4,4,4,4]. A random member of A will be closer to a random member of B than to another random member of A.
I probably would say that that is because your two sets A and B do not carve reality at its joints. What I think army1987 intended to talk about is “real” sets, where a “real” set is defined as one that carves reality at its joints in one form or another.
Er, no, I was just mistaken. (And forgot to retract the great-grandparent—done now.) For a pair of sets who do carve reality at (one of) its joints but still is like that, try A = {(10, 0), (30, 0), (50, 0), (70, 0)} and B = {(40, 1), (40, 1), (40, 1), (40, 1)}.
(What I was thinking were cases were A = {10, 20, 30, 40} and B = {11, 21, 31, 41}, where it is the case that “two random members of A are more alike than a random member of A and a random member of B”, and my point was that “Two random men are more alike than a random man and a random woman” doesn’t rule out {men} and {women} being like that.)
Ah, okay then. That makes sense.
There will be some real sets that are similar to Nominull’s (well, natural numbers are a subset of reals, eh?), however army1987 did emphasize the any, so Nominull’s correction was well warranted.
Let A = “humans” and B = “male humans.”
I believe what Manon meant is that the difference in this case between two random members of the same class exceeds the difference between the average members of each class.
What about cases in which group B is a subset of Group A?
Most people are members of more than just one group.
So?
Soooooo, real humans might be a mite more complicated than that, such that your summary does not usefully cover inferences about people.
I don’t see where I assumed that the groups were disjoint. My point was that “Two random men are more alike than a random man and a random woman”, while technically true, isn’t particularly informative about men and women.
Ah, my mistake. I thought you were saying that given your proposition is (asserted to be true), the idea that two random men are more alike than a random man and woman must be meaningfully true.
Not necessarily—for example, if all the members of both groups are on a one-dimensional space, both groups have the same mean, and Group B had much smaller variance than Group A… But still.