Ok, I guess I'm always confused why |S| on a set doesn't take the values into account, whereas |x| on a vector does take its values into account; how can mathematicians of all people be so inconsistent?
There is a joke saying "a mathematician says X, writes Y on the board and means Z". The really amusing(?) thing is that other mathematicians still (sort of) perfectly understands Z. Once you have enough experience you fill in the blanks automatically.
Math exposition is tricky: too few details and you're just floating in the sky, to many details and the audience loses sight of the forest for all the trees. You can go (more or less) all formal, but it's a pain for the writer and a pain for the experienced reader.
If it's any consolation, the punchline to the joke is that it often is small/big lie: the other mathematicians reads "Y" and goes WTF!? And then 1 minute, 1 hour, 1 day, or one week later says "aaah, that's what he/she meant! I guess it was 'obvious' all along". :-)
Mathematicians use inconsistent notations all the time. Symbols meaning slightly different things based on the type of the arguments are among the benign cases.
I stopped reading earlier, when they used superscript without explaining its meaning. Its clearly meant for someone with more domain expertise than me, with my hazy recollections of college math.
Math exposition is tricky: too few details and you're just floating in the sky, to many details and the audience loses sight of the forest for all the trees. You can go (more or less) all formal, but it's a pain for the writer and a pain for the experienced reader.
If it's any consolation, the punchline to the joke is that it often is small/big lie: the other mathematicians reads "Y" and goes WTF!? And then 1 minute, 1 hour, 1 day, or one week later says "aaah, that's what he/she meant! I guess it was 'obvious' all along". :-)
For the present case, see https://en.wikipedia.org/wiki/Vertical_bar#Mathematics.