Sum-product, unit distances, and number fields
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Sum-product, unit distances, and number fields
1–10 of 21 posts
Re: Sum-product, unit distances, and number fields
#2Stopped reading when they used |·| without explaining its meaning.
Re: Sum-product, unit distances, and number fields
#3Stopped reading when they used |·| without explaining its meaning.
Size of a finite set. It’s common notation in this field.
Re: Sum-product, unit distances, and number fields
#4Stopped reading when they used |·| without explaining its meaning.
Size of a finite set. It’s common notation in this field.
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?
Re: Sum-product, unit distances, and number fields
#5Earlier quoted context omitted.
Size of a finite set. It’s common notation in this field.
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?
Who do you think invented operator overloading?
Re: Sum-product, unit distances, and number fields
#6Earlier quoted context omitted.
Size of a finite set. It’s common notation in this field.
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?
Mathematicians use inconsistent notations all the time. Symbols meaning slightly different things based on the type of the arguments are among the benign cases.
For the present case, see https://en.wikipedia.org/wiki/Vertical_bar#Mathematics.
Re: Sum-product, unit distances, and number fields
#7Earlier quoted context omitted.
Size of a finite set. It’s common notation in this field.
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?
[deleted]
Re: Sum-product, unit distances, and number fields
#8Stopped reading when they used |·| without explaining its meaning.
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.
Re: Sum-product, unit distances, and number fields
#9Earlier quoted context omitted.
Size of a finite set. It’s common notation in this field.
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, too 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". :-)
Re: Sum-product, unit distances, and number fields
#10Earlier quoted context omitted.
Size of a finite set. It’s common notation in this field.
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?
I dunno I think it makes sense. For a vector x, the length |x| says something about its size relative to other vectors. For a set S, the cardinality |S| says something about its size relative to other sets.
The vector is always defined in a vector field which has a given dimension, and usually the dimension isn't that interesting. Typically it's either the same between the vectors you consider, or the vectors have one of a few fixed number of dimensions. Meanwhile the length of vectors is an interesting quantity.
For sets, since the values can be anything, nothing or everything in between, you can't really define many interesting functions or operations that work on the elements of sets in general. Meanwhile, the number of elements in a set is an interesting quantity.
Anyway, just my take, though I never did take much math.