Sometimes I think mathematicians are amazingly creative. Leonhard Euler's solution to the Konigsberg Bridge problem is elegant. Cantor's proof that the set of irrational numbers is uncountably infinite is elegant. Much of ancient Greek solid geometry in elegant.
Other times I think mathematicians aren't creative at all. I offer as one example the symbology of vertical lines. We use vertical lines to mean entirely different things depending on the context; do we lack so much creativity that we couldn't come up with different symbols?
| | . A pair of vertical lines.
That symbol is used 4 entirely different ways that I know of, and there's plenty of math I haven't even heard of. Perhaps it's even used other ways as well.
In a case like |x|, the vertical lines mean "absolute value" of a number, be that number real or imaginary.
In a case like |z| it represents the magnitude of vector z.
It could also represent the determinant of a matrix, det A:
| 3 5 7|
| 2 4 6|
| 1 4 9|
or |A| for short.
Lastly, at least in my experience, it represents the cardinality of a set--that is, how many items are in the set. |1, 3, 5, 7, 9, 11|=6, or |Z+| = אo, which means that the set of positive integers (Z+) is countably infinite, symbolized by "aleph-naught".
Maybe, instead of worrying that mathematicians aren't creative, I should celebrate the efficiency of using the same symbol so many different ways. That's a much more positive way to look at things. Yes, that's what I think I'll do.
But those are all variants on the same thing, aren't they? Some measure of the "size" of an entity. In one dimension, the absolute value is the distance from the origin. The magnitude of a vector is its size. The cardinality of a set is its size. The determinant is the outlier here in that it's not a conventional "size" of a matrix (it's not a matrix norm; the norm is usually represented with paired vertical lines). But there is a way (probably more than one, but I can't think of another quickly) to see the determinant relating to size: the absolute value of the determinant of a 3x3 matrix is the volume of the parallelepiped defined by the three row vectors.
ReplyDeleteThere is some similarity, yes, but it strikes me as very superficial. I don't see how the magnitude of a vector in 4-space relates to aleph-naught.
ReplyDeleteIn logic, especially computer programming, and C language in particular, vertical lines symbolize the OR function.
ReplyDeleteThere are a somewhat finite number of shorthand notations one can use for representing an almost infinite number of ideas. The some are reused is basically a necessity. Generally, you can easily deduce the meaning of the particular shorthand (your ||) in this case by the context that goes with the notation.
ReplyDeleteFor instance, even the English language finds itself repeating words but with different meanings called homonyms. (see https://en.wikipedia.org/wiki/List_of_true_homonyms)
I am surprised that you didn't mention all the meaning of lower-case sigma. A quick Google search yielded 26 different definitions. But again, each meaning is discernible by how it is used and which field is using it.
So I guess I am surprised by your surprise of lack of creativity. I think it is very creative to assign different meanings to the same thing while ensuring the intended definition is (usually) obvious through use.