it's hard enough to learn math already, especially with all those downright unnecessarily alien-looking symbols, without having to worry about 15 different ways in which mathematicians (and physicists) might use the same symbol.

for example, in http://en.wikipedia.org/wiki/Omega, it lists 9 different uses for Omega in math alone (5 for physics).

math is supposed to be an exact science, and it's a little disconcerting for mathematicians to re-use the same symbols for many different mathematical domains and contexts. it's sloppy and ambiguous and it just shows that if the symbology is that arbitrary anyway you might as well just use normal letters and names.

My idea is to either a) invent a plethora of new symbols to use in mathematical contexts - one for every unique calling for a symbol - or b) use symbols that already exist somewhere by the dozens, for example, the Katakana seems like an excellent choice.
an alternative idea would, of course, be either to a) just use normal latin letters for math, and define their meaning through context like we always do anyway, or b) use names, sort of akin to sin, cos and tan.

it would due if we use names, to bind the letters together somehow, maybe in an artistic way, but at least to distinguish their letters from individual coefficients and to multiply names side-by-side without it looking awkward. like invent a (simple) calligraphical way of wrapping all the letters in on themselves into one being, and at least write the letters in cursive.
alternatively we could invent a new symbol to open and close names, which looks elegant in normal usage, something vaguely like a brace/bracket/chevron

another possibility, if we construct new symbols, is to invent something where we construct whole symbols out of combinations of atomic, dynamically integratable sub-symbols representing individual phonems in a symbol's pronunciation.

ideas for closing symbols:

A) something like angle brackets (not inequality signs) but there's a small circle circumstribing the angle of each bracket.
B) something like angle brackets but with bars joining the two on the top and bottom so the word is completely closed in.
C) names are closed in by diamonds ,but every name can only be two characters.
D) names are enclosed in isosceles trangles but every name can have no more or less than three characters, and one goes in each corner, the triangle is oriented so the bottom is flat
E) every name has exactly four (latin) letters, and they're always arranged in a 2x2 grid pattern, but the letters are scrunched so close together so as to mildly give the effect of one conglomeration of letters while still being apart enough to be readable. choice of font here may be crucial.
    instead of having it in a square shape have it in a diamond shape. then consolidate the symbol by adding a thin cross/plus shape that bisects the letters horizontally and vertically and meets in the middle of it all.
    or enclose it in a diamond like in option C but with four characters instead of 2.
    or make it the original square 2x2 shape, but consolidate it with a thin diamond drawing instead. so the diamond runs through them all vertically and its vertexes meet between the letters.
F) use angle brackets on the top and bottom of the name instead of the right and left of it. limit names to 2 or 3 letters.
G) something like normal (non-angle) brackets, but the top and bottom horizontal pieces actually go in at an angle, rike 30 - 45 degrees.

more ideas:
make new symbols out of supermposing the same letter upon itself 2 or 3 times, slightly displaced from each other. but that only gives us at most 52 new symbols.
do the same thing but also can rotate the letter, in 4 different orientations. or perhaps 8.
do the same thing but also allow to invert the letter.
possibly allow greek letters to be drawn into the mix while we do these things.
some international standarsd committee could hire an artist come up with symbols for math. the artist would already be a math dilattante or would be intelligent and willing to explore the arena.
  and i think in this case we'd have to hold votes.

another possibility: function names have free reign to have as many characters as they want, because they end in (),. therefore other variables like pi could be defined like this: pi(). (not that i'm suggesting we do away with the pi symbol.)
problem: how do you assign a value of 3 to "meh()"?
solution: math doesn't actually assign values. a thing is equal or it's not. it's like constraint programming. therefore meh() = 3 is identical to meh() == 3.
problem: it still makes us explicitly use a multiplication symbol to multiply consecutive entities.
solution: none
problem: it's ugly, counterintuitive and inelegant.
solution: none