What do your symbols *, 0, 1, 2, and = mean? Do they mean the same as in standard arithmetic? If so, then there is no "thing" that would fit those equations in a consistent way. At least, that's true if you accept the Peano axioms and Zermelo–Fraenkel set theory.
Answers & Comments
If ☽ * 0 = 1 then
☽ * 0 + ☽ * 0 = 2
☽ * (0 + 0) = 2
☽ * 0 = 2
1 = 2
So if such a number ☽ exists then the world of mathematics collapses .
The graphing calculator would wouldn't it?
What do your symbols *, 0, 1, 2, and = mean? Do they mean the same as in standard arithmetic? If so, then there is no "thing" that would fit those equations in a consistent way. At least, that's true if you accept the Peano axioms and Zermelo–Fraenkel set theory.
Mathematicians would ignore it because it has no utility.
Suppose I create a new letter of the alphabet,⊗. Would anything break? No. People would ignore it.