Uh, symbols and math??? Uh, as I read mathematical physics, I get the impression that certain symbols in certain equations are accepted throughout physics as already defined. But in math, we have, e.g.,
"For the set of real numbers R, some positive integer n, and R^n, with both R and R^n with the usual topologies and sigma algebras, we have function f: R^n --> R, Lebesgue measurable and >= 0."
That is, in writing in math, a popular but implicit standard is, each symbol used, even if as common as R and n, is defined before being used.
Sooooo, right there in the math writing, the symbols are defined. Or, right, an author might assume that the reader knows what "the usual topology" or "Lebesgue measurable" are but does not assume the reader knows what the symbols mean. I.e., the issue is not something about helping readers with their knowledge of math but, instead, just being clear about the symbols. Or, can assume the reader DOES know about the real numbers but does NOT assume that R is the set of real numbers -- and correctly so because R might be the set of rational numbers, some group (from abstract algebra), nearly anything.
Again for physics, E = mc^2 and J = ns^2 are not the same, not within the standards, maybe even if say J is energy in Joules, n is mass, and s is the speed of light!!!
The standard is based on the expectation that you should be able take a definition or a theorem out of its context and it should still make sense. The same symbols are often used for different things in different contexts, while concepts (even advanced ones) tend to remain unambiguous.
R is unambiguous if it's in blackboard bold, but otherwise it can mean almost anything. n is likely interpreted as a non-negative integer if left undefined, but you usually need to establish explicitly if it can be 0.
"For the set of real numbers R, some positive integer n, and R^n, with both R and R^n with the usual topologies and sigma algebras, we have function f: R^n --> R, Lebesgue measurable and >= 0."
That is, in writing in math, a popular but implicit standard is, each symbol used, even if as common as R and n, is defined before being used.
Sooooo, right there in the math writing, the symbols are defined. Or, right, an author might assume that the reader knows what "the usual topology" or "Lebesgue measurable" are but does not assume the reader knows what the symbols mean. I.e., the issue is not something about helping readers with their knowledge of math but, instead, just being clear about the symbols. Or, can assume the reader DOES know about the real numbers but does NOT assume that R is the set of real numbers -- and correctly so because R might be the set of rational numbers, some group (from abstract algebra), nearly anything.
Again for physics, E = mc^2 and J = ns^2 are not the same, not within the standards, maybe even if say J is energy in Joules, n is mass, and s is the speed of light!!!