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In its broadest possible sense:

A symbol is an object used to represent another object.

In a narrower and more "mathematical" sense, a symbol is a sign of a particular shape to which is assigned a meaning, and is used to represent a concept or identify a particular object.

It is generally much more convenient to use a symbol than the plain speaking that it replaces, because it is invariably more compact. One character can replace a large number of words. As definitions become more complex, the symbols tend to convey more information -- but by the same coin, understanding exactly what a symbol means becomes more difficult.

Symbols may mean different things in different contexts. A symbol that means something in one context may mean something completely different in another. This is because the number of different concepts is greater than human ingenuity can create symbols for, and some of them naturally have to be used more than once.

This does not matter as long as, before we use any symbol, we define exactly what we mean by it. Some symbols are standard and rarely need defining, but in some contexts there are subtle differences to the exact meaning of a "standard" symbol. Therefore all fields of mathematics generally introduce themselves with a rash of definitions, many of which are symbols.

Also known as

A symbol is sometimes known under its older term ideogram.

The term is in apposition to the term phonogram, which is a sign standing directly for a sequence of sounds that are then interpreted according to the natural language in which the phonogram is written.

Both an ideogram and a phonogram can be considered as symbols.

In certain contexts, the word label can be used.

Historical Note

The use of symbols was one of the characteristics that distinguish symbolic logic from classical logic.

Also see

In formal systems, it is usual to name certain collections of symbols after the objects they represent.

For example, the language of predicate logic has predicate symbols and function symbols to represent predicates and functions, respectively.