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Formula that represents a mathematical object

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax. Many authors distinguish an expression from a formula, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects. For example, is an expression, while is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example takes the value false if x is given a value less than –1, and the value true otherwise.

Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions. In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression 1 + 2 × 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See also Operations § Calculators). The semantic rules may declare that certain expressions do not designate any value (for instance when they involve division by 0); such expressions are said to have an undefined value, but they are well formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator to designate an internal direct sum.

Formal languages allow formalizing the concept of well formed expressions. In the 1930s, a new type of expressions, called lambda expressions, were introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. They form the basis for lambda calculus, a formal system used in mathematical logic and the theory of programming languages. The equivalence of two lambda expressions is undecidable. This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (Richardson's theorem).