Mathematical notation

Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.

Mathematical language also is hard for beginners. Even common words, such as or and only, have more precise meanings than in everyday speech. Mathematicians, like lawyers, strive to be as unambiguous as possible. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings, and mathematical jargon includes technical terms such as "homeomorphism" and integrable. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

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Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis). The level of rigor expected in mathematics has varied over time; the Greeks expected detailed arguments, but by the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since errors can be made in a computation, such proofs may not be sufficiently rigorous.

Axioms in traditional thought were 'self-evident truths', but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

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Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. The complexity of such notation ranges from relatively simple symbolic representations, such as 1 and 2; to conceptual symbols, such as + and dy/dx; to equations, functions, and variables. A mathematical notation is a writing system used for recording concepts in mathematics. The notation uses symbols or symbolic expressions which are intended to have a precise semantic meaning.
In the history of mathematics, these symbols have denoted numbers, shapes, patterns, and change. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing mathematics as a language.
The media used for writing are recounted below, but common materials currently include paper and pencil, or perhaps computer screen and keyboard, as well as board and chalk. One key point behind mathematical notation is the systematic adherence to mathematical concepts as recounted below. (But see also some related concepts: Topic (linguistics), Logical argument, Cogency, Mathematical logic, Model theory, and Major themes in mathematics.)

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A mathematical expression is a sequence of symbols which can be evaluated. For example, if the symbols represent numbers, the expressions are evaluated according to a conventional order of operations which provides for calculation, if possible, of any expressions within parentheses, followed by any multiplications and divisions done from left to right, finally any additions or subtractions done from left to right. In a computer language, these rules are implemented by the compilers. For more on expression evaluation, see the computer science topics: eager evaluation, lazy evaluation, and evaluation operator.

Precision is necessary so that we can know what we are investigating. Suppose that we have statements, denoted by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the symbols refer to those denoted objects, perhaps in a model. The semantics of that object has a heuristic side and a deductive side. In either case, we might want to know the properties of that object, which we might then list in an intentional definition.

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Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols. This mathematical notation might include annotation such as "All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function" "A mapping from the real numbers to the complex numbers"

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