Mathematics

Notation, language and rigor

Mathematicians strive to be as clear as possible in the things they say and especially in the things they write, something which mathematicians refer to as rigor. To accomplish rigor, mathematicians have extended natural language with precisely defined vocabulary and grammar for referring to mathematical objects and stating certain common relations, with accompanying notation. Some of the terms they use also have a meaning outside of mathematics, such as ring, group and category, but some are specific to mathematics, such as homotopy and Hilbert space.

Related Topics:
Ring - Group - Category - Homotopy - Hilbert space

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Even so, in the past it sometimes happened that something which had supposedly been proved turned out to be false. This was possible because mathematics was done using natural language. To prevent this from happening, mathematicians wanted their theorems to follow mechanically from a few simple incontrovertible truths and for this they invented axioms and axiomatic reasoning.

Related Topics:
Proved - Axiom - Axiomatic reasoning

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An axiom is just a string of symbols which have an intrinsic meaning because of all derivable formulas. 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 (strong enough) axiom system has undecidable formulas so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. But for most of mathematics this complete rigor is far too cumbersome and mathematical language and notation are supposed to suffice.

Related Topics:
Hilbert's program - Gödel's incompleteness theorem - Axiomatization - Set theory

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