Notation, language, and rigor
Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics
was written out in words, a painstaking process that limited mathematical
In the 18th century, Euler was responsible for many of the notations in use today. 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 can also be hard for beginners. Words such as or and only have more
precise meanings than in everyday speech. Additionally, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. 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".
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.The level of rigor
expected in mathematics has varied over time: the Greeks expected detailed
arguments, but at 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 large
computations are hard to verify, 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 impossible.
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.