## 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

discovery.

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 GĂ¶del'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.

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