Maths History-Notation

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 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.

Maths History - History

History

• The evolution of mathematics might be seen as an ever-increasing series of abstractions,
  or alternatively an expansion of subject matter.
• The first abstraction was probably that of numbers.
• The realization that two apples and two oranges have something in common was a
  breakthrough in human thought.
• In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Arithmetic
  (addition, subtraction,multiplication and division), naturally followed.
• Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.
• Numeralsystems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus. The Indus Valley civilization developed the modern decimal system, including the concept of zero.
• From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics into the studies of quantity, structure,space, and change.
• since been greatly extended, and there has been a fruitful interaction between
  mathematics and science, to the benefit of both.
• Mathematical discoveries have been made throughout history and continue to be made today.
• According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American
  Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year.
• The overwhelming majorityof works in this ocean contain new mathematical theorems and
  their proofs."

Maths History - Etymology

• The word "mathematics" (Greek: μαθηματικά or mathēmatiká)
  comes from the Greek μάθημα (máthēma),
• which means learning, study, science, and additionally came to have the narrower and
• more technical meaning "mathematical study", even in Classical times.
• Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical.
• In particular, μαθηματικὴτέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.
• The apparent plural form in English, like the French plural form les mathématiques (and
  the less commonly used singular derivative la mathématique), goes back
  to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα
  μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things
  mathematical".
• In English, however, the noun mathematics takes
  singular verb forms.
• It is often shortened to math in English-speaking
  North America and maths elsewhere.