Mathematics (from Greek μάθημα máth...
Mathematics (from Greek μάθημα máthēma "knowledge, study, learning") is the study of quantity, space, structure, and change.

Mathematicians resolve the truth or falsity of conjectures by mat...
Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.

The research required to solve mathematical problems can take yea...
The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

However, mathematical proofs are less formal and painstaking than...
However, mathematical proofs are less formal and painstaking than proofs in mathematical logic.

Since the pioneering work of Giuseppe Peano (1858-1932), David Hi...
Since the pioneering work of Giuseppe Peano (1858-1932), David Hilbert (1862-1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.

When those mathematical structures are good models of real phenom...
When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics...
Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.

Practical mathematics has been a human activity for as far back a...
Practical mathematics has been a human activity for as far back as written records exist.

Rigorous arguments first appeared in Greek mathematics, most nota...
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.

Mathematics developed at a relatively slow pace until the Renaiss...
Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.

Galileo Galilei (1564-1642) said, 'The universe cannot be read un...
Galileo Galilei (1564-1642) said, 'The universe cannot be read until we have learned the language and become familiar with the characters in which it is written.

It is written in mathematical language, and the letters are trian...
It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

Without these, one is wandering about in a dark labyrinth'....
Without these, one is wandering about in a dark labyrinth'.

The mathematician Benjamin Peirce (1809-1880) called mathematics ...
The mathematician Benjamin Peirce (1809-1880) called mathematics "the science that draws necessary conclusions".

David Hilbert said of mathematics: "We are not speaking here of a...
David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense.

Mathematics is not like a game whose tasks are determined by arbi...
Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.

Rather, it is a conceptual system possessing internal necessity t...
Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.

" Albert Einstein (1879-1955) stated that "as far as the laws of ...
" Albert Einstein (1879-1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".

More recently, Marcus du Sautoy has called mathematics 'the Queen...
More recently, Marcus du Sautoy has called mathematics 'the Queen of Science.

Mathematics is used throughout the world as an essential tool in ...
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences.

Applied mathematics, the branch of mathematics concerned with app...
Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory.

Mathematicians also engage in pure mathematics, or mathematics fo...
Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind.

There is no clear line separating pure and applied mathematics, a...
There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

The word "mathematics" comes from the Greek μάθ ...
The word "mathematics" comes from the Greek μάθημα (máthēma), which means in ancient Greek what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson.

The word máthēma comes from μανθά&#...
The word máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn.

The word "mathematics" in Greek came to have the narrower and mor...
The word "mathematics" in Greek came to have the narrower and more technical meaning "mathematical study", even in Classical times.

Its adjective is μαθηματι...
Its adjective is μαθηματικός (mathēmatikós), meaning related to learning or studious, which likewise further came to mean mathematical.

In particular, μαθηματι&#...
In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.

In Latin, and in English until around 1700, the term "mathematics...
In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800.

This has resulted in several mistranslations: a particularly noto...
This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of "mathematici" meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The apparent plural form in English, like the French plural form ...
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 (384-322BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.

It is often shortened to maths or, in English-speaking North Amer...
It is often shortened to maths or, in English-speaking North America, math.

The evolution of mathematics might be seen as an ever-increasing ...
The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter.

The first abstraction, which is shared by many animals, was proba...
The first abstraction, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count physical objects, prehist...
In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.

Since numeracy pre-dated writing, further steps were needed for r...
Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.

[citation needed] Numeral systems have been many and diverse, wit...
[citation needed] Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.

Mayan numerals The earliest uses of mathematics were in trading,...
Mayan numerals The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time.

More complex mathematics did not appear until around 3000 BC, whe...
More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.

The systematic study of mathematics in its own right began with t...
The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.

Mathematics has since been greatly extended, and there has been a...
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Sevryuk, in the January 2006 issue of the Bulletin of the America...
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 datab...
9 million, and more than 75 thousand items are added to the database each year.