Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[1] algebra,[2] geometry,[1] and analysis,[3] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
"Math" and "Maths" redirect here. For other uses, see Mathematics (disambiguation) and Math (disambiguation).
Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.[4]
Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications.[5][6]
Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[7] Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.[8] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,[9] which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.
Etymology
The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt",[10] "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in Classical times.[b] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".[14] In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art".[10]
Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.[15]
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 change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.[16]
The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means 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, inherited from Greek.[17] In English, the noun mathematics takes a singular verb. It is often shortened to maths[18] or, in North America, math.[19]
Relationship with astrology and esotericism
Some renowned mathematicians have also been considered to be renowned astrologists; for example, Ptolemy, Arab astronomers, Regiomantus, Cardano, Kepler, or John Dee. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, Theodor Zwinger wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences [of stars]" to foresee their "conjunctions and oppositions".[170] As of 2023, astrology is no longer considered a science, but pseudoscience.[171]
The most prestigious award in mathematics is the Fields Medal,[216][217] established in 1936 and awarded every four years (except around World War II) to up to four individuals.[218][219] It is considered the mathematical equivalent of the Nobel Prize.[219]
Other prestigious mathematics awards include:[220]
A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[228] This list has achieved great celebrity among mathematicians,[229] and at least thirteen of the problems (depending how some are interpreted) have been solved.[228]
A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.[230] To date, only one of these problems, the Poincaré conjecture, has been solved.[231]