While most mathematicians don't typically concern themselves with the questions raised by Platonism, some more philosophically-minded ones do identify as Platonists, even in contemporary times.[51]. [19] Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. {\displaystyle (\mathbb {Z} )} [22] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[23] trigonometry (Hipparchus of Nicaea, 2nd century BC),[24] and the beginnings of algebra (Diophantus, 3rd century AD).[25]. (

Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories. and later expanded to integers

( Mathematicians refer to this precision of language and logic as "rigor".

Calculus, consisting of the two subfields infinitesimal calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities (variables).

Today, all sciences pose problems studied by mathematicians, and conversely, results from mathematics often lead to new questions and realizations in the sciences. This allows one to use algebra (and later, calculus) to solve geometrical problems.

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Presently "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts. Mathematicians develop mathematical hypotheses, known as conjectures, using trial and error with intuition too, similarly to scientists. It is often shortened to maths or, in North America, math. A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be proved. [70] To date, only one of these problems, the Poincar conjecture, has been solved.

The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. [30], The apparent plural form in English, like the French plural form les mathmatiques (and the less commonly used singular derivative la mathmatique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathmatik ( ), used by Aristotle (384322BC), 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 Greek. Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.

This principle, which is foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.

This has resulted in several mistranslations. Such curves can be defined as graph of functions (whose study led to differential geometry). [55] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

[56] Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs.

Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretisation with special focus on rounding errors.

( For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number as a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms). The latter applies to every mathematical structure (not only algebraic ones).

This in turn opened up both fields to greater abstraction and spawned entirely new subfields. Stated in 1742 by Christian Goldbach, it remains unproven to this day despite considerable effort. By questioning the truth of that postulate, this discovery joins Russel's paradox as revealing the foundational crisis of mathematics.

Mathematics (from Ancient Greek ; mthma:'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic, number theory),[1] formulas and related structures (algebra),[2] shapes and the spaces in which they are contained (geometry),[1] and quantities and their changes (calculus and analysis).[3][4][5]. The book containing the complete proof has more than 1,000 pages.

[d] Algorithms - especially their implementation and computational complexity - play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The rigor expected in mathematics has varied over time: the ancient Greeks expected detailed arguments, but in Isaac Newton's time, the methods employed were less rigorous (not because of a different conception of mathematics, but because of the lack of the mathematical methods that are required for reaching rigor). (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.). Perhaps even more surprising is when ideas flow in the other direction, and even the "purest" mathematics lead to unexpected predictions or applications. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.[53]. Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers.

[definition needed] Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;[11] the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available).

[45] However, some authors emphasize that mathematics differs from the modern notion of science in a major way: it does not rely on empirical evidence.[46][47][48][49]. When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inferencewith model selection and estimation; the estimated models and consequential predictions should be tested on new data. Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. To allow deductive reasoning, some basic assumptions need to be admitted explicitly as axioms. The most prestigious award in mathematics is the Fields Medal,[58][59] established in 1936 and awarded every four years (except around World War II) to up to four individuals.

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.

Only very exceptional results are accepted as not fitting into one axiomatic system or another.[41].

[60][61] It is considered the mathematical equivalent of the Nobel Prize.[61]. 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. This, in turn, gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. Other prestigious mathematics awards include: A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.

The ancient philosopher Plato argued this was possible because material reality reflects abstract objects that exist outside time. [21] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics. In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind.

Many easily-stated number problems have solutions that require sophisticated methods from across mathematics. In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour.

In practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences, notably deductive reasoning from assumptions. N [8] Since its beginning, mathematics were essentially divided into geometry, and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[b] and infinitesimal calculus were introduced as new areas.

[10], Applied mathematics is the study of mathematical methods used in science, engineering, business, and industry.

[68], A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000.

Inaccurate predictions imply the need for improving or changing mathematical models, not that mathematics is wrong in the models themselves. Analytic geometry allows the study of curves that are not related to circles and lines. This enables the extraction of quantitative predictions from experimental laws.

Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians. [42] Experimental mathematics and computational methods like simulation also continue to grow in importance within mathematics. This object of algebra was called modern algebra or abstract algebra.

It is fundamentally the study of the relationship of variable that depend on each other.

String theory, on the other hand, is a proposed framework for unifying much of modern physics that has inspired new techniques and results in mathematics.

[34] Some just say, "Mathematics is what mathematicians do. For example, the movement of planets can be accurately predicted using Newton's law of gravitation combined with mathematical computation. Before the Renaissance, mathematics was divided into two main areas: arithmetic regarding the manipulation of numbers, and geometry regarding the study of shapes. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. In particular, mathmatik tkhn ( ; Latin: ars mathematica) meant "the mathematical art.". Mathematical notation led to algebra, which, roughly speaking, consists of the study and the manipulation of formulas.

This led to the controversy over Cantor's set theory. The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. Mathematicians strive to develop their results with systematic reasoning in order to avoid mistaken "theorems". Number theory began with the manipulation of numbers, that is, natural numbers One prominent example is Fermat's last theorem.