Mathematics 1

Mathematics. The term is derived from Ancient Greek μάθημα; máthēma: knowledge, study, learning. The study of mathematics includes such topics as: 1. Number Theory (arithmetics and numbers); 2. geometry; 3. math-based Programming and modeling; 4. Mathematical Proofs; 5. Algebra; 6. Quadratic formulas; 7. Rubik's cube; 8. Calculus and analysis; 9. Discrete mathematics; 10. Combinatorics; 11. Mathematical Logic and Set Theory; 12. Applied mathematics; 13. Statistics and Decision Sciences; and 14. Computational mathematics. It should be noted that there is no consensus about the exact epistemology of mathematics.

1. Number Theory. This includes the distribution of prime numbers as a central point of study. In addition, number theory covers several subareas such as analytics, algebraics, the geometry of numbers (method oriented), diophantine equations, and the problem-oriented transcendence theory.

2. Geometry. This is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles, and circles, which were developed mainly for the needs of surveying and architecture. The basic statements are not subject to proof because they are self-evident (postulates) or part of the definition of the subject of study (axioms or hypotheses). 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. The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes, and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space.

Euclidean geometry was developed without a change of methods or scope until the 17th century when René Descartes introduced what is now called Cartesian coordinates. This was a major paradigm change since instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using numbers (their coordinates) and for the use of algebra and, later, calculus for solving geometrical problems. This split geometry into two parts that differ only by their methods, synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.

A significant 19th-century event was the discovery of non-Euclidean geometries, those that abandoned the parallel postulate. This joins Russel's paradox, as revealing the foundational crisis of mathematics, by questioning the truth of that postulate. This aspect of the problem was solved by systematizing the axiomatic method (viewing propositions as assumed without proof, being self=evident) and adopting that the validity of the chosen axioms (hypotheses) is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering invariant properties under specific space transformations.

This multiplied the number of subareas and generalizations of geometry to include: Projective geometry, introduced in the 16th century by Girard Desargues; Affine geometry, the study of properties relative to parallelism and independent from the concept of length; Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions; Manifold theory, the study of shapes that are not necessarily embedded in a larger space; Riemannian geometry, the study of distance properties in curved spaces; Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials; Topology, the study of properties that are kept under continuous deformations; Algebraic topology, the use in topology of algebraic methods, mainly homological algebra; Discrete geometry, the study of finite configurations in geometry; Convex geometry, or the

study of convex sets; and, Complex geometry, or the geometry obtained by replacing real numbers with complex numbers.

Analytic geometry allows the study of new shapes, particularly curves unrelated to circles and lines; these curves are defined either as a graph of functions (whose investigation led to differential geometry) or by implicit equations, often polynomial equations (which spawned algebraic geometry). In addition, analytic geometry makes it possible to consider spaces of higher than three dimensions that model more than physical space.

3. Math-Based Programming and Modeling. Mathematics is widely used in science for modeling phenomena. This enables the extraction of qualitative and quantitative or mixed-method probabilities and associated predictions from experimental laws.

Mathematics is essential in many fields, including natural sciences, engineering, medicine, finance, computer science, and social sciences. 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 mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later.

4. Mathematical Proofs. In the history of mathematics, the concept of a proof and its associated mathematical rigor first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed relatively slowly until the Renaissance, when algebra and infinitesimal calculus were added to arithmetic and geometry as the main areas of mathematics.

Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid increase in the development of mathematics. For example, at the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. This, in turn, gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. An example of this is the Mathematics Subject Classification, which lists more than sixty first-level areas of mathematics. Among them are the following: Pythagorean theorem, Conic Sections, Elliptic curve, Triangle on a paraboloid, Torus, Fractal, and Algebra.

5. Algebra. This may be viewed as the art of manipulating equations and formulas. Diophantus (3rd century) and Al-Khwarizmi (9th century) were two main precursors of algebra. The first one solved some relations between unknown natural numbers (equations) by deducing new relations until obtaining the solution. The second one introduced systematic methods for transforming equations (such as moving a term from one side of an equation to the other side). The term algebra is derived from the Arabic word that he used for naming one of these methods in the title of his main treatise.

6. Quadratic Formulas. These may be expressed as concise solutions in the form of quadratic equations. Algebra began to be a specific area only with François Viète (1540–1603), who introduced letters (variables) to represent unknown or unspecified numbers. This allows describing concisely the operations that have to be done on the numbers represented by the variables.

Until the 19th century, algebra consisted mainly of studying linear equations, also known as linear algebra, and polynomial equations in a single unknown were called algebraic equations (a term that is still in use, although it may be ambiguous). However, during the 19th century, variables began to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, operations acting on the elements of the set, and rules that these operations must follow. So, the scope of algebra evolved into essentially the study of algebraic structures. This object of algebra was called modern or abstract algebra; the latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.

7. Rubik's Cube.This is the study of possible moves within a concrete application of group theory. Some types of algebraic structures have beneficial and often fundamental properties in many areas of mathematics. Their study became autonomous parts of algebra and include: group theory; field theory; vector spaces, whose study is essentially the same as linear algebra; ring theory; commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry; homological algebra, Lie algebra and Lie group theory; and, Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the object of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra, to allow the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.

8. Calculus and Analysis. Previously called infinitesimal calculus, calculus was introduced in the 17th century by Newton and Leibniz, independently and simultaneously. It is fundamentally the study of the relationship of two changing quantities, called variables, in the case that one depends on the other. Euler expanded Calculus in the 18th century with the introduction of the concept of a function and many other results. Presently "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

The analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. The analysis includes many subareas, sharing some with other areas of mathematics; they include: Multivariable calculus; Functional analysis, where variables represent varying functions; Integration, measure theory, and potential theory, all strongly related to Probability theory; Ordinary differential equations; Partial differential equations; and, Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

9. Discrete Mathematics. This area of study represents a recently-emerging wide range of mathematics that aggregates several existing areas that deal with finite mathematical structures and processes where continuous variations are not found. These areas have in common that, because of the discrete aspect, the standard methods of calculus and mathematical analysis do not apply directly.[c] These areas have in common that algorithms, their implementation and their computational complexity play a major role. Despite the many different objects of study, they share often similar methods.

Discrete mathematics includes: Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting: configurations of geometric shapes; graph theory and hypergraphs; coding theory, including error correcting codes and a part of cryptography; Matroid theory; discrete probabilities; game theory (although continuous games are also studied, most common games, such as chess and poker are discrete); discrete optimization, including combinatorial optimization, integer programming, constraint programming; four color theorem and optimal sphere packing which are two major problems of discrete mathematics that have been solved since the second half of the 20th century. The open problem P=NP is important for discrete mathematics, since its solution would impact most parts of discrete mathematics, regardless of the solution.

10. Combinatorics. Combinatorics may be viewed primarily as the art of enumerating a prescribed set of objects. The history of combinatorics began in ancient societies that excavated combinatorial techniques. The usage of the term combinatorics in the modern mathematical sense was coined by Leibiniz in the 17th century, although Euler added many of its modern tools, such as generating functions.

Combinatorics has been used to study enumeration problems arising in pure mathematics within algebra, number theory, probability theory, topology and geometry, as well as many areas of applied math. Due to the wide variety of objects that may be enumerated, the theory is often subdivided based on either the type of objects under consideration or the methods used, including: algebraic combinatorics, analytic combinatorics; arithmetic combinatorics; combinatorial design theory; enumerative combinatorics; extremal combinatorics; geometric combinatorics; infinitary combinatorics; probabilistic combinatorics; topological combinatorics; and Ramsey theory. In addition, combinatorics is frequently used in graph theory, as well as the analysis of algorithms.

11. Mathematical Logic and Set Theory. These subjects have belonged to mathematics since the end of the 19th century. 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.

Before Cantor studied infinite sets, mathematicians were reluctant to consider infinite collections and considered infinity to result from an endless enumeration. Cantor's work offended many mathematicians not only by considering infinite sets but by showing that this implies different sizes of infinity (see Cantor's diagonal argument) and the existence of mathematical objects that cannot be computed or even explicitly described (for example, Hamel bases of the real numbers over the rational numbers). This led to the controversy over Cantor's set theory.

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigor. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting," "a point is a shape with a zero-length in every direction," and "a curve is a trace left by a moving point," etc.

This became the foundational crisis of mathematics.[13] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. 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 "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers. Many mathematicians have opinions on this nature and use their opinion—sometimes called "intuition"—to guide their studies and proofs.

This approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc., as mathematical objects and proving theorems about them. For example, Gödel's incompleteness theorems assert that, in every theory that contains the natural numbers, there are true theorems (that are provable in a more extensive theory) but not provable inside the theory.

This approach to studying the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted an intuitionistic logic that excludes the law of excluded middle.

These problems and debates led to a broad expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory, and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants, and other computer science aspects contributed to the expansion of these logical theories.

12. Applied Mathematics. This area concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. Applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models."

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

13. Statistics and Decision Sciences. Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modeling and the theory of inference—with model selection and estimation; the estimated models and consequential predictions should be tested on new data.

Statistical theory studies decision problems from the viewpoint of minimizing risk (expected loss) of an action. For example, focusing on parameter estimations, hypothesis testing, and selecting the best outcome. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.

14. Computational Mathematics. This area of study proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. For example, this applies to numerical analysis studies and techniques used to analyze problems by using what is referred to as functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with a particular focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.