Mathematics

Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of "figures and numbers". A formal and rigorous definition would be: "A science that deals with the analysis of similarities and dissimilarities between systems". Mathematics is widely acknowledged as probably the only perfect science, in the sense that the solutions it gives (from proven theorems) are absolute and stand the test of time. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that mathematical knowledge is revised and updated in a different way; though arguably founded on experiment in some manner, it is not comparable to the natural sciences in this respect. Mathematics is usually regarded as a tool for other sciences, even though the development of mathematics is not necessarily done with other sciences in mind. The specific structures that are investigated by mathematicians often do have their origin in the natural sciences, most commonly in physics. Contemporary mathematics also relates closely to applications in computer science and communication theory. In the formalist view, widely accepted as a description by professionals in the field, mathematics is defined as the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation. Mathematics might accordingly be seen as an extension of spoken and written natural languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. There are other views, and some are described in the article on the philosophy of mathematics. Mathematics itself is usually considered absolute, without any reference. Mathematicians define and investigate some structures for reasons purely internal to mathematics; they may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians work for purely aesthetic reasons, viewing mathematics more as an art rather than as a practical or applied science; this is the same kind of motivation as poets and philosophers may experience, and no more explicable. Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions. Mathematics is often abbreviated as math (American English) or maths (British English).

Overview and History of Mathematics

The word "mathematics" comes from the Greek µ???µa (máthema) which means "science, knowledge, or learning"; µa??µat???? (mathematikós) means "fond of learning". The major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change. The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vectors, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space. The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and less dimensions), later also generalized to non-Euclidean geometries which play a central role in general relativity. Several long standing questions about ruler-and-compass constructions were finally settled by Galois theory. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity. Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior. In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic, and model theory were developed. When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory, and algorithmic information theory. Many of these questions are now investigated in theoretical computer science. Discrete mathematics is the common name for those fields of mathematics useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.

Mathematics is not...

While physics deals with an extremely wide variety of systems, there are certain theories that are used by physics as a whole, and not by any single field. Each of these theories is believed to be basically correct, within a certain domain of validity. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These theories continue to be areas of active research; for instance, a remarkable aspect of classical mechanics known as chaos was discovered in the 20th century, three centuries after its formulation by Isaac Newton. However, few physicists expect any of them to prove fundamentally misguided. Therefore, they are used as the basis for research into more specialized topics, and any contemporary student of physics, regardless of his or her specialization, is generally expected to be well-versed in all of them.

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