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Belongs to subject Mathematics

Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Mathematical discoveries continue to be made today. The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. Haskell Curry defined mathematics simply as "the science of formal systems". There is not even consensus on whether mathematics is an art or a science. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science. Mathematical research often seeks critical features of a mathematical object. The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. Before that, mathematics was written out in words, limiting mathematical discovery. Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Some disagreement about the foundations of mathematics continues to the present day. Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science, as well as to category theory. Theoretical computer science includes computability theory, computational complexity theory, and information theory. {\displaystyle p\Rightarrow q}

Mathematical logic Set theory Category theory Theory of computation

Real numbers are generalized to complex numbers. Rational numbers Real numbers Complex numbers

Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Group theory Graph theory Order theory

In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Measure theory

Chaos theory

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. The term 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 science, engineering, and other areas of mathematical practice. 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.

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. 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.

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for 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.

Game theory Probability theory Mathematical finance Mathematical physics Mathematical chemistry Mathematical biology Mathematical economics Control theory

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