Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change. There are a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.
What mathematics does
Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof.
When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.
Origins of mathematics
Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
Further development of mathematics
Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Historical views of mathematics
Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth."
Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".
Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".
David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."
Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
French mathematician Claire Voisin states "There is creative drive in mathematics, it's all about movement trying to express itself."
Applications and value of mathematics
Mathematics is used throughout the world as an essential tool in many fields, including:
Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries. This has led to the development of entirely new mathematical disciplines, such as statistics and game theory.
Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
Fields or branches of mathematics
Mathematics can, broadly speaking, be subdivided into the study of:
- Quantity, (numbers and arithmetic)
- change (analysis)
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),
Computational mathematics (maths that often requires long repetitive calculations) include: