Concept

Mathematics

Overview

Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.

There are many areas of mathematics, including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study of shapes and spaces that contain them), analysis (the quantitative study of approximation and convergence), and set theory (presently used as a foundation for all mathematics).

Mathematics involves the description and manipulation of abstract objects that are either abstractions from nature or purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove the properties of objects through proofs, which consist of a succession of applications of deductive rules to already established results. These results, called theorems, include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is widely used to model empirical phenomena, its results are established by deductive proof rather than by experiment. The relationship between mathematical truth, logic, and reality is a subject of philosophical debate. Some areas of mathematics, such as game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application but often find practical applications later.

Mathematical written records first appeared in Ancient Egypt and Mesopotamia, but the concept of proof and its associated mathematical rigor began in Ancient Greek mathematics, exemplified in Euclid's Elements. Mathematics was primarily divided into geometry and arithmetic until the 16th and 17th centuries, when algebra and infinitesimal calculus evolved into new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematic use of the axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

In the original texts

Passages from public-domain books where Mathematics is discussed.

As one of the chief triumphs of modern mathematics consists in having discovered what mathematics really is, a few more words on this subject may not be amiss. It is common to start any branch of mathematics--for instance, Geometry--with a certain number of primitive ideas, supposed incapable of definition, and a certain number of primitive propositions or axioms, supposed…
Bertrand Russell, Mysticism and Logic and Other Essays
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The nature of abstract mathematics (the general division of which will be examined in the following chapter) is clearly and exactly determined. It is composed of what is called the _Calculus_,[2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The _Calculus_ has the solution of all questions relating to…
Auguste Comte, The philosophy of mathematics
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The main ideas which lie at the base of mathematics are not at all recondite. They are abstract. But one of the main objects of the inclusion of mathematics in a liberal education is to train the pupils to handle abstract ideas.
Alfred North Whitehead, The organisation of thought, educational and scientific
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Mathematics originally signified any kind of discipline or learning, but now it is taken for that science which teaches or contemplates whatever is capable of being numbered or measured. That part of the mathematics which relates to numbers only, is called _arithmetic_; and that which is concerned about measure in general, whether length, breadth, motion, force, &c., is called _geometry_. As to the usefulness of…
Benjamin Franklin, Memoirs of Benjamin Franklin
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When we think of mathematics, we have in our mind a science devoted to the exploration of number, quantity, geometry, and in modern times also including investigation into yet more abstract concepts of order, and into analogous types of purely logical relations. The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example…
Alfred North Whitehead, Science and the modern world
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Mathematics

Overview

In the original texts

Philosophers who discussed Mathematics

Related ideas

Opposing perspectives

Place in the tradition