History of mathematics
- See Timeline of mathematics for a timeline of events in mathematics. See mathematician for a list of biographies of mathematicians.
- Also see The Nine Chapters on the Mathematical Art for information about the development of mathematics in China.
Historically, 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, firstly 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 vector, 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, but 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 coordinate system, 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 doing just 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.
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.
The development of the concept of number
Mathematics did not start with the concept of the complex numbers. It took many years and much discussion to get this far. Roughly speaking over time mathematicians have broadened the definition of number. Opinions differ as to how to treat the complex numbers philosophically.
Many of the extentions of the concept of number can be seen as responses to equations that would otherwise have had no solution. In each of the extentions given below we start with an equation and then give the extention to the system which allows the equation to be solved. We start with the notion of positive integers and 0 although it should be noted that some ancient mathematics did not have the concept of 0. Also note that it was assumed that the normal algebraic operations return only one value (division by zero is not defined).
- X + 1 = This equation requires the existence of negative numbers such as - 1 for its solution. The word negative was originally used by those who opposed the introduction of such numbers.
- 5X = 3 This equation requires the existence of fractional numbers for its existence. If we allow the solution of all equations of the form mX = n then we get the rational numbers (m and n are both integers).
- X * X - 2 = 0 has no rational solution. Mathematicians responded by introducting radicals which allowed many polynomials to be solved.
- X * X + 1 = 0 is the equation that introduces us to the complex numbers. Many people argued that it was just an imaginary construct to solve the cubic and shouldn't be considered 'real'. This is the origin of the terms imaginary and real. However it was found that a whole new beautiful world of complex numbers opened up if you did allow them. To represent a solution to this equation mathematicians chose the letter i. Even with all of these extensions of the naturals we are still not finished.
In order to construct the complex numbers we need only one more assumption: Any set of real numbers with an upper bound has a least upper bound. This fills in the real line with all of the irrational numbers that cannot be derived merely from the equations above. It is worth noting that this is an entirely different type of extension. This is because of the cardinality of complex numbers is greater than that of the rationals. Once this is done all polynomial equations can be solved (although this can be done in smaller fields than the complex numbers).
Mathematicians today rarely view the development of the complex numbers in this way (the preferred teaching method does not emphasize this stepwise development) but it demonstrates the tension in mathematics between the rigorous and the creative which is the main power behind much of modern mathematics.
Interestingly the independence of the continuum hypothesis can be seen as an inability to prove whether or not certain real numbers should be thought to exist.
- John J O'Connor and Edmund F Robertson: MacTutor History of Mathematics, http://www-groups.dcs.st-andrews.ac.uk/~history/. Contains biographies, timelines and historical articles about mathematical concepts.
- Jeff Miller: Earliest uses of various mathematical symbols, http://members.aol.com/jeff570/mathsym.html
- Jeff Miller: Earliest known uses of some of the words of mathematics, http://members.aol.com/jeff570/mathword.html
- Ian Pearce: History of Indian mathematics, http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/index.html
- History of Mathematics, public domain article
- Important publications in history of mathematics
- Fred Rickey: History of calculus, http://www.dean.usma.edu/math/people/rickey/hm/default.htm