Mathematics is often defined as the study of certain topics, such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the 'purest' mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries.
- 1 History
- 2 Notation, language, and rigor
- 3 Overview of fields of mathematics
- 4 Major themes in mathematics
- Main article: History of mathematics
Notation, language, and rigor
Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales.
The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation, which like musical notation has a definite content, and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said, rather bitchily, that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary.
Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis).
Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Overview of fields of mathematics
The major disciplines within mathematics first 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 (i.e. algebra, geometry and analysis). In addition to these three main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic and other simpler systems (foundations) and to the empirical systems of the various sciences (applied mathematics).
The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized 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 everyday numbers. Long-standing questions about ruler-and-compass constructions were finally settled by Galois theory. 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 fewer dimensions), later also generalized to non-Euclidean geometries which play a central role in general relativity. 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; topology, the greatest growth area in the twentieth century, has a focus on the concept of continuity. Both the group theory of Lie groups and topology reveal the intimate connections of space, structure and change.
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 generalize 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; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.
In order to clarify the foundations of mathematics, the fields first of mathematical logic and then set theory were developed. Mathematical logic, which divides into recursion theory, model theory and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally 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 where chance plays a part. It is used in all sciences. Numerical analysis investigates methods for efficiently solving a broad range of mathematical problems numerically on computers, beyond human capacities, and taking rounding errors and other sources of error into account to obtain credible answers.
Major themes in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.
This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.
- Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base
- Ways to express and handle change in mathematical functions, and changes between numbers.
- Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions
- Pinning down ideas of size, symmetry, and mathematical structure.
- Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory
- A more visual approach to mathematics.
- Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry
- Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.
- Applied mathematics uses the full knowledge of mathematics to solve real-world problems.
- Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory
See list of theorems for more
- These theorems have interested mathematicians and non-mathematicians alike.
- Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – Church-Turing thesis– Riemann hypothesis – Continuum hypothesis – Central limit theorem – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Fundamental theorem of projective geometry – Gauss-Bonnet theorem.
See list of conjectures for more
- These are open questions that are topics of active, current research.
- Goldbach's conjecture – Twin prime conjecture – Collatz conjecture – Poincaré conjecture – classification theorems of surfaces – P=NP
- In a slightly different category is the continuum hypothesis; some mathematicians consider the issue closed on the grounds that it's independent of ZFC, whereas others are actively trying to determine whether it's true.
Foundations and methods
- Approaches to understanding the nature of mathematics also influence the way mathematicians study their subject.
- Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Foundations of mathematics – Set theory – Symbolic logic – Model theory – Category theory – Logic – Reverse Mathematics – Table of mathematical symbols
History and the world of mathematicians
See also list of mathematics history topics
- History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues