Online Encyclopedia
Mathematics
Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of "figures and numbers". In the formalist view, it is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics. However, mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. Some mathematicians refer to their subject as the Queen of the Sciences. Mathematics is considered absolute, without any reference.
Mathematics is often abbreviated as math (American English) or maths (British English).
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Overview and history of mathematics
See the article on the history of mathematics for details.
The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning".
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, first 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 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 less dimensions), 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 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 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 Censored page . 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.
In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.
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.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.
Topics in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of subfields and topics reflects one organizational view of mathematics.
Quantity
In general, these topics and ideas present explicit measurements of sizes of numbers or sets, or ways to find such measurements.
- Number – Natural number – Pi – 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
Change
These topics give ways to measure change in mathematical functions, and changes between numbers.
- Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems and chaos theory – List of functions
Structure
These branches of mathematics measure size and symmetry of numbers, and various constructs.
- Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory
Space
These topics tend to quantify a more visual approach to mathematics than others.
- Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry
Discrete mathematics
Topics in discrete mathematics deal with branches of mathematics with objects that can only take on specific, separated values.
- Combinatorics – Naive set theory – Probability – Theory of computation – Finite mathematics – Cryptography – Graph theory – Game theory
Applied mathematics
Fields in applied mathematics use knowledge of mathematics to real world problems.
Famous theorems and conjectures
These theorems have interested mathematicians and non-mathematicians alike.
- Fermat's last theorem – Goldbach's conjecture – Twin Prime Conjecture – Gödel's incompleteness theorems – Poincaré conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – Scholz Conjecture – Church-Turing thesis
Important theorems
These are theorems that have changed the face of mathematics throughout history.
- Riemann hypothesis – Continuum hypothesis – P=NP – Pythagorean theorem – Central limit theorem – Fundamental theorem of calculus – Fundamental theorem of algebra – Fundamental theorem of arithmetic – Fundamental theorem of projective geometry – classification theorems of surfaces – Gauss-Bonnet theorem
Foundations and methods
Such topics are approaches to mathematics, and 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 – Theorem-proving – Logic – Reverse Mathematics – Table of mathematical symbols
History and the world of mathematicians
- 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
Mathematics and other fields
- Mathematics and architecture – Mathematics and education – Mathematics of musical scales
Mathematical coincidences
Mathematical tools
Old:
New:
- Calculators and computers
- Programming languages
- Computer algebra systems (listing)
- Internet shorthand notation
- statistical analysis software
Quotes
Referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequences thereof are then logically derived, Bertrand Russell said:
- Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
This may explain why John Von Neumann once said:
- In mathematics you don't understand things. You just get used to them.
About the beauty of Mathematics, Bertrand Russell said in Study of Mathematics:
- Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
Elucidating the symmetry between the creative and logical aspects of mathematics, W.S. Anglin observed, in Mathematics and History:
- Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
Mathematics is not...
Mathematics is not numerology. Although numerology uses modular arithmetic to boil names and dates down to single digit numbers, numerology arbitrarily assigns emotions or traits to numbers without bothering to prove the assignments in a logical manner. Mathematics is concerned with proving or disproving ideas in a logical manner, but numerology is not. The interactions between the arbitrarily assigned emotions of the numbers are intuitively estimated rather than calculated in a thoroughgoing manner.
Mathematics is not accountancy. Although arithmetic computation is crucial to the work of accountants, they are mainly concerned with proving that the computations are true and correct through a system of doublechecks. The proving or disproving of hypotheses is very important to mathematicians, but not so much to accountants. Advances in abstract mathematics are irrelevant to accountancy if the discoveries can't be applied to improving the efficiency of concrete bookkeeping.
Mathematics is not physics, despite the number of historical and philosophical relations between the two.
Mathematics is not a science, in the proper connotation of the word. Some thinkers see mathematicians as scientists, regarding physical experiments as inessential or mathematical proofs as equivalent to experiments. But they themselves do not see mathematics as a science, since it does not require experimental test of its theories and hypotheses. In either case, the fact that mathematics is such a useful tool in describing the universe is a central issue in its philosophy. But in any case, mathematics is essential to science, being its most important function the role it plays in the expression of scientific models.
Bibliography
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973);
External links
- A mathematics, science, technology, and physics discussion forum
- Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics.
- A mathematical thesaurus maintained by the NRICH project at the University of Cambridge (UK), Connecting Mathematics
- Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics, focusing on classical mathematics.
- Planet Math. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL license, allowing article exchange with Wikipedia. Uses TeX markup.
- Stefanov, Alexandre: Textbooks in Mathematics. A list of free online textbooks and lecture notes in mathematics.
- Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. A huge collection of articles on various math topics with more than 400 illustrated with Java applets.
- Mathforge. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- Metamath. A site and a language, that formalize math from its foundations.
- Mathematics Quizzes & Worksheets: Grade 1 - Grade 2 - Grade 3 - Grade 4 - Grade 5