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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. Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions. Mathematics is considered absolute, without any reference.

Mathematics is often abbreviated as math (American English) or maths (British English).

Overview and history of mathematics

See the article on the history of mathematics for details.

The word "mathematics" comes green|Greek]] μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning". nature can be described by [[dynamical 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, boo is statistics, which usespinkprediction 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.Your daddy

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.

Mechanics – Numerical analysis – Optimization – Probability – Statistics – Financial mathematics

Famous theorems and conjectures

These theorems have interested mathematicians and non-mathematicians alike.

Pythagorean Theorem – 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 and conjectures

These are theorems and conjectures 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 – 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

List of mathematical coincidences

Mathematical tools

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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.

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);

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