Mathematics

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Mathematics is often defined as the study of 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.

The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in American English.

Contents

History

Main article: History of mathematics

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.

Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to measure land and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of number, space and change.

Mathematics since has been been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, aesthetics, and pure and applied mathematics

Mathematics arises wherever there are difficult problems that merit careful mental investigation. At first these were found in commerce, land measurement and later astronomy. Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language. In the past Newton invented infinitesimal calculus and Feynman his Feynman path integral, by a mixture of reasoning and physical insight; it happens too with today's string theory.

Mathematics is relevant for the subject which inspired it, and can be applied to solve further problems in that same area. There is also mathematics that joins the general stock of concepts, and requires to be put on a 'common denominator' with existing ideas. During the nineteenth century this distinction hardened up, into applied mathematics as opposed to pure mathematics.

Mathematics interests mathematicians because of its elegance, the intrinsic aesthetics or inner beauty, which is hard for anyone to articulate. Simplicity and generality are valued. These seemingly incompatible properties may combine in a piece of mathematics: a unifying generalization for several subfields, or a helpful tool for common calculations. Pure mathematics, it may seem, has value only in its beauty. But a part of mathematics, which was considered only of interest to mathematicians, has frequently later become applied mathematics because of some new insight or field of study opening up, as if it anticipated later needs.

Notation, language, and rigor

Main article: Mathematical notation

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 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.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)."

If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [1]

In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

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.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.

1, 2, \ldots -1, 0, 1, \ldots \frac{1}{2}, \frac{2}{3}, 0.125,\ldots \pi, e, \sqrt{2},\ldots i, 3i+2, e^{i\pi/3},\ldots
Natural numbers Integers Rational numbers Real numbers Complex numbers
NumberNatural numberIntegersRational numbersReal numbersComplex numbersHypercomplex numbersQuaternionsOctonionsSedenionsHyperreal numbersSurreal numbersOrdinal numbersCardinal numbersp-adic numbersInteger sequencesMathematical constantsNumber namesInfinityBase

Change

Ways to express and handle change in mathematical functions, and changes between numbers.
36 \div 9 = 4 \int 1_S\,d\mu=\mu(S)
Arithmetic Calculus Vector calculus Analysis
\frac{d^2}{dx^2} y = \frac{d}{dx} y + c
Differential equations Dynamical systems Chaos theory
ArithmeticCalculusVector calculusAnalysisDifferential equationsDynamical systemsChaos theoryList of functions

Structure

Pinning down ideas of size, symmetry, and mathematical structure.
Abstract algebra Number theory Group theory
Topology Category theory Order theory
Abstract algebraNumber theoryAlgebraic geometryGroup theoryMonoidsAnalysisTopologyLinear algebraGraph theoryUniversal algebraCategory theoryOrder theoryMeasure theory

Spatial relations

A more visual approach to mathematics.
Topology Geometry Trigonometry Differential geometry Fractal geometry
TopologyGeometryTrigonometryAlgebraic geometryDifferential geometryDifferential topologyAlgebraic topologyLinear algebraFractal geometry

Discrete mathematics

Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.
[1,2,3][1,3,2]
[2,1,3][2,3,1]
[3,1,2][3,2,1]
Combinatorics Naive set theory Theory of computation Cryptography Graph theory
CombinatoricsNaive set theoryTheory of computationCryptographyGraph theory

Applied mathematics

Applied mathematics uses the full knowledge of mathematics to solve real-world problems.
Mathematical physicsMechanicsFluid mechanicsNumerical analysisOptimizationProbabilityStatisticsFinancial mathematicsGame theoryMathematical biologyCryptographyInformation theory

Important theorems

These theorems have interested mathematicians and non-mathematicians alike.
See list of theorems for more
Pythagorean theoremFermat's last theoremGödel's incompleteness theoremsFundamental theorem of arithmeticFundamental theorem of algebraFundamental theorem of calculusCantor's diagonal argumentFour color theoremZorn's lemmaEuler's identityChurch-Turing thesisclassification theorems of surfacesGauss-Bonnet theoremQuadratic reciprocityRiemann-Roch theorem.

Important conjectures

See list of conjectures for more

These are some of the major unsolved problems in mathematics.
Goldbach's conjectureTwin Prime ConjectureRiemann hypothesisPoincaré conjectureCollatz conjectureP=NP? – open Hilbert problems.

Foundations and methods

Approaches to understanding the nature of mathematics also influence the way mathematicians study their subject.
Philosophy of mathematicsMathematical intuitionismMathematical constructivismFoundations of mathematicsSet theorySymbolic logicModel theoryCategory theoryLogicReverse MathematicsTable of mathematical symbols

History and the world of mathematicians

See also list of mathematics history topics

History of mathematicsTimeline of mathematicsMathematiciansFields medalAbel PrizeMillennium Prize Problems (Clay Math Prize)International Mathematical UnionMathematics competitionsLateral thinkingMathematical abilities and gender issues

Mathematics and other fields

Mathematics and architectureMathematics and educationMathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.

Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.

Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. Indeed, theorems may be encapsulated in formal languages, and computers may then be used to prove them, by means of automated theorem provers. Such techniques, however, are as likely to generate mathematics as the proverbial thousand monkeys, seated at typewriters, are likely to write a play worthy of Shakespeare. A computer cannot reliably tell an interesting theorem from a boring one.

See also

Bibliography

  • Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
  • 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.
  • Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary 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).
  • Pappas, Theoni, The Joy Of Mathematics (1989).

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