Citizendium, the Citizens' Compendium
cz
http://en.citizendium.org/wiki/Welcome_to_Citizendium
MediaWiki 1.24.1
first-letter
Media
Special
Talk
User
User talk
Citizendium Pilot
Citizendium Pilot Talk
File
File talk
MediaWiki
MediaWiki talk
Template
Template talk
Help
Help talk
Category
Category talk
CZ
CZ Talk
TI
TI Talk
WYA
WYA Talk
Forum
Forum Talk
Widget
Widget talk
Mathematics
0
18831
move=sysop
100680191
100489349
2010-06-11T03:44:16Z
Sandy Harris
32079
TOC
{{subpages}}
{{TOC|right}}
'''Mathematics''', or '''maths''' (British), '''math''' (American), is the discipline that deals with concepts such as [[quantity]], [[structure]], [[space]] and [[change]]. It evolved, through the use of [[abstraction (mathematics)|abstraction]] and [[logic|logical]] [[reasoning]], from [[counting]], [[calculation]], [[measurement]] and the study of the [[shape]]s and [[motion (physics)|motion]]s of physical objects. [[Mathematician]]s explore such concepts, aiming to formulate new [[conjecture]]s and establish their truth by [[Rigour|rigorous]] deduction from appropriately chosen [[axiom]]s and [[definition]]s. <ref> Philip E. B. Jourdain, ''The Nature of Mathematics'', in ''The World of Mathematics'', James R. Newman, editor, Dover, 2003, ISBN 0486432688. </ref>
Knowledge and use of basic mathematics is widespread, as it has been throughout history. Refinements of basic ideas are visible in ancient mathematical texts originating in [[ancient Egypt]], [[Mesopotamia]] and [[History of India|ancient India]]. More rigorous methods were later introduced by the [[ancient Greeks]]. Development continued in short bursts until the [[Renaissance]], when mathematical innovations interacted with new [[scientific discoveries]] to yield an acceleration in understanding that continues to the present day. <ref> Howard Eves, ''An Introduction to the History of Mathematics'', Sixth Edition, Saunders, 1990, ISBN 0030295580. </ref>
Mathematics is used in many fields, including [[science]], [[engineering]], [[medicine]] and [[economics]]. The application of mathematics to such fields, often dubbed [[applied mathematics]], inspires and makes use of new mathematical discoveries and has sometimes led to the development of entirely new disciplines. Mathematicians also engage in [[pure mathematics]] without any practical applications in mind, although applications are often discovered later. <ref> Ivars Peterson, ''Mathematical Tourist, New and Updated Snapshots of Modern Mathematics'', Owl Books, 2001, ISBN 0805071598. </ref>
==Etymology==
The word "mathematics" (Greek: μαθηματικά) comes from the [[Ancient Greek language|Greek]] μάθημα (''máthēma''), which means ''learning'', ''study'', ''science'', and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (''mathēmatikós''), ''related to learning'', or ''studious'', which likewise further came to mean ''mathematical''. In particular, (μαθηματικὴ τέχνη,''mathēmatikḗ tékhnē''), in [[Latin]] ''ars mathematica'', meant ''the mathematical art''.
The apparent plural form in [[English language|English]], like the [[French language|French]] plural form ''les mathématiques'' (and the less commonly used singular derivative ''la mathématique''), goes back to the Latin neuter plural ''mathematica'' ([[Cicero]]), based on the Greek plural τα μαθηματικά (''ta mathēmatiká''), used by [[Aristotle]], and meaning roughly "all things mathematical".<ref>''[[The Oxford Dictionary of English Etymology]]'', 1983 reprint. ISBN 0-19-861112-9.</ref><ref>[[Oxford English Dictionary]], second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.</ref>
Despite the form and etymology, the word ''mathematics'', like the names of arts and sciences in general, is used as a singular [[mass noun]] in English today. The common English-language abbreviations perpetuate this singular/plural idiosyncrasy, as the word becomes ''math'' in [[North America]], while it is ''maths'' elsewhere (including [[British English|Britain]], [[Hiberno-English|Ireland]], [[Australia]] and many other [[Commonwealth of Nations|Commonwealth]] countries).
==History==
:''Main article: [[History of mathematics]]''
The evolution of mathematics might be seen to be an ever-increasing series of [[abstraction]]s, or alternatively an expansion of subject matter. One of the earliest abstractions is the representation of [[number]]s. The realization that two apples and two oranges can be named and symbolized with the same entity is a breakthrough in human thought.
In addition to recognizing how to [[Counting|count]] ''physical'' objects, [[Prehistory|prehistoric]] peoples also recognized how to count ''abstract'' quantities, like [[time]] — [[day]]s, [[season]]s, [[year]]s. [[Arithmetic]] ([[addition]], [[subtraction]], [[multiplication]] and [[division (mathematics)|division]]), naturally followed. Monolithic monuments testify to knowledge of [[geometry]].
Further steps need [[writing]] or some other system for recording numbers such as [[Tally sticks|tallies]] or the knotted strings called [[quipu]] used by the [[Inca empire]] to store numerical data. [[Numeral system]]s have been many and diverse.
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to [[taxation]] and [[commerce]], to understand the relationships among numbers, to [[land measurement|measure land]], and to predict [[astronomy|astronomical events]]. These needs can be roughly related to the broad subdivision of mathematics, into the studies of ''quantity'', ''structure'', ''space'', and ''change''.
Mathematics since has 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. According to Mikhail B. Sevryuk, in the January 2006 issue of the [[Bulletin of the American Mathematical Society]], "The number of papers and books included in the [[Mathematical Reviews]] database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical [[theorem]]s and their [[mathematical proof|proof]]s."<ref>{{cite journal| author = Mikhail B. Sevryuk| year = 2006| month = January| title = Book Reviews| journal = [[Bulletin of the American Mathematical Society]]| volume = 43| issue = 1| pages = 101-109| url = http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf| format = PDF| accessdate = 2006-06-24}}</ref>
==Inspiration, pure and applied mathematics, and aesthetics==
{{main|Mathematical beauty}}
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in [[commerce]], [[land measurement]] and later [[astronomy]]; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. [[Isaac Newton|Newton]] was one of the [[infinitesimal calculus]] inventors, [[Feynman]] invented the [[Feynman path integral]] using a combination of reasoning and physical insight, and today's [[string theory]] also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what [[Eugene Wigner]] has called "[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|the unreasonable effectiveness of mathematics]]."
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between [[pure mathematics]] and [[applied mathematics]]. Within applied mathematics, two major areas have split off and become disciplines in their own right, [[statistics]] and [[computer science]].
Many mathematicians talk about the ''elegance'' of mathematics, its intrinsic [[aesthetics]] and inner [[beauty]]. [[Simplicity]] and [[generality]] are valued. There is beauty also in a clever proof, such as [[Euclid]]'s proof that there are infinitely many [[prime number]]s, and in a numerical method that speeds calculation, such as the [[fast Fourier transform]]. [[G. H. Hardy]] in ''[[A Mathematician's Apology]]'' expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.
==Notation, language, and rigor==
{{main|Mathematical notation}}
Most of the mathematical notation we use today was not invented until the [[16th century]]. <ref>[http://members.aol.com/jeff570/mathsym.html Earliest Uses of Various Mathematical Symbols] (Contains many further references)</ref> Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.
Mathematical language also is hard for beginners. Words such as ''or'' or ''only'' have more precise meanings than in everyday speech. Also confusing to beginners, words such as ''[[open set|open]]'' and ''[[field (mathematics)|field]]'' have been given specialized mathematical meanings, and [[mathematical jargon]] includes technical terms such as ''[[homeomorphism]]'' and ''[[integrability|integrable]]''. It was said that [[Henri Poincaré]] was only elected to the [[Académie française]] so that he could tell them how to define ''[[automorphic|automorphe]]'' in their dictionary. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Rigor is fundamentally a matter of [[mathematical proof]]. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "[[theorem]]s", based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in [[mathematical analysis]]). The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of [[Isaac Newton]] the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about [[computer-assisted proof]]s. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.
[[Axiom]]s in traditional thought were 'self-evident truths', but that conception is problematic. 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 (sufficiently powerful) axiomatic system has undecidable formulas; and so a final [[axiomatization]] of mathematics is impossible. 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.
==Mathematics as science==
[[Carl Friedrich Gauss]] referred to mathematics as "the Queen of the Sciences".<ref>[[Wolfgang Sartorius von Waltershausen|Waltershausen, Wolfgang Sartorius von]]: ''Gauss zum Gedächtniss'', 1856. (''Gauss zum Gedächtnis'' 1965 reprint by Sändig Reprint Verlag H. R. Wohlwend: ISBN 3-253-01702-8, [http://www.amazon.de/exec/obidos/ASIN/B0000BN5SQ ASIN: B0000BN5SQ]).</ref> In the original Latin ''Regina Scientiarum'', as well as in [[German language|German]] ''Königin der Wissenschaften'', the word corresponding to ''science'' means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to ''natural'' science is of later date. If one considers [[science]] to be strictly about the physical world, then mathematics, or at least [[pure mathematics]], is not a science. [[Albert Einstein]] has stated that ''"as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality''"<ref>{{cite journal
| last = Einstein
| first = Albert
| authorlink = Albert Einstein
| title = Sidelights on Relativity (Geometry and Experience)
| publisher = P. Dutton., Co
| date = 1923}}</ref>. [[Karl Popper]] believed that mathematics was not experimentally [[Falsifiability|falsifiable]] and thus not a science. However, other thinkers, notably [[Imre Lakatos]], have applied a version of falsificationism to mathematics itself. In his 2002 book ''[[A New Kind of Science]]'', [[Stephen Wolfram]] argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
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.<ref>{{cite paper|url=http://info.med.yale.edu/therarad/summers/ziman.htm|date=1968|title=Public Knowledge:An essay concerning the social dimension of science|author= Ziman, J.M., F.R.S.}}</ref> 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 (knowledge)|Intuition]] and [[experiment|experimentation]] also play a role in the formulation of [[conjecture]]s in both mathematics and the (other) sciences. [[Experimental mathematics]] continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the [[scientific method]].
The opinions of mathematicians on this matter are varied. While some in [[applied mathematics]] feel that they are scientists, those in pure mathematics often feel that they are working in an area more akin to [[logic]] and that they are, hence, fundamentally [[Philosophy|philosophers]]. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven [[liberal arts]]; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and [[engineering]] has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is ''created'' (as in art) or ''discovered'' (as in science). It is common to see [[University|universities]] divided into sections that include a division of ''Science and Mathematics'', indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the [[philosophy of mathematics]].
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the [[Fields Medal]],<ref>"The Fields Medal is now indisputably the best known and most influential award in mathematics."{{cite paper|url=http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf|date=2001|title=Some Trends in Modern Mathematics and the Fields Medal|author=Monastyrsky, Michael|publisher=Canadian Mathematical Society|accessdate=2006-07-28}}</ref> established in 1936 and now awarded every 4 years. It is usually considered the equivalent of science's [[Nobel prize]]. Another major international award, the [[Abel Prize]], was introduced in 2003. Both of these are awarded for a particular body of work, either innovation in a new area of mathematics or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "[[Hilbert's problems]]", was compiled in 1900 by German mathematician [[David Hilbert]]. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "[[Clay Mathematics Institute#The Millennium Prize problems|Millennium Prize Problems]]", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the [[Riemann hypothesis]]) is duplicated in Hilbert's problems.
==Fields of mathematics==
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict [[astronomy|astronomical]] events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., [[arithmetic]], [[algebra]], [[geometry]], and [[mathematical analysis|analysis]]). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to [[Mathematical logic|logic]], to [[set theory]] ([[Foundations of mathematics|foundations]]), to the empirical mathematics of the various sciences ([[applied mathematics]]), and more recently to the rigorous study of [[uncertainty]].
===Quantity===
The study of quantity starts with [[number]]s, first the familiar [[natural number]]s and [[integer]]s ("whole numbers") and arithmetical operations on them, which are characterized in [[arithmetic]]. The deeper properties of integers are studied in [[number theory]], whence such popular results as [[Fermat's last theorem]]. Two famous unsolved problems in number theory are the [[twin prime conjecture]] and [[Goldbach's conjecture]].
As the number system is further developed, the integers are recognised as a [[subset]] of the [[rational numbers]] ("fractions"). These, in turn, are contained within the [[real numbers]], which are used to represent continuous quantities. Real numbers are generalised to [[complex number]]s. These are the first steps of a hierarchy of number systems that include the [[quaternions]] and [[octonions]]. Consideration of numbers larger than all finite natural numbers leads to the concept of [[transfinite numbers]]. In this formalism, infinite [[cardinal number]]s, the [[aleph number]]s, allow meaningful comparison of the size of infinitely large sets.
:{| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="20"
| <math>1, 2, 3\,\!</math> || <math>-2, -1, 0, 1, 2\,\!</math> || <math> -2, \frac{2}{3}, 1.21\,\!</math> || <math>-e, \sqrt{2}, 3, \pi\,\!</math> || <math>2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!</math>
|-
| [[Natural number]]s|| [[Integer]]s || [[Rational number]]s || [[Real number]]s || [[Complex number]]s
|}
===Structure===
Many mathematical objects, such as [[set]]s of numbers and [[function (mathematics)|function]]s, exhibit internal structure. The structural properties of these objects are investigated in the study of [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]] and other abstract systems, which are themselves such objects. This is the field of [[abstract algebra]]. An important concept here is that of [[vector (spatial)|vector]]s, generalized to [[vector space]]s, and studied in [[linear algebra]]. The study of vectors combines three of the fundamental areas of mathematics; quantity, structure, and space. [[Vector calculus]] expands the field into a fourth fundamental area, that of change.
:{| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="15"
| <!--[[Image:Elliptic curve simple.png|96px]]--> || <!--[[Image:Rubik float.png|96px]] -->|| <!--[[Image:GroupDiagramD6.png|96px]]--> || <!--[[Image:Lattice of the divisibility of 60.svg|96px]]-->
|-
| [[Number theory]] || [[Set theory]] || [[Abstract algebra]] || [[Group theory]] || [[Order theory]]
|}
===Space===
The study of space originates with [[geometry]] - in particular, [[Euclidean geometry]]. [[Trigonometry]] combines space and number, and encompasses the well-known [[Pythagorean theorem]]. The modern study of space generalizes these ideas to include higher-dimensional geometry, [[Non-euclidean geometry|non-Euclidean geometries]] (which play a central role in [[general relativity]]) and [[topology]]. Quantity and space both play a role in [[analytic geometry]], [[differential geometry]], and [[algebraic geometry]]. Within differential geometry are the concepts of [[fiber bundles]] and calculus on [[manifold]]s. Within algebraic geometry is the description of geometric objects as solution sets of [[polynomial]] equations, combining the concepts of quantity and space, and also the study of [[topological groups]], which combine structure and space. [[Lie group]]s are used to study space, structure, and change. [[Topology]] in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing [[Poincaré conjecture]] and the controversial [[four color theorem]], whose only proof, by computer, has never been verified by a human.
:{| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="15"
| <!--[[Image:Pythagorean.svg|96px]]--> || <!--[[Image:Taylorsine.svg|96px]]--> || <!--[[Image:OsculatingCircle.png|96px]]--> || <!--[[Image:Torus.jpg|96px]]--> || <!-- [[Image:Koch curve.png|96px]]-->
|-
|[[Geometry]] || [[Trigonometry]] || [[Differential geometry]] || [[Topology]] || [[Fractal geometry]]
|}
===Change===
Understanding and describing change is a common theme in the [[natural science]]s, and [[calculus]] was developed as a powerful tool to investigate it. [[function (mathematics)|Functions]] arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and real-valued functions is known as [[real analysis]], with [[complex analysis]] the equivalent field for the complex numbers. The [[Riemann hypothesis]], one of the most fundamental open questions in mathematics, is drawn from complex analysis. [[Functional analysis]] focuses attention on (typically infinite-dimensional) [[space#Mathematical spaces|space]]s of functions. One of many applications of functional analysis is [[quantum mechanics]]. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as [[differential equation]]s. Many phenomena in nature can be described by [[dynamical system]]s; [[chaos theory]] makes precise the ways in which many of these systems exhibit unpredictable yet still [[deterministic system (mathematics)|deterministic]] behavior.
{| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="20"
| <--[[Image:Integral_as_region_under_curve.svg|96px]]--> || <!--[[Image:Vectorfield_jaredwf.png|96px]]--> || <math>\frac{d^2}{dx^2} y = \frac{d}{dx} y + c</math> || <!-- [[Image:Limitcycle.jpg|96px]]--> || <!--[[Image:LorenzAttractor.png|96px]]-->
|-
| [[Calculus]] || [[Vector calculus]]|| [[Differential equation]]s || [[Dynamical system]]s || [[Chaos theory]]
|}
===Foundations and philosophy===
In order to clarify the [[foundations of mathematics]], the fields of [[mathematical logic]] and [[set theory]] were developed.
Mathematical logic is concerned with setting mathematics on a rigid [[axiom]]atic framework, and studying the results of such a framework. As such, it is home to [[Gödel's incompleteness theorems#Second incompleteness theorem|Gödel's second incompleteness theorem]], perhaps the most widely celebrated result in logic, which (informally) implies that there are always true theorems which cannot be proven. Modern logic is divided into [[recursion theory]], [[model theory]], and [[proof theory]], and is closely linked to [[theoretical computer science|theoretical]] [[computer science]].
:{| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="15"
| <math> P \Rightarrow Q \,</math>|| <!--[[Image:Venn_A_intersect_B.png|128px]] --> || <!--[[Image:MorphismComposition-01.png|96px]]-->
|-
| [[Mathematical logic]] || [[Set theory]] || [[Category theory]] ||
|}
===Discrete mathematics===
[[Discrete mathematics]] is the common name for the fields of mathematics most generally useful in [[theoretical computer science]]. This includes [[Computability theory (computation)|computability theory]], [[computational complexity theory]], and [[information theory]]. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model - the [[Turing machine]]. Complexity theory is the study of tractability by computer; some problems, although theoretically soluble by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as [[data compression|compression]] and [[Entropy in thermodynamics and information theory|entropy]].
As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "[[Complexity classes P and NP|P=NP?]]" problem, one of the [[Millennium Prize Problems]]. It is widely believed that the answer to this problem is no.
:{| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="15"
| <math>\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}</math> || <!--[[Image:DFAexample.png|96px]] -->|| <!--[[Image:Caesar3.png|96px]] -->|| <!--[[Image:6n-graf.svg|96px]]-->
|-
| [[Combinatorics]] || [[Theory of computation]] || [[Cryptography]] || [[Graph theory]]
|}
===Applied mathematics===
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the [[science]]s, [[business]], and other areas. 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 role. Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) [[Numerical analysis]] investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of [[rounding error]]s or other sources of error in computation.
:[[Mathematical physics]] • [[Mechanics|Analytical mechanics]] • [[Fluid mechanics|Mathematical fluid dynamics]] • [[Numerical analysis]] • [[Optimization (mathematics)|Optimization]] • [[Probability]] • [[Statistics]] • [[Mathematical economics]] • [[Financial mathematics]] • [[Game theory]] • [[Mathematical biology]] • [[Cryptography]] • [[Operations research]]
==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:
*misunderstanding of the implications of [[mathematical rigor]];
*attempts to circumvent the usual criteria for publication of [[mathematical paper]]s in a [[learned journal]] after [[peer review]], often in the belief that the journal is biased against the author;
*lack of familiarity with, and therefore underestimation of, the existing literature.
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 [[Pierre de Fermat|Fermat]] and [[Marin Mersenne|Mersenne]].
=== Relationship between mathematics and physical reality ===
Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense.
Thus while most systems of axioms are derived from our perceptions and experiments, they are not dependent on them. Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led Eugene Wigner to write an essay, ''[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]''.
=== What mathematics is not ===
Mathematics is not about unrestricted [[theorem proving]], any more than [[literature]] is about the construction of [[grammar|grammatically correct]] [[sentence (linguistics)|sentence]]s.
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 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.
==References==
<div class="references-small">
<references/>
*[[Morris Kline|Kline, Morris]], ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
</div>
8u693jnc8st6e6i9883z9i6c2r29xxx
wikitext
text/x-wiki