1 History
#_#A7sx:z q(p2xoX 2 Inspiration, aesthetics and pure and applied mathematics
m7J}9v$b#N w 3 Notation, language and rigor *I9f MwU(P5l L[5r
4 Is mathematics a science?
9CB8Mq*^GNh(rWu,I:} 5 Overview of fields of mathematics
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6.8 Important theorems and conjectures
O3l9a~de_:Qw 6.9 Foundations and methods
)ND'Ul5X ?:dRG 6.10 History and the world of mathematicians
,e{} V"OU#| S 8 See also /q^8i~3o {}"u/C4g
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History
+a,X#w!K$~Y"L[t Main article: History of mathematics hQ ny7}B c"WL
The evolution of mathematics can be seen to be an ever increasing series of abstractions. 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. From counting, naturally followed arithmetic (e.g. addition, subtraction, multiplication and division).
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I1xMbd Overview of fields of mathematics0dcr+S}7K-d,J*Y
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).`PcO;F
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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.)`L b:J5v;_{}F0R,e| -sY1V b_%QNY
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.1@5n&ys5{}&C[ o|gW
Spatial relations:m4A x%^K7h*X(L
A more visual approach to mathematics. _*Q,o;z(Z"b
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Topology Geometry Trigonometry Differential geometry Fractal geometry
1qA+oq6xI P#r{} s*]2f[7Q^q Combinatorics Naive set theory Theory of computation Cryptography Graph theory
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Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory ~3P0cA&?;x{}Qo8V
Applied mathematics
)q5o1j~j I/clk Applied mathematics uses the full knowledge of mathematics to solve real-world problems.
g3j:D9B S a Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory [.c*v?(N YqXR
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@|%r#g*?*w Famous theorems and conjecturesjt ] Y6ew
These theorems have interested mathematicians and non-mathematicians alike.
q%Ur `;Le H'O@ Pythagorean theorem – Fermat's last theorem – Goldbach's conjecture – Twin Prime Conjecture – Gφdel's incompleteness theorems – Poincarι conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – Church-Turing thesis – Collatz conjecture +}xYo y0w
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Important theorems and conjectures
*H,[,m,P)Y%[ See list of theorems, list of conjectures for more
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e\cL5L;P These are theorems and conjectures that have changed the face of mathematics throughout history. !Y2pK Shxxt
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
MN#oe}5p#G [edit]{};``$o jG;k _ Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Foundations of mathematics – Set theory – Symbolic logic – Model theory – Category theory – Logic – Reverse Mathematics – Table of mathematical symbols uPDk9dv
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G@iGzQ,a+Ht History and the world of mathematicians
6P-F.l QG%D$u See also list of mathematics history topics
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!L uL_ LkD'u 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
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~5p6nl/_Ze&kc Mathematics and other fields
c3^6i"W3` Mathematics and architecture – Mathematics and education – Mathematics of musical scales
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Common misconceptions,Ub0C[a T
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.q%H;LU(m2N*\vn
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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:
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misunderstanding of the implications of mathematical rigour;
%g&y/EILCn attempts to get round the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
_}&C'Jsf lack of familiarity with, and therefore underestimation of, the existing literature. G%c.g?&H
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.3e-s+^u'E}1r e Q?B
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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.
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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.
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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.
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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. %U,k6Wq8S
Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language. 8H(ocw-p'^+I
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. -n S+p2S8|qf6Gsm
Kline, M., Mathematical Thought from Ancient to Modern Times (1973). 7jS7iRgl
Pappas, Theoni, The Joy Of Mathematics (1989). ]7uh)I5z#k&r:R+@y
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K/Nge,Q] External links$t^h(?6\L?.v
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Z)?0u$bM`6V*L1})z Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. A collection of articles on various math topics, with interactive Java illustrations
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