Mathematics

In General > s.a. history of mathematics.
* Idea: The study of patterns; It arises from interaction of empirical facts and abstract ideas.
* Debates: (a) Pure vs applied mathematics, what is what? (b) Does mathematics create or discover? (See positions of Cantor and Kronecker, respectively); For the second point of view, see Chern on fiber bundles [@ in Yang PNYAS(77)].
* And reasoning: Reasoning is usually based on logic, but one can shift the emphasis to combinatorics.
* Interconnections: Examples are algebraic topology, and the Langlands program.

Major Programs, Problems, Areas > s.a. conjectures; mathematical physics.
* Hilbert's Program: In 1900, Hilbert proposed 23 problems that seemed to him to be the most difficult and rewarding ones in mathematics; 2000, Three of them are not yet fully solved, and one of them, the Riemann Hypothesis, is a completely open question.
* Langlands Program: A vast mathematical vision formulated by Robert Langlands to unite whole areas of math; The theory of automorphic forms and its connection with L-functions and other fields; A special case is the Shimura-Taniyama-Weil conjecture.
* 2000: The Clay Mathematics Institute has offered a new list of seven outstanding problems (including the Poincaré Conjecture and the Navier-Stokes problem, whether the equations develop singularities), offering $1M for a verified solution to each.
@ Hilbert's Program: Hilbert MN(1900), BAMS(02), reprinted BAMS(00); Kantor MI(96) [status], Ilyashenko BAMS(02) [16th]; Gray 00; Yandell 02 [problems and solvers].
@ Langlands Program: Frenkel BAMS(04); Frenkel ht/05-ln [and quantum field theory].
@ Clay Institute problems: Smith m.DG/06-wd [Navier-Stokes claim, withdrawn].
> Main areas: see algebra; analysis; Arithmetic; combinatorics; differential equations; geometry; logic; number theory; probability; set theory; topology.
> More specific topics: see inequalities; matrices; proof theory; Relations; Solvability.

Foundations > s.a. numbers [rational, real].
* Peano's axioms:
(i) for all x in N, 0 x + 1;
(ii) for all x, y in N, x + 1 = y + 1 only if x = y;
(iii) M N, and M Ø implies that M has a smallest element with respect to <;
(iv) x y precisely when there exists z in N such that x + z = y;
(v) + and · satisfy, for all x, y in N: x + (y+1) = (x+y) + 1; x + 0 = x; x · (y+1) = x · y + x; x · 0 = 0.
@ References: Engeler 93 [short]; Chaitin AS(02) [randomness and paradoxes], SA(06)mar [limitations].

Philosophy
* "Intuitionism seeks to break up and to disfigure mathematics" [@ Hilbert 35, p188].
* "The universe of mathematics grows out of the world about us like dreams out of the events of the day" [@ Stein 69].
@ References: in Wigner CPAM(60); Hersh AiM(79); Field 89; Maddy 93; Brown 99; Shapiro 00.

General References
@ Books: Courant & Robbins 41; Polya 62, 68; Bochner 66; Saaty & Weyl ed-69; Stein 69; Iyanaga & Kawada 80; Kramer 81; Campbell & Higgins ed-84; Dunham 90.
@ Method: Polya 57; Van Gasteren 90; Tao BAMS(07) ["good mathematics"].
@ I, books: B W Jones 70 [nonstandard]; Davis & Hersch 81; Dieudonné 92; Casti 95; Devlin 94, 99, 00.
@ I, short topics: Honsberger 73, 76; Newman 82; Davis & Chinn 85; Ekeland 88; Peterson 90; Barrow 92; Stewart 92; Devlin 03 [unsolved problems].
@ Reference books: Smith 59; Hazewinkel 87–00.
@ Conceptual: Friedman & Flagg AAM(90) [complexity of mathematical concepts]; Sherry SHPSA(06) [mathematical reasoning].
@ Teaching: Polya 57; Wickelgren 74; Krantz 94; Burton 04 [learning]; Bass BAMS(05) [mathematicians and math education].
@ Innumeracy: Paulos 90; NS(91)mar30, p44 [need for mathematics].
@ Quotations: Gaither & Cavazos-Gaither 98.
@ Related topics: Hellman; Knuth BAMS(79) [typography]; Ruelle BAMS(88); Gardner SA(98)aug [recreational]; Renteln & Dundes NAMS(05) [humor].

Applications
@ References: Kline 59; Casti 89 [models of nature]; Schwarz PhSc(95) [psychology]; Hersch 97; Zebrowski 99 [and physical universe]; Rotman 00 [as an activity]; Lakoff & Núñez 00 [and cognitive science]; Arianrhod 05 [as language]; Deem PT(07)jan [biology].

Online Resources
> Encyclopedias: see Concise Math Encyclopedia; MathPages; MathWorld; Internet Encyclopedia of Science; PlanetMath.org; Platonic Realms.
> Other resources: see Earliest Use of Math Terms; MathSciNet [reviews].

"The Great Architect of the Universe now begins to appear as a pure mathematician,"
Sir James Jeans (1877 – 1946), expressing his surprise that quantum theory works.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 22 jul 2008