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 general__:
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.

* __Hilbert's 6th Problem__:
"To treat by means of axioms, those physical sciences in which mathematics plays an important part."

* __Langlands Program__:
A vast mathematical vision formulated by Robert Langlands to unite whole areas
of mathematics; 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 Millennium Problems (including the
Riemann Hypothesis, Poincaré Conjecture, P vs NP Problem, and Navier-Stokes Equation, whether the equations
develop singularities), offering $1M for a verified solution to each.

@ __General references__: Glimm BAMS(10)
[challenges and opportunities].

@ __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].

@ __Hilbert's Program, 6th Problem__: Dass a0909,
IJNS?-a1002,
Pra(11)-a1006;
Schreiber a1311-proc [classical field theory from cohesive homotopy type theory];
Gorban PTRS(18)-a1803 [intro].

@ __Langlands Program__: Frenkel BAMS(04);
Frenkel ht/05-ln [and quantum field theory];
Frenkel a0906-talk [geometric, and gauge theory];
Frenkel BAMS(13) [trace formulas and geometrization].

@ __Millennium Problems__: Smith m.DG/06-wd [Navier-Stokes claim, withdrawn];
Chatterjee JFA-a1602 [Yang-Mills free energy];
> s.a. CMI page.

__Main areas__: see
algebra; analysis;
Arithmetic; combinatorics;
differential equations;
geometry; logic; number
theory; probability;
set theory; topology.

__More specific topics__:
see inequalities; Mathematical Constants; matrices;
proof
theory; Relations; series; Solvability.

**Foundations** > s.a. numbers [rational, real].

* __Peano's axioms__:

(i) for all *x* in \(\mathbb N\), 0 ≠ *x* + 1;

(ii) for all *x*, *y* in \(\mathbb N\), *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 \(\mathbb N\) such that *x* + *z* = *y*;

(v) + and · satisfy, for all *x*, *y* in \(\mathbb N\): *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 [realism];
Brown 99;
Shapiro 00,
00,
05;
Linnebo & Uzquiano BJPS(09) [acceptable abstraction principles];
Baker BJPS(09) [explanations in science
and existence of mathematical entities];
Cellucci SHPSA(13);
Werndl PhM-a1310 [justifying definitions].

**General References**

@ __Books__: Courant & Robbins 41; Pólya 62, 68; Bochner 66; Saaty & Weyl
ed-69; Stein 69; Iyanaga & Kawada 80; Kramer 81; Campbell & Higgins
ed-84; Dunham 90; Bajnok 13 [abstract mathematics, II]; Kohar 16 [discrete mathematics].

@ __Method__: Pólya 57; Van Gasteren 90; Tao BAMS(07) ["good mathematics"]; Kjeldsen & Carter SHPSA(12) [growth of mathematical knowledge]; Roytvarf 13; Mazur 15 [history of mathematical notation].

@ __I, books__: Jones 70 [non-standard]; Davis & Hersch 81; Dieudonné
92; Casti 95; Devlin 94, 99, 00; Joyner 08 [toys]; Gallier 11 [discrete mathematics]; Herrmann 12.

@ __I, short topics__: Honsberger 73, 76; Newman 82; Davis & Chinn
85; Ekeland 88; Peterson 88, 90; Barrow 92; Stewart 92; Devlin 03 [unsolved problems]; Tubbs 08; Havil 10 [paradoxes]; Stewart 12 [17 equations that changed the world]; Mackenzie 12 [the story of mathematics in 24 equations]; Beardon 16.

@ __Reference books__: Smith 59; Hazewinkel 87–00.

@ __Doing mathematics__: Krieger 15; Meier & Smith 17 [and proofs].

@ __Conceptual__:
Friedman & Flagg AAM(90) [complexity of mathematical concepts]; Lucas 99; Rotman 00; Sherry SHPSA(06)
[mathematical reasoning]; Josephson a1307-in [is mathematical truth a human construct?]; Tallant BJPS(13) [pretense theories of mathematics fail]; Lev PPNL(17)-a1409 [standard vs finite mathematics]; Dantas ch(16)-a1506-FQXi [as tactics of self-referential systems]; > s.a. Meaning.

**Teaching, Applications, and Related Topics**

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

@ __Teaching__: Pólya 57; Wickelgren 74; Krantz 94; Burton 04 [learning];
Bass BAMS(05)
[mathematicians and math education]; Hewson 09 [bridge to university-level mathematics]; Li 11 [problems]; Hiriart-Urruty 16 [non-standard exercises].

@ __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];
Silva et al JPA(10) [network of mathematical knowledge];
news nat(11)jul
+ plus(11)jul [unplanned impact].

**Online Resources**

> __Encyclopedias__: see MathPages; MathWorld;
Internet
Encyclopedia of Science; PlanetMath.org; Platonic
Realms; Springer Online
Encyclopaedia of Mathematics.

> __Other resources__:
see Earliest
Use of Math Terms; MathSciNet [reviews].

"In mathematics the art of proposing a question must be held of higher value than solving it,"

Georg Cantor (1845 – 1918)

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 mar 2018