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];
Wolpert & Kinney a2012-FQXi [mathematics as a fundamentally stochastic process].

@ __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];
news sn(19)nov [10th problem, contribution by Julia Robinson].

@ __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Smith m.DG/06-wd [Navier-Stokes claim, withdrawn];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];
Jaffe & Xue a2007 [little-known anecdotes];
> 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) For all *x*, *y* in \(\mathbb N\), *x* ≤
*y* precisely when there exists *z* in \(\mathbb N\)
such that *x* + *z* = *y*;

(v) The operations + 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__: "Intuitionism seeks to
break up and to disfigure mathematics" [@ Hilbert 35,
p188]; > s.a. time [and physics];
Wikipedia page.

* "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**
> s.a. Machine Learning.

@ __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 14 feb 2021