Mathematics| 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Smith m.DG/06-wd [Navier-Stokes claim, withdrawn];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)
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 14 feb 2021