Proof
Theory |

**In General**

* __Methods of proof__: By
contradiction, by construction, by induction, by exhaustion; Disproof by counterexample.

* __Constructive vs contradiction__:
(Jan Brouwer) If numbers and other mathematical objects do not exist in a Platonic
realm–if they are constructed–then
the
only acceptable existence proofs must be recipes for constructing them;
1960s (Errett A Bishop), Most of classical mathematical
analysis can
be proved constructively–but one has to abandon the idea of
an infinite
set.

* __Checking proofs__: Computer-based
proofs are very difficult/tedious to check; Even in traditional proofs, reviewers
rarely check every step, instead
focusing mostly on the major points, and relying also on the author's track
record; In the end, they either believe the proof or not (according
to mathematics
historian Akihiro Kanamori "It's like osmosis; More and more
people say it's a proof and you believe them"); > s.a. computation.

> __Online resources__:
see Wikipedia page.

**References** > s.a. conjectures; Theorems.

@ __Intros__: Lakatos 63 [classic on mathematical discovery and methodology, Worrall & Zahar ed-15]; Solow 82; Pohlers 89, 08; Hayes AS(07).

@ __Textbooks, II__: Penner 99 [and combinatorics]; Beck & Geoghegan 10; Gerstein 12; Cunningham 13 [logical introduction]; Oliveira & Stewart 15.

@ __Other general references__: Takeuti 75; Schütte 77 [not standard framework]; Simmons 00 [III, and computation]; Toelstra & Schwichtenberg 00 [III]; Hanna et al ed-09; Stillwell 10.

@ __Related topics__: Maclane Syn(97) [physicists and mathematics]; Hughes AM(06)
[combinatorics as opposed to syntax].

@ __Proof by computer__: Horgan SA(93)oct; MacKenzie 01; Simpson LMP(04)
[overview]; > s.a.
computation.

@ __Quantum proofs__: Aaronson & Kuperberg
qp/06 [vs classical proofs].

**Physics-Related and Other Topics** > see philosophy
of science [explanations].

@ __References__: Cohen JSP(09)
[example of failure of common opinions and intuitive arguments]; Bolotin a1509 [examples of non-constructive proofs in quantum theory].

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