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; 1960's (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.

Related Topics > see philosophy of science [explanations].

References
@ General: Takeuti 75; Schütte 77 [not standard framework]; Simmons 00 [III].
@ Simple: Solow 82; Pohlers 89; Hayes AS(07).
@ 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; Woos 03; Simpson LMP(04) [overview]; > s.a. computation.
@ Quantum proofs: Aaronson & Kuperberg qp/06 [vs classical proofs].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 11 jun 2008