Friday, 25 November 2011

The Axiom of Choice and Mathematical Proof...(humour!)

Probably the most controversial axiom in mathematics, but quite simple XKCD explains :-)

Proof by Intimidation needs to be added to the canonical list of "Proof by..."
Aside: an an undergrad many years ago I read a fascinating paper on reactive systems and after 3 or 4 pages of dense, complicated mathematics there was the statement: Proof left as exercise to gives you some idea of the calibre of the author that he could get away with that (and no, I'm not telling who....)
anyway, a quick search using an popular search engine reveals a list which I'll just put a few choice examples:

Proof Techniques:

Proof by example
The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.
Proof by intimidation
``Trivial'' or ``obvious.''
Proof by omission
``The reader may easily supply the details'', ``The other 253 cases are analogous''
Proof by obfuscation
A long plotless sequence of true and/or meaningless syntactically related statements.
Proof by personal communication
``Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication].''
Proof by reference to talk
``At the special NSA workshop on computer vision, Binford proved that SHGC's could be recognized in polynomial time.''
Proof by reference to inaccessible literature
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Icelandic Philological Society, 1883. This works even better if the paper has never been translated from the original Icelandic.
Proof by flashy graphics
A moving sequence of shaded, 3D color models will convince anyone that your object recognition algorithm works. An SGI workstation is helpful here.
Proof by misleading or uninterpretable graphs
Almost any curve can be made to look like the desired result by suitable transformation of the variables and manipulation of the axis scales. Common in experimental work.
Proof by vigorous handwaving
Works well in a classroom, seminar, or workshop setting.
Proof by cumbersome notation
Best done with access to at least four alphabets, special symbols, and the newest release of LaTeX.
Proof by abstract nonsense
A version of proof by intimidation. The author uses terms or theorems from advanced mathematics which look impressive but are only tangentially related to the problem at hand. A few integrals here, a few exact sequences there, and who will know if you really had a proof?
Disproof by ``not invented here''
We have years of experience with this equipment at MIT and we have never observed that effect.
Proof by personal communication I remember as being something like:
"I met Knuth/Scott/Gödel/... in the corridor the other day and he thought it sounded ok"...

Note the the "Proof by Intimidation" above is slightly different to the XKCD example....both work....hmmm, might try this in my talk on the Theory of Privacy next week....did I tell you about that? Later....

I guess we could also add the "Proof by UML" from Software Engineering, this is explained in the excellent Death by UML [1]  paper.

Proof by Abstract Nonsense.....did someone mention category theory? ;-)

[1] Alex E. Bell. 2004. Death by UML Fever. Queue 2, 1 (March 2004), 72-80. DOI=10.1145/984458.984495

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