The following is a list of some common proof techniques that are often extremely useful. I received the original list long time ago via e-mail. Recently, or rather over time, Michael Ashley, Deepak Azad, Roman Beslik, Atanas Boev, John Chase, Maxwell Davenport, Christophe de Dinechin, Hedi Driss, Dan Echlin, Ron Garret, Manuel Hoffmann, Arthur Keller, Vivek Joy Kozhikkottu, John McGoff, Scott A. McIntyre, John Norberg, Filip Opálený, Alex Papanicolao, József Pék, Arnold G. Reinhold, Noemi Reynolds, Greg Rose, Lucas Scharenbroich, John Snider, Ki Song, Moritz Voss, David N. Werner and Thomas Zaslavsky suggested some additions. I also added my own experiences from teaching techniques of proofs in a first year mathematics units for several years.

Unfortunately all these proof techniques are invalid and, hence,
*one should not use them* in assignments, workshops, exams,
papers, etc.

- 1. Original List
- 1.1 Proof by example
- 1.2 Proof by intimidation
- 1.3 Proof by vigorous handwaving
- 1.4 Proof by cumbersome notation
- 1.5 Proof by exhaustion
- 1.6 Proof by omission
- 1.7 Proof by obfuscation
- 1.8 Proof by wishful citation
- 1.9 Proof by funding
- 1.10 Proof by eminent authority
- 1.11 Proof by personal communication
- 1.12 Proof by reduction to the wrong problem
- 1.13 Proof by reference to inaccessible literature
- 1.14 Proof by importance
- 1.15 Proof by accumulated evidence
- 1.16 Proof by cosmology
- 1.17 Proof by mutual reference
- 1.18 Proof by metaproof
- 1.19 Proof by picture
- 1.20 Proof by vehement assertion
- 1.21 Proof by ghost reference
- 1.22 Proof by forward reference
- 1.23 Proof by semantic shift
- 1.24 Proof by appeal to intuition
- 2. From my own experience
- 2.1 Proof by acceptance
- 3. Entry suggested by Michael Ashley
- 3.1 Proof by erasure
- 4. Entry suggested by Deepak Azad
- 4.1 Proof by placing a bet
- 5. Entry suggested by Roman Beslik
- 5.1 Proof by indeterminate literature
- 6. Entry suggested by Atanas Boev
- 6.1 Proof by overwhelming size
- 7. Entry suggested by John Chase
- 7.1 Proof by beautiful typesetting
- 8. Entries suggested by Maxwell Davenport
- 8.1 Proof by group complaint
- 8.2 Proof by convincing knowledge of related facts
- 9. Entries suggested by Christophe de Dinechin
- 9.1 Proof by voting
- 9.2 Proof by prohibitive cost
- 9.3 Proof by religious belief
- 9.4 Proof by astonishment
- 9.5 Proof by invalid negation
- 9.6 Proof by reverse recurrence
- 9.7 Proof by computer
- 9.8 Proof by partial enumeration
- 10. Entry suggested by Hedi Driss
- 10.1 Proof by Hot Air
- 11. Entries suggested by Dan Echlin
- 11.1 Proof by Selective Hearing
- 11.2 Proof by Accelerated Course
- 12. Entries suggested by Ron Garret
- 12.1 Proof by horror story
- 12.2 Proof by rhyming
- 13. Entry suggested by Manuel Hoffmann
- 13.1 Proof by sketch/illustration
- 14. Entries suggested by Arthur Keller
- 14.1 Proof by use of puns and homonyms
- 14.2 Proof by invalid induction
- 14.3 Proof by assuming a contradiction
- 14.4 Proof by introduction of useful statement
- 14.5 Proof by parent
- 14.6 Proof by repetition
- 14.7 Or the opposite
- 14.7.1 Proof denial by continually sowing doubt
- 14.7.2 Proof denial by citing vested interest
- 15. Entries suggested by Vivek Joy Kozhikkottu
- 15.1 Proof by “Its too good to be true”
- 15.2 Proof by lack of space
- 15.3 Proof by pity
- 16. Entry suggested by John McGoff
- 16.1 Proof by page turning
- 17. Entry suggested by Scott A. McIntyre
- 17.1 Proof by legislation
- 18. Entry suggested by John Norberg
- 18.1 Proof by Elephant
- 19. Entry suggested by Filip Opálený
- 19.1 Proof by the break
- 20. Entry suggested by Alex Papanicolaou
- 20.1 Proof by exercise
- 21. Entry suggested by József Pék
- 21.1 Proof by condescence and reassurance
- 22. Entries suggested by Arnold G. Reinhold
- 22.1 Proof by ethical exclusion
- 22.2 Proof by legal intimidation
- 22.3 Proof by demonstrating equivalence to a problem thought to be hard
- 22.4 Proof by unclaimed reward
- 22.5 Proof by Never-Ending Revision
- 23. Entries suggested by Noemi Reynolds
- 23.1 Proof by being a nice person
- 23.2 Proof by appeal to humanity
- 23.3 Proof by establishing and then tearing down a straw man
- 23.4 Proof by circular argument
- 24. Entry suggested by Greg Rose
- 24.1 Proof by Sketch or Outline
- 25. Entries suggested by Lucas Scharenbroich
- 25.1 Proof by haste
- 25.2 Proof by mercy
- 26. Entry suggested by John Snider
- 26.1 Proof by Lack of Counter Example
- 27. Entry suggested by Ki Song
- 27.1 Proof by Exhaustion of Grader
- 28. Entry suggested by Moritz Voss
- 28.1 Proof by Recess
- 29. Entries suggested by David N Werner
- 29.1 Proof by lyricism
- 29.2 Interdisciplinary proof
- 29.3 Proof by word order
- 29.4 Proof by sneak preview
- 30. Entries suggested by Thomas Zaslavsky
- 30.1 Proof by semantic shift
- 30.2 Proof by slippery definitions
- 30.3 Proof by countervailing errors
- 30.4 Proof by overwhelming errors

“The other 253 cases are analogous.”

“...”

[I learnt this from an applied maths lecturer at ANU in the late 1970's; and he actually used it! He would write using chalk with one hand, and an eraser in the other.]

They draw a little figure of how they think, e.g. a graph looks like, and deduce facts by this illustration. This way often special cases (like empty graph, isolated nodes, ...) are ignored.

and

A cow have four more legs than no cow. No cow has five legs.

Therefore, a cow has nine legs.

"If it walks like an elephant, talks like an elephant, it must be an elephant."

"...and the proof why this is true we'll show after the break."

"As we proved before the break, this theorem is true for..."

Person A: “The vast majority of primes are odd.”

Person B: “Person A says all primes are odd. But 2 is prime and not odd so A's statement is wrong.”

The student basically tried to enumerate every single group of that order, and he/she basically went through each group to show that the property holds for them.

The grader wrote on the page: “Proof by Exhaustion of Grader.”

-between 10 and 15 minutes pass-

“...as proven before recess, we can now ...”

Author: Berwin A Turlach

Date Last modified:Wed May 7 18:34:43 WST 2014

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URL: http://www.maths.uwa.edu.au/~berwin/humour/invalid.proofs.html