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Text File  |  2003-06-21  |  8KB  |  153 lines
  1. How to prove anything 
  2.                               How to prove anything
  3.  
  4.                               HOW TO PROVE ANYTHING (source unknown)6-15-89
  5.  
  6.                               Proof by example:
  7.                               The author gives only the case n = 2 and suggests 
  8.                               that it contains most of
  9.                               the ideas of the general proof.
  10.  
  11.                               Proof by intimidation:
  12.                               "Trivial."
  13.                               "Intuitively obvious to the most casual observer."
  14.  
  15.                               Proof by vigorous handwaving:
  16.                               Works well in a classroom or seminar setting.
  17.  
  18.                               Proof by cumbersome notation:
  19.                               Best done with access to at least four alphabets 
  20.                               and special symbols.
  21.  
  22.                               Proof by exhaustion:
  23.                               An issue or two of a journal devoted to your proof 
  24.                               is useful.
  25.  
  26.                               Proof by omission:
  27.                               "The reader may easily supply the details . . . "
  28.                               "The other 253 cases are analogous . . . "
  29.  
  30.                               Proof by obfuscation:
  31.                               A long plotless sequence of true and/or 
  32.                               meaningless syntactically related
  33.                               statements.
  34.  
  35.                               Proof by wishful citation:
  36.                               The author cites the negation, converse, or 
  37.                               generalization of a theorem
  38.                               from the literature to support his claims.
  39.  
  40.                               Proof by funding:
  41.                               How could three different government agencies be 
  42.                               wrong?
  43.  
  44.                               Proof by eminent authority:
  45.                               "I saw Karp in the elevator and he said it was 
  46.                               probably NP-complete."
  47.  
  48.                               Proof by personal communication:
  49.                               "Eight-dimensional colored cycle stripping is 
  50.                               NP-complete."
  51.                               [Karp, personal communication]."
  52.  
  53.                               Proof by reduction to the wrong problem:
  54.                               "To see that infinite-dimensional colored cycle 
  55.                               stripping is decidable, we
  56.                               reduce it to the halting problem."
  57.  
  58.                               Proof by reference to inaccessible literature:
  59.                               The author cites a simple corollary of a theorem 
  60.                               to be found in a
  61.                               privately circulated memoir of the Slovenian 
  62.                               Philological Society, 1883.
  63.  
  64.                               Proof by importance:
  65.                               "A large body of useful consequences all follow 
  66.                               from the proposition in
  67.                               question."
  68.  
  69.                               Proof by accumulated evidence:
  70.                               "Long and diligent search has not revealed a 
  71.                               counterexample."
  72.  
  73.                               Proof by cosmology:
  74.                               "The negation of the proposition is unimaginable 
  75.                               or meaningless." (Popular
  76.                               for proofs of the existence of God.)
  77.  
  78.                               Proof by mutual reference:
  79.                               In reference A, Theorem 5 is said to follow from 
  80.                               Theorem 3 in reference B,
  81.                               which is shown to follow from Corollary 6.2 in 
  82.                               reference C, which is an easy
  83.                               consequence of Theorem 5 in reference A.
  84.  
  85.                               Proof by metaproof:
  86.                               A method is given to construct the desired proof. 
  87.                               The correctness of the
  88.                               method is proved by any of these techniques.
  89.  
  90.                               Proof by picture:
  91.                               A more convincing form of proof by example. 
  92.                               Combines well with proof by
  93.                               omission.
  94.  
  95.                               Proof by vehement assertion:
  96.                               It is useful to have some kind of authority 
  97.                               relation to the audience.
  98.  
  99.                               Proof by ghost reference:
  100.                               Nothing even remotely resembling the cited theorem 
  101.                               appears in the
  102.                               reference given.
  103.  
  104.                               Proof by forward reference:
  105.                               Reference is usually to a forthcoming paper of the 
  106.                               author, which is often
  107.                               not as forthcoming as at first.
  108.  
  109.                               Proof by semantic shift:
  110.                               Some of the standard but inconvenient definitions 
  111.                               are changed for the
  112.                               statement of the result.
  113.  
  114.                               Proof by appeal to intuition:
  115.                               Cloud-shaped drawings frequently help here.
  116.  
  117.                               Now, the lab session: A group of scientists of all 
  118.                               kinds gathered one day
  119.                               to prove that all odd numbers were prime numbers.
  120.  
  121.                               The mathematician was first: "One is a prime 
  122.                               number, three is a prime
  123.                               number, five is a prime number, seven is a prime 
  124.                               number. Thus by induction,
  125.                               all odd numbers are prime."
  126.  
  127.                               Then came the physicist: "One is a prime number, 
  128.                               three is a prime number,
  129.                               five is a prime number, seven is a prime number. 
  130.                               Nine is NOT a prime number.
  131.                               Eleven is a prime nummber, thirteen is a prime 
  132.                               number. Well, nine falls within
  133.                               experimental error, so all odd numbers are prime."
  134.  
  135.                               Finally the chemist: "One is a prime number, three 
  136.                               is a prime number, five
  137.                               is a prime number, seven is a prime number, nine 
  138.                               is a prime number . . .
  139.  
  140.  
  141.  
  142.                                
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