Proving the “obvious”


As Eric Temple Bell said, “‘Obvious’ is the most dangerous word in mathematics.” That being the case, it is often true that we have trouble proving statements that seem self-evident. Many times we are indeed tempted to say “clearly” or “obviously” or “it is trivial” or “the details have been left to the reader” or “this easily follows from Theorems 4.8, 5.1, and Definition 5.8”. For a full list of invalid proof techniques, visit this hilarious site. Here are a few samples (it’s a LONG list!), quoted from  full list on their site:

  • Proof by intimidation (“Trivial.”)
  • 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 vigorous hand waving (Works well in a classroom or seminar setting. )
  • Proof by exhaustion (An issue or two of a journal devoted to your proof is useful. )
  • Proof by importance (A large body of useful consequences all follow from the proposition in question.)
  • Proof by accelerated course (We don’t have time to prove this… )

Choosing the level of rigor for a proof is often difficult–depending on the mathematical context, and the audience. I’m taking a graduate class in Analysis right now, so I definitely think about this a lot! In fact, I might add one more to the list:

  • Proof by beautiful typesetting (Because the proof looks good and is typed in \LaTeX, it must be right.)

At least,  I hope my professor feels that’s a valid technique :-).

2 thoughts on “Proving the “obvious”

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