

The Earth's volume is 1.1*10^27 cm^3. For comparison, a sugar cube is about 1 cm^3. So if you wanted to make the Earth from sugar cubes, you'd need about 1.1*10^27 of them. This is one of those fun little facts you can try to use to impress people at parties, but it probably won't work.

36 Methods of Mathematical Proof



Proof by obviousness
"The proof is so clear that it need not be mentioned."

Proof by general agreement
"All in favor?. . . "

Proof by imagination
"Well, we'll pretend it's true. . .

Proof by convenience
"It would be very nice if it were true, so . . .

Proof by necessity
"It had better be true, or the entire structure of mathematics would crumble to the ground."

Proof by plausibility
"It sounds good, so it must be true."

Proof by intimidation
"Don't be stupid; of course it's true."

Proof by lack of sufficient time
"Because of the time constraint, I'll leave the proof to you."


Proof by postponement
"The proof for this is long and arduous, so it is given in the appendix."

Proof by accident
"Hey, what have we here?!"

Proof by insignificance
"Who really cares, anyway?"

Proof by mumbo-jumbo
" (B Ì P ) , $ (C Î W )

Proof by profanity
(example omitted)

 Proof by definition
"We define it to be true."

Proof by tautology
'It's true because it's true."

Proof by plagiarism
"As we see on page 289......"

Proof by lost reference
"I know I saw it somewhere......"

Proof by calculus
"This proof requires calculus, so we'll skip it."

Proof by terror
When intimidation fails ...

Proof by lack of interest
"Does anyone really want to see this?"

Proof by illegibility
(scribble, scribble) QED

Proof by logic
"If it is on the problem sheet, then it must be true!"

Proof by majority rule
Only to be used if general agreement is impossible

Proof by clever variable choice
"Let A be the number such that this proof works. . "

Proof by tessellation
"This proof is the same as the last."

Proof by divine word
"And the Lord said, 'Let it be true,' and it was true."

Proof by stubbornness
"I don't care what you say-it is true!"

Proof by simplification
"This proof reduces to the statement 1 + 1 = 2."

Proof by hasty generalization
"Well, it works for 17, so it works for all reals."

Proof by deception
"Now everyone turn their backs. . ."

Proof by supplication
"Oh please, let it be true."

Proof by poor analogy
"Well, it's just like . . . "

Proof by avoidance
Limit of proof by postponement as it approaches infinity

Proof by design
If it's not true in today's math, invent a new system in which it is.

Proof by authority
"Well, Don Knuth says it's true, so it must be!"

Proof by intuition
"I just have this gut feeling. . ."

 Useful proof methods  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 exhaustion  An issue or two of a journal devoted to your proof is useful.

 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 wishful citation  The author cites the negation, converse, or generalization of a theorem from the literature to support his claims.

 Proof by funding  How could three different government agencies be wrong? Or, to play the game a different way: how could anything funded by those bozos be correct?

 Proof by democracy  A lot of people believe it's true: how could they all be wrong?

 Proof by market economics  Mine is the only theory on the market that will handle the data.

 Proof by eminent authority  ``I saw Marty Fischler in the elevator and he said that was tried in the 70's and doesn't work."

 Proof by cosmology  The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God and for proofs that computers cannot think.

 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 reduction to the wrong problem  ``To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem.''

 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 ghost reference  Nothing even remotely resembling the cited theorem appears in the reference given. Works well in combination with proof by reference to inaccessible literature.

 Proof by forward reference  Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.

 Proof by importance  A large body of useful consequences all follow from the proposition in question.

 Proof by accumulated evidence  Long and diligent search has not revealed a counterexample.

 Proof by mutual reference  In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to follow from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.

 Proof by metaproof  A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques. A strong background in programming language semantics will help here.

 Proof by picture  A more convincing form of proof by example. Combines well with proof by omission.

 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 vehement assertion  It is useful to have some kind of authority relation to the audience, so this is particularly useful in classroom settings.

 Proof by repetition  Otherwise known as the Bellman's proof: ``What I say three times is true.''

 Proof by appeal to intuition  Cloud-shaped drawings frequently help here.

 Proof by vigorous handwaving  Works well in a classroom, seminar, or workshop setting.

 Proof by semantic shift  Some of the standard but inconvenient definitions are changed for the statement of the result.

 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 finding a bad apple  One bad apple spoils the whole bunch. Among the many proponents of this theory, we have found one who is obviously loony; so we can discredit the entire theory. (Often used in political contexts.)

 Disproof by slippery slope (or thin end of wedge, if you are British)  If we accepted [original proposal], we'd have to accept [slightly modified proposal], and eventually this would lead to [radically different and clearly objectionable proposal].

 Disproof by ``not invented here''  We have years of experience with this equipment at SRI and we have never observed that effect.

 Proof by lack of space  "The proof is not detailled due to lack of space in this proceedings..." works well in conjunction with proof by forward reference.

A variation of this list seems to have been around for quite a while. I lifted this one from Margaret Fleck and added a few of my own. Back to Tuan's research page