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Please note that as Proof has been shot dead, all information below has been rendered obsolete.
Methods of proof
There are several methods of proof which are commonly used:
Proof by Revenge
"2+2=5" "no it doesn't" "REVENGE!"
Proof by Adding a Constant
2 = 1 if we add a constant C such that 2 = 1 + C.
Multiplicative Identity Additive Identity
Multiply both expressions by zero, e.g.,
- 1 = 2
- 1 × 0 = 2 × 0
- 0 = 0
Since the final statement is true, so is the first.
See also Proof by Pedantry.
Proof by Altering (or Destroying) the Original Premise (or Evidence)
- A = 1 and B = 1 and A + B = 3
- [Long list of confusing statements]
- [Somewhere down the line, stating B = 2 and covering up the previous definition]
- [Long list of other statements]
- A + B = 3
Works best over long period of time.
Proof by Analogy
Draw a poor analogy. Say you have two cows. But one is a bull. After the proper gestation period, 1 + 1 = 3.
Proof by Anti-proof
If there is proof that has yet to be accounted for in your opponent's argument, then it is wholly discreditable and thus proof of your own concept. It also works if you claim to be unable to comprehend their proof. Example:
- I can't see how a flagellum can evolve by itself, therefore the theory of evolution is incorrect, therefore someone must have put them together, therefore convert now!
Note: This generally works equally well in both directions:
- I can't see how someone could have put a flagellum together, therefore the theory of Creation is incorrect, therefore it must have evolved by itself, therefore Let's Party!
Proof by August
Since August is such a good time of year, no one will disagree with a proof published then, and therefore it is true. Of course, the converse is also true, i.e., January is crap, and all the logic in the world will not prove your statement then.
Proof by Assumption
An offshoot of Proof by Induction, one may assume the result is true. Therefore, it is true.
Proof by Axiom
Assert an axiom A such that the proposition P you are trying to prove is true. Thus, any statement S contradicting P is false, so P is true. Q.E.D.
Proof by Belief
"I believe assertion A to hold, therefore it does. Q.E.D."
Proof by Bijection
This is a method of proof made famous by P. T. Johnstone. Start with a completely irrelevant fact. Construct a bijection from the irrelevant fact to the thing you are trying to prove. Talk about rings for a few minutes, but make sure you keep their meaning a secret. When the audience are all confused, write Q.E.D. and call it trivial. Example:
- To prove the Chinese Remainder Theorem, observe that if p divides q, we have a well-defined function. Z/qZ → Z/qZ is a bijection. Since f is a homomorphism of rings, φ(mn) = φ(m) × φ(n) whenever (n, m) = 1. Using IEP on the hyperfield, there is a unique integer x, modulo mn, satifying x = a (mod m) and x = b (mod n). Thus, Q.E.D., and we can see it is trivial.
Proof by B.O.
This method is a fruitful attack on a wide range of problems: don't have a shower for several weeks and play lots of sports.
Proof by Calling the Other Guy an Idiot
"I used to respect his views, but by stating this opinion, he has now proven himself to be an idiot." Q.E.D.
Proof by Arbitration
Often times in mathematics, it is useful to create abitrary "Where in the hell did that come from?" type theorems which are designed to make the reader become so confused that the proof passes as sound reasoning.
Proof by Faulty Logic
Math professors and logicians sometimes rely on their own intuition to prove important mathematical theorems. The following is an especially important theorem which opened up the multi-disciplinary field of YouTube.
- Let k and l be the two infinities: mainly, the negative infinity and the positive infinity. Then, there exists a real number c, such that k and l cease to exist. Such a s is zero. We conclude that the zero infinity exists and is in between the postive and negative infinities. This theorem opens up many important ideas. For example, primitive logic would dictate that the square root of infinity, r, is a number less than r.
"I proved, therefore I am proof." – Isaac Newton, 1678, American Idol.
Proof by Canada
Like other proofs, but replace Q.E.D. with Z.E.D. Best when submitted with a bowl of Kraft Dinner.
Proof by Cantona
Conduct the proof in a confident manner in which you are convinced in what you are saying is correct, but which is absolute bollocks – and try to involve seagulls in some way. Example:
- If sin x < x … for all x > 0 … and when … [pause to have a sip of water] … the fisherman … throws sardines off the back of the trawler … and x > 0 … then … you can expect the seagulls to follow … and so sin x = 0 for all x.
Proof by Cases
AN ARGUMENT MADE IN CAPITAL LETTERS IS CORRECT. THEREFORE, SIMPLY RESTATE THE PROPOSITION YOU ARE TRYING TO PROVE IN CAPITAL LETTERS, AND IT WILL BE CORRECT!!!!!1 (USE TYPOS AND EXCLAMATION MARKS FOR ESPECIALLY DIFFICULT PROOFS)
Proof by Chocolate
By writing what seems to be an extensive proof and then smearing chocolate to stain the most crucial parts, the reader will assume that the proof is correct so as not to appear to be a fool.
Proof by Complexity
Remember, something is not true when its proof has been verified, it is true as long as it has not been disproved. For this reason, the best strategy is to limit as much as possible the number of people with the needed competence to understand your proof.
Be sure to include very complex elements in your proof. Infinite numbers of dimensions, hypercomplex numbers, indeterminate forms, graphs, references to very old books/movies/bands that almost nobody knows, quantum physics, modal logic, and chess opening theory are to be included in the thesis. Make sentences in Latin, Ancient Greek, Sanskrit, Ithkuil, and invent languages.
Refer to the Cumbersome Notation to make it more complex.
Again, the goal: nobody must understand, and this way, nobody can disprove you.
Proof by (a Broad) Consensus
If enough people believe something to be true, then it must be so. For even more emphatic proof, one can use the similar Proof by a Broad Consensus.
Proof by Contradiction (reductio ad absurdum)
- Assume the opposite: "not p".
- Bla, bla, bla …
- … which leads to "not p" being false, which contradicts assumption (1). Whatever you say in (2) and (3), (4) will make p true.
Useful to support other proofs.
Proof by Coolness (ad coolidum)
Let C be the coolness function
- C(2 + 2 = 4) < C(2 + 2 = 5)
- Therefore, 2 + 2 = 5
Let ACB be A claims B.
- C(Y) > C(X)
- Therefore Q unless there is Z, C(Z) > C(Y) and (ZC¬Q)
Let H be the previous demonstration, N nothingness, and M me.
- C(M) < C(N)
- Therefore ¬H
and all this is false since nothingness is cooler.
Let J be previous counter-argument and K be HVJ.
- Substitude K for H in J
- Therefore ¬K
- ¬K implies ¬J and ¬H
- ¬J implies H
- Therefore H and ¬H
- Therefore non-contradiction is false ad coolidum
- Therefore C(Aristotle) < C(M)
Proof by Cumbersome Notation
Best done with access to at least four alphabets and special symbols. Matrices, Tensors, Lie algebra and the Kronecker-Weyl Theorem are also well-suited.
Proof by Default
Proof by Definition
Define something as such that the problem falls into grade one math, e.g., "I am, therefore I am".
Proof by Delegation
"The general result is left as an exercise to the reader."
Proof by Dessert
The proof is in the pudding.
Philosophers consider this to be the tastiest possible proof.
Proof by Diagram
Reducing problems to diagrams with lots of arrows. Particularly common in category theory.
Proof by Disability
Proof conducted by the principle of not drawing attention to somebody's disability – like a speech impediment for oral proofs, or a severed tendon in the arm for written proofs.
Proof by Disgust
State two alternatives and explain how one is disgusting. The other is therefore obviously right and true.
- Do we come from God or from monkeys? Monkeys are disgusting. Ergo, God made Adam.
- Is euthanasia right or wrong? Dogs get euthanasia. Dogs smell and lick their butts. Ergo, euthanasia is wrong.
- Is cannibalism right or wrong? Eeew, blood! Ergo, cannibalism is wrong.
Proof by Dissent
If there is a consensus on a topic, and you disagree, then you are right because people are stupid. See global warming sceptics, creationist, tobacco companies, etc., for application of this proof.
Proof by Distraction
Be sure to provide some distraction while you go on with your proof, e.g., some third-party announces, a fire alarm (a fake one would do, too) or the end of the universe. You could also exclaim, "Look! A distraction!", meanwhile pointing towards the nearest brick wall. Be sure to wipe the blackboard before the distraction is presumably over so you have the whole board for your final conclusion.
Don't be intimidated if the distraction takes longer than planned – simply head over to the next proof.
An example is given below.
- Look behind you!
- … and proves the existence of an answer for 2 + 2.
- Look! A three-headed monkey over there!
- … leaves 5 as the only result of 2 + 2.
- Therefore 2 + 2 = 5. Q.E.D.
This is related to the classic Proof by "Look, a naked woman!"
Proof by Elephant
Q: For all rectangles, prove diagonals are bisectors. A: None: there is an elephant in the way!
Proof by Engineer's Induction
- See also: Proof by Induction.
Suppose P(n) is a statement.
- Prove true for P(1).
- Prove true for P(2).
- Prove true for P(3).
- Therefore P(n) is true for all .
Proof by Exhaustion
This method of proof requires all possible values of the expression to be evaluated and due to the infinite length of the proof, can be used to prove almost anything since the reader will either get bored whilst reading and skip to the conclusion or get hopelessly lost and thus convinced that the proof is concrete.
Proof by Eyeballing
Quantities that look similar are indeed the same. Often drawing random pictures will aid with this process.
Corollary: If it looks like a duck and acts like a duck, then it must be a duck.
Proof by Flutterby Effect
Proofs to the contrary that you can (and do) vigorously and emphatically ignore therefore you don't know about, don't exist. Ergo, they can't and don't apply.
Corollary: If it looks like a duck, acts like a duck and quacks like a duck, but I didn't see it (and hey, did you know my Mom ruptured my eardrums), then it's maybe … an aadvark?
Proof by Gun
A special case of Proof by Intimidation: "I have a gun and you don't. I'm right, you're wrong. Repeat after me: Q.E.D."
Proof by Global Warming
If it doesn't contribute to Global Warming, it is null and void.
Proof by God
Also a special case of proof. "Don't question my religion, or you are supremely insensitive and God will smite you." Similar to Proof by Religion, but sanctioned by George W. Bush.
Proof by Hand Waving
- See main article: Hand waving.
Commonly used in calculus, hand waving dispenses with the pointless notion that a proof need be rigorous.
Proof by Hitler Analogy
The opposite of Proof by Wikipedia. If Hitler said,
- "I like cute kittens."
then – automatically – cute kittens are evil, and liking them proves that you caused everything that's wrong in the world for the last 50 years.
Simple Proof by Hubris
I exist, therefore I am correct.
Proof by Hypnosis
Try to relate your proof to simple harmonic motion in some way and then convince people to look at a swinging pendulum.
Proof by Imitation
Make a ridiculous imitation of your opponent in a debate. Arguments cannot be seriously considered when the one who proposes them was laughed at a moment before.
Make sure to use puppets and high-pitched voices, and also have the puppet repeat "I am a X", replacing X with any minority that the audience might disregard: gay, lawyer, atheist, creationist, zoophile, paedophile … the choice is yours!
Proof by Immediate Danger
Having a fluorescent green gas gently seep into the room through the air vents will probably be beneficial to your proof.
Proof by Impartiality
If you, Y, disagree with X on issue I, you can invariably prove yourself right by the following procedure:
- Get on TV with X.
- Open with an ad hominem attack on X and then follow up by saying that God hates X for X's position on I.
- When X attempts to talk, interrupt him very loudly, and turn down his microphone.
- Remind your audience that you are impartial where I is concerned, while X is an unwitting servant of Conspiracy Z, e.g., the Liberal Media, and that therefore X is wrong. Then also remind your audience that I is binary, and since your position on I is different from X's, it must be right.
- That sometimes fails to prove the result on the first attempt, but by repeatedly attacking figures X1, X2, …, Xn – and by proving furthermore (possibly using Proof by Engineer's Induction) that Xn is wrong implies Xn+1 is wrong, and by demonstrating that you cannot be an Xi because your stance on I differs due to a change in position i, demonstrating that while the set of Xi's is countable, the set containing you is uncountable by the diagonal argument, and from there one can apply Proof by Consensus, as your set is infinitely bigger – you can prove yourself right.
A noted master of the technique is Bill O'Reilly.
Proof by Induction
Proof by Induction claims that
where is the number of pages used to contain the proof and is the time required to prove something, relative to the trivial case.
For the common, but special case of generalising the proof,
where is the number of pages used to contain the proof, is the number of things which are being proved and is the time required to prove something, relative to the trivial case.
The actual method of constructing the proof is irrelevant.
Proof by Intimidation
One of the principal methods used to prove mathematical statements. Remember, even if your achievements have nothing to do with the topic, you're still right. Also, if you spell even slightly better, make less typos, or use better grammar, you've got even more proof. The exact statement of proof by intimidation is given below.
Suppose a mathematician F is at a position n in the following hierarchy:
- Fields Medal winner
- Tenured Professor
- Non-tenured professor
- Graduate Student
- Undergraduate Student
If a second mathematician G is at any position p such that p < n, then any statement S given to F by G is true.
Alternatively: Theorem 3.6. All zeros of the Riemann Zeta function lie on the critical line (have a real component of 1/2).
Proof: "… trivial …"
Proof by Irrelevant References
A proof that is backed up by citations that may or may not contain a proof of the assertion. This includes references to documents that don't exist. (Cf. Schott, Wiggenmeyer & Pratt, Annals of Veterninary Medicine and Modern Domestic Plumbing, vol. 164, Jul 1983.)
Proof by Jack Bauer
If Jack Bauer says something is true, then it is. No ifs, ands, or buts about it. End of discussion.
This is why, for example, torture is good.
Proof by Lecturer
It's true because my lecturer said it was true. QED.
Proof by Liar
If liar say, that he is a liar, he lies, because liars always lie, so he is not liar.
Simple, ain't it?
Proof by Kim G. S. Øyhus' Inference
- and .
- Ergo, .
- Therefore I'm right and you're wrong.
Proof by LSD
Wow! That is sooo real, man!
Proof by Margin Too Small
"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
Proof by Mathematical Interpretive Dance
Proof by Misunderstanding
"2 is equal to 3 for sufficiently large values of 2."
Proof by Mockery
Let the other state his claim in detail, wait he lists and explain all his argument and, at any time, explose in laughter and ask, "No, are you serious? That must be a joke. You can't really think that, do you?" Then you leave the debate in laughter and shout, "If you all want to listen to this parody of argument, I shan't prevent you!"
Proof by Narcotics Abuse
Spike the drinks/food of all people attending with physcoaltering or hallucinogenic chemicals.
Proof by Obama
Yes, we can.
Proof by Obfuscation
A long, plotless sequence of true and/or meaningless syntactically related statements.
Proof by Omission
Make it easier on yourself by leaving it up to the reader. After all, if you can figure it out, surely they can. Examples:
- The reader may easily supply the details.
- The other 253 cases are analogous.
- The proof is left as an exercise for the reader.
- The proof is left as an exercise for the marker (guaranteed to work in an exam).
Proof by Ostention
- 2 + 2 = 5
Proof by Outside the Scope
All the non-trivial parts of the proof are left out, stating that proving them is outside the scope of the book.
Proof by Overwhelming Errors
A proof in which there are so many errors that the reader can't tell whether the conclusion is proved or not, and so is forced to accept the claims of the writer. Most elegant when the number of errors is even, thus leaving open the possibility that all the errors exactly cancel each other out.
Proof by Ødemarksism
- See also: Proof by Consensus.
- The majority thinks P.
- Therefore P is true (and dissenters should be silenced in order to reduce conflict from diversity).
The silencing of dissenters can be made easier with convincing arguments.
Proof by Penis Size
My dick's much bigger than yours, so I'm right.
Corollary: You don't have a penis, so I'm right.
Proof by Pornography
Include pornographic pictures or videos in the proof – preferably playing a porno flick exactly to the side of where you are conducting the proof. Works best if you pretend to be oblivious to the porn yourself and act as if nothing is unusual.
Proof by Process of Elimination
so 2 + 2 = 5
Proof by Promiscuity
I get laid much more than you, so I'm right.
Proof by Proving
Well proven is the proof that all proofs need not be unproven in order to be proven to be proofs. But where is the real proof of this? A proof, after all, cannot be a good proof until it has been proven. Right?
Proof by Question
If you are asking me to prove something, it must be true. So why bother asking?
Proof by Realization
A form of proof where something is proved by realizing that is true. Therefore, the proof holds.
Proof by Reduction
Show that the theorem you are attempting to prove is equivalent to the trivial problem of not getting laid. Particularly useful in axiomatic set theory.
Proof by Reduction to the Wrong Problem
Why prove this theorem when you can show it's identical to some other, already proven problem? Plus a few additional steps, of course …
- Example: "To prove the four colour theorem, we reduce it to the halting problem."
Proof by Religion
Related to Proof by Belief, this method of attacking a problem involves the principle of mathematical freedom of expression by asserting that the proof is part of your religion, and then accusing all dissenters of religiously persecuting you, due to their stupidity of not accepting your obviously correct and logical proof. See also Proof by God.
Proof by Repetition
AKA the Socratic method.
If you say something is true enough times, then it is true. Repeatedly asserting something to be true makes it so. To repeat many times and at length the veracity of a given proposition adds to the general conviction that such a proposition might come to be truthful. Also, if you say something is true enough times, then it is true. Let n be the times any given proposition p was stated, preferably in different forms and ways, but not necessarily so. Then it comes to pass that the higher n comes to be, the more truth-content t it possesses. Recency bias and fear of ostracism will make people believe almost anything that is said enough times. If something has been said to be true again and again, it must definitely be true, beyond any shadow of doubt. The very fact that something is stated endlessly is enough for any reasonable person to believe it. And, finally, if you say something is true enough times, then it is true. Q.E.D.
Exactly how many times one needs to repeat the statement for it to be true, is debated widely in academic circles. Generally, the point is reached when those around die through boredom.
- E.g., let A = B. Since A = B, and B = A, and A = B, and A = B, and A = B, and B = A, and A = B, and A = B, then A = B.
Proof by Restriction
If you prove your claim for one case, and make sure to restrict yourself to this one, you thus avoid any case that could compromise you. You can hope that people won't notice the omission.
Example: Prove the four-color theorem.
- Take a map of only one region. Only 1 color is needed to color it, and 1 ≤ 4. End of the proof.
If someone questions the completeness of the proof, others methods of proofs can be used.
Proof by the Rovdistic Principle
- See also: Proof by Belief.
- I like to think that 2 + 2 = 5.
- Therefore, 2 + 2 = 5. Q.E.D.
Proof by Russian Reversal
In Soviet Russia, proof gives YOU!
Proof by Self-evidence
Claim something and tell how self-evident it is: you are right!
Proof by Semantics
Proof by semantics is simple to perform and best demonstrated by example. Using this method, I will prove the famous Riemann Hypothesis as follows:
We seek to prove that the Riemann function defined off of the critical line has no non-trivial zeroes. It is known that all non-trivial zeroes lie in the region with 0 < Re(z) < 1, so we need not concern ourselves with numbers with negative real parts. The Riemann zeta function is defined for Re(z) > 1 by sum over k of 1/kz, which can be written 1 + sum over k from 2 of 1/kz.
Consider the group (C, +). There is a trivial action theta from this group to itself by addition. Hence, by applying theta and using the fact that it is trivial, we can conclude that sum (1/kz) over k from 2 is the identity element 0. Hence, the Riemann zeta function for Re(z) > 0 is simply the constant function 1. This has an obvious analytic continuation to Re(z) > 0 minus the critical line, namely that zeta(z) = 1 for all z in the domain.
Hence, zeta(z) is not equal to zero anywhere with Re(z) > 0 and Re(z) not equal to 1/2. Q.E.D.
Observe how we used the power of the homonyms "trivial" meaning ease of proof and "trivial" as in "the trivial action" to produce a brief and elegant proof of a classical mathematical problem.
Proof by Semitics
If it happened to the Jews and has been confirmed by the state of Israel, then it must be true.
Proof by Staring
- x2 − 1 = (x + 1)(x − 1)
This becomes obvious after you stare at it for a while and the symbols all blur together.
Proof by Substitution
One may substitute any arbitrary value for any variable to prove something. Example:
- Assume that 2 = P.
- Substitute 3 for P.
- Therefore, 2 = 3. Q.E.D.
Proof by Superior IQ
- See also: Proof by Intimidation.
If your IQ is greater than that of the other person in the argument, you are right and what you say is proven.
Proof by Surprise
The proof is accomplished by stating completely random and arbitrary facts that have nothing to do with the topic at hand, and then using these facts to mysteriously conclude the proof by appealing to the Axiom of Surprise. The most known user of this style of proof is Walter Rudin in Principles of Mathematical Analysis. To quote an example:
Theorem: If and is real, then .
Proof: Let be an integer such that , . For , . Hence, . Since , . Q.E.D.
Walter Rudin, Principles of Mathematical Analysis, 3rd Edition, p. 58, middle.
Proof by Tarantino
Proof by Tension
Try to up the tension in the room by throwing in phrases like "I found my wife cheating on me … with another woman", or "I wonder if anybody would care if I slit my wrists tomorrow". The more awkward the situation you can make, the better.
Proof by TeX
Proof by … Then a Miracle Happens
Similar to Proof by Hand Waving, but without the need to wave your hand.
Example: Prove that .
- … then a miracle happens.
- . Q.E.D.
Proof by Triviality
The Proof of this theorem/result is obvious, and hence left as an exercise for the reader.
Proof by Uncyclopedia
Uncyclopedia is the greatest storehouse of human knowledge that has ever existed. Therefore, citing any fact, quote or reference from Uncyclopedia will let your readers know that you are no intellectual lightweight. Because of Uncyclopedia's steadfast adherence to accuracy, any proof with an Uncyclopedia reference will defeat any and all detractors.
(Hint: In any proof, limit your use of Oscar Wilde quotes to a maximum of five.)
Proof by Volume
If you shout something really, really loud often enough, it will be accepted as true.
Also, if the proof takes up several volumes, then any reader will get bored and go do something more fun, like math.
Proof by War
My guns are much bigger than yours, therefore I'm right.
See also Proof by Penis Size.
Proof by Wolfram Alpha
If Wolfram Alpha says it is true, then it is true.
Proof by Wikipedia
If the Wikipedia website states that something is true, it must be true. Therefore, to use this proof method, simply edit Wikipedia so that it says whatever you are trying to prove is true, then cite Wikipedia for your proof.
Proof by Yoda
If stated the proof by Yoda is, then true it must be.
Proof by Your Mom
You don't believe me? Well, your mom believed me last night!
Proof by Actually Trying and Doing It the Honest W– *gunshot*
Let this be a lesson to you do-gooders.
Proof by Reading the Symbols Carefully
Proving the contrapositive theorem: Let (p→q), (¬q→¬p) be true statements. (p→q) if (and only if) (¬q→¬p).
The symbols → may also mean shooting and ¬ may also represent a gun. The symbols would then be read as this:
If statement p shoots statement q, then statement q possibly did not shoot statement p at all, because statement q is a n00b player for pointing the gun in the opposite direction of statement p.
Also, if statement q didn't shoot statement p on the right direction in time (due to n00biness), p would then shoot q.
Oh! I get it now. The power of symbol reading made the theorem sense. Therefore, the theorem is true.
The Ultimate Proof
However, despite all of these methods of proof, there is only one way of ensuring not only that you are 100% correct, but 1000 million per cent correct, and that everyone, no matter how strong or how argumentative they may be, will invariably agree with you. That, my friends, is being a girl. "I'm a girl, so there", is the line that all men dread, and no reply has been discovered which doesn't result in a slap/dumping/strop being thrown/brick being thrown/death being caused. Guys, when approached by this such form of proof, must destroy all evidence of it and hide all elements of its existence.
Some other terms one may come across when working with proofs:
A method of proof attempted at 3:00 A.M. the day a problem set is due, which generally seems to produce far better results at that time than when looked at in the light of day.
Q.E.D. stands for "Quebec's Electrical Distributor", commonly known as Hydro Quebec. It is commonly used to indicate where the author has given up on the proof and moved onto the next problem.
Can be substituted for the phrase "So there, you bastard!" when you need the extra bit of proof.
When handling or working with proofs, one should always wear protective gloves (preferably made of LaTeX).
The Burden of Proof
In recent years, proofs have gotten extremely heavy (see Proof by Volume, second entry). As a result, in some circles, the process of providing actual proof has been replaced by a practice known as the Burden of Proof. A piece of luggage of some kind is placed in a clear area, weighted down with lead weights approximating the hypothetical weight of the proof in question. The person who was asked to provide proof is then asked to lift this so-called "burden of proof". If he cannot, then he loses his balance and the burden of proof falls on him, which means that he has made the fatal mistake of daring to mention God on an Internet message board.