Difference between revisions of "Proof"

A proof is a concept commonly used in the treatment of mathematics, philosophy and taxidermy. One can measure the proof by taking twice the concentration of alcohol by volume at 15.5 degrees celsius.

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 Assumption

An offshoot of proof by induction, one may assume the result is true. Therefore it is true.

Proof by Cumbersome Notation

Best done with access to at least four alphabets and special symbols.

Matrices, Tensors, Lie algebra and Kronecker-Weyl Theorem are also well-suited.

Proof by Default

The proposition is true due to the lack of a counterexample. For when you know you are right and that you don't give a shit about what others may think of you.

Proof by Definition

Define something as such that the problem falls into grade one math.

• "I am, therefore I am."

Proof by Diagram

Reducing problems to diagrams with lots of arrows. Particularly common in category theory.

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 presumeably 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.

1. Look behind you!
2. ... and proves the existence of an answer for 2 + 2.
3. Look! A three-headed monkey over there!
4. ... leaves 5 as the only result of 2 + 2.
5. Therefore 2 + 2 = 5. QED.

Proof by the rovdistic principle

1. I like to think that 2 + 2 = 5.
2. Therefore, 2 + 2 = 5. QED.

Proof by Engineer's Induction

Suppose P(n) is a statement.

• Prove true for P(1).
• Prove true for P(2).
• Prove true for P(3).

Therefore true for all ${\displaystyle n\in \mathbb {N} }$

Proof by Eyeballing

Quantities that look similar are indeed the same. Often times, drawing random pictures will aid with this process.

Corollary: If it looks like a duck and acts like a duck, it must be a duck.

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 Mathematical Interpretive Dance

An archaic form of "proof by hand waving", it is one of the most popular forms of proof in lectures and seminars in most universities today. See Mathematical Interpretive Dance.

Proof by Induction

Proof by induction claims that

${\displaystyle {\mathcal {E}}={-{d\Phi _{B}} \over dt}}$

where

${\displaystyle {\mathcal {E}}}$ is the number of pages used to contain the proof
ΦB is the time required to prove something, relative to the trivial case

For the common, but special case, of generalising the proof,

${\displaystyle {\mathcal {E}}=-N{{d\Phi _{B}} \over dt}}$

where

${\displaystyle {\mathcal {E}}}$ is the number of pages used to contain the proof
N is the number of things which are being proved
ΦB is the time required to prove something, relative to the trivial case

The actual method of constructing the proof is irrelevant.

Proof by Inspection

A form of proof where something is proved by realising that is true. Therefore, the proof holds.

Proof by Intimidation

One of the principal methods used to prove mathematical statements. The exact statement of proof by intimidation is given below.

Suppose a mathematician, F, at a position, n, in the following hierarchy:

1. Fields Medal winner
2. Tenured Professor
3. Non-tenured professor
4. Post-doc

and a second mathematician, G, at any position p such that p > n. Then any statement S given to G by F is true.

This is a general offshoot of Liouville's Theorem, the proof of which is left to the reader.

Hint: Use the hyperensemble theory and the Slim Shady Algorithm.

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 Misunderstanding

• "2 is equal to 3 for large values of 2"

Proof by Wikipedia

If the Wikipedia website state something is true, it must be

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.

• "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 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.

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...

Proof by Repetition

If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true. If you say something is true enough times, it is true.

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 you die through boredom.

Proof by Superior I.Q.

If your I.Q. is greater than that of the other person in the argument, you are right and what you say is proven.

Proof by Volume

If you shout something really really loud often enough, it will be accepted as true.

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?

Also if the proof takes up several volumes, then any reader will get bored and go do something more fun, like math.

Consider ${\displaystyle p}$

1. Assume the opposite: "not ${\displaystyle p}$"
2. bla bla bla
3. ...
4. Which leads to "not ${\displaystyle p}$" being false, wich contradicts with assumption (1)

Whatever you say in (2) and (3), (4) will make ${\displaystyle p}$ true. Useful to support other proofs

${\displaystyle 2=1}$, if we add a constant ${\displaystyle C}$ that ${\displaystyle 2=1+C}$.

Proof by Surprise

Proof 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 your proof by appealing to the Axiom of Surprise. The most known user of this syle of proof is Walter Rudin in Principles of Mathematical Analysis' To quote an example...
"Theorem: If ${\displaystyle p>0}$ and ${\displaystyle \alpha }$ is real, then ${\displaystyle \lim \limits _{n\to \infty }{\frac {n^{\alpha }}{(1+p)^{n}}}=0}$.
Proof: Let ${\displaystyle k}$ be an integer such that ${\displaystyle k>\alpha }$, ${\displaystyle k>0}$. For ${\displaystyle n>2k}$, ${\displaystyle (1+p)^{n}>{n \choose k}p^{k}={\frac {n(n-1)\cdots (n-k+1)}{k!}}p^{k}>{\frac {n^{k}p^{k}}{2^{k}k!}}}$. Hence, ${\displaystyle 0<{\frac {n^{\alpha }}{(1+p)^{n}}}<{\frac {2^{k}k!}{p^{k}}}n^{\alpha -k}}$. Since ${\displaystyle \alpha -k<0}$, ${\displaystyle n^{\alpha -k}\to 0}$. QED" Walter Rudin, Principles of Mathematical Analysis, 3rd Edition, p.58 middle.

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 Outside the Scope

All the non-trivial parts of the proof is left out, stating that proving them is outside the scope of the book.

Proof by Nimalin Says So

If Nimalin says so then it's true

Terminology

Some other terms one may come across when working with proofs:

Exhaustive Proof

A method of proof attempted at 3:00 am 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.

Safety

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.