## 2018-09-06

### Powerful proof techniques

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There's proof by contradiction. There's proof by construction. For the brave of heart, there's proof by induction, and for the untiring, proof by exhaustion.

But sometimes you need something more. And for those cases, Strata of the World proudly presents the following collection of powerful proof techniques.

Proof by the algebra is boring: "... and the result follows after some uninteresting algebraic manipulations." An excellent choice when you're running out of space, need to cut words, or don't remember the intermediate steps.

Proof by boring algebra. Let's be honest, if your proof involves (or can be modified to involve) thirty lines of algebra, how many people are going to go through the details? To maximize the effect, write out the steps with a small font and no line spacing, use pixellated text, and separate it from the rest of the document by putting it in a textbox or appendix. If you suspect this to be insufficient, sprinkle remarks like "the details are not relevant to the main discussion", or "the intrepid reader is warned that the proof is arduous and somewhat tricky" into the surrounding text. Use this if you need to conceal some algebraic sleight-of-hand to "prove" your not-quite-a-theorem, or if you simply want to avoid annoying questions about it.

(Note: extremely determined readers may still decide to figure it out, but they are likely to spend at least an hour trying to work it out before they think to question its validity. This is when you make your getaway.)

Proof by circular reference: Write "We prove B by seeing that it follows from A, which we know to be true" in one place, and "We prove A by seeing that it follows from B, which we know to be true" somewhere else. This is especially effective when at least one of the statements is outside the main body of the text (appendix, textbox, etc.).

Proof by the proof is complicated: "To prove A, we invoke theorem X, the proof of which is beyond the scope of this discussion."

A special case of the above: proof by the proof involves calculus and is therefore omitted. This is a time-honored tactic used by many high-school physics and chemistry books.

Proof by elegance: "One possible solution to problem A is $$X$$. $$X$$ is an extremely elegant solution because [reason], and therefore is correct." A good "reason" might be that $$X$$ is the first thing you thought of. If all else fails, mumble something about "symmetry".

Proof by making a hella assumptions: "This can be shown rigorously if we first assume that $$\Delta x$$ is small, $$s^2 ≈ s$$, $$\sin (\theta) ≈ \theta ≈ 0$$, $$\alpha < \beta$$, and there is a full moon tomorrow."

Is your result shaky even after making all possible assumptions and simplifications? Simply appeal to proof by usefulness: "Though A is not proved here, assuming it allows us to solve/prove B, C, D, and E."

Proof by authority: "This result holds because [the teacher]/[the professor]/[the Mathematical Establishment]/[God]/[Richard Feynman] says so."

If proof by authority is too crass, replace it with a proof by impressive citation: "Theorem A holds (Einstein 1905b; Euclid 302 BCE; Euler 1779e; Hawking, et. al. 1997; Gödel 1931a; Turing 1953; Wiles 1993; Euler 1764g; Erdös 1964; Tao, et. al. 2005; Euler 1778$$\omega$$)".

If mathematical induction is not up to the task, why not try proof by electromagnetic induction? "This result holds. Doubters of this result will be administered electric shocks until they desist their doubting."

A more peaceful means of alternative induction is proof by cult induction: "This proof is the truth, and the whole truth. All those who want to be saved must agree, immediately transfer all their funds to [bank account details], and join us in the Kalahari Desert, where we await the alien messiah."

And, of course, the classic: proof by the proof is left as an exercise for the reader.

Inspired by conversations with friends, as well as the process of writing a longer math paper for school. More examples of this genre of humor can be found online, for example here. See also this article, specifically page 28 (page 5 of the PDF).