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(Message started by: Garzahd on Oct 30th, 2002, 5:24pm)
Title: Proof methods
Post by Garzahd on Oct 30th, 2002, 5:24pm
Partially off-topic, but I was reminded of this when discussing Trianglia and Proof By Lack of a Counterexample. Enjoy.

A Guide to Formal Proof Methods for Mathematics Supervisors
-----------------------------------------------------------

Compiled from experience by many at Cambridge University and someone called Dana Angluin who I know nothing about.  [] means "End of Proof"

Things start to go wrong at school, when you are first introduced to the "Proof by Division by Zero". By University, you have gotten a little wiser, so they introduce some variants:

Proof by Bullsh*t - Write down hundreds of symbols in a fast and
               frenzied manner while mumbling a constant stream of
               vaguely technical jargon. Ensure that at least one line
               looks vaguely like the answer, and underline it several
               times. Move quickly onto the next question.

Proof by Misdirection - Waffle on until the students become
               disinterested and start staring into space, and then
               write anything you fancy on a piece of paper, which you
               quickly cast aside, asking "Do you see that now?"

Proof by Writing the Answer Down - Scribble down the answer without any
               working whatsoever, while hinting at how trivial the
               question is. (A crib sheet under the desk may be
               useful.)

Proof by Changing the Question - This is a very slick method for use
               with questions containing fancy mathematical terminology
               - by declaring sagely that the wording of the question
               implies some cunning simplification, you can make the
               hardest problems completely trivial.

Proof by "Surely You Can See That Makes..." - Plough through the problem
               until you are completely stuck and then do as for Proof
               by Writing the Answer Down.

Proof by Student Continuation - As above, but leave
               the last part to the students. (I sometimes think that
               some lecturers would give this proof for the 4-colour
               theorem, given half a chance.)

Proof by Example - Demonstrate that it works for a few carefully chosen
               numbers you know in advance will not give the students
               even a hint of the difficulty in finding a general
               proof.
               Alternatively, state WLOG (without loss of generality)
               and then prove a specific example that you know works.

Proof by Intimidation - Ask the audience if anybody can't see how to
               prove it. Nobody will put their hand up.

Proof by Omission - Cultivate such a tedious lecturing style during the
               first lecture, so that no-one arrives for any subsequent
               lectures, so that proofs are not required.

Proof by TeX - Apologize for not being a TeX wizard, but claim that a
               random series of squiggles, and diverse other marks on
               the handout constitute a proof.

Proof by Indirection - Find a book which contains the proof, and explain
               that the proof is trivial, but dedicated students can
               find it in <%EXPENSIVE MATHEMATICAL TEXT>

Proof by Xerox - As for proof by TeX, except that you rely on the run-
               down toner cartridge in the photocopier to hide the
               errors.

Proof by Reduction - Take any four A4 sheets of lines of mathematical
               working, and use a photocopier to reduce to 1/4 size,
               with the contrast set incorrectly, and assert that these
               sheets constitute an elegant, if lengthy proof.

Proof by Intuition - Promise a proof to be given next year, and next
               year say that the thing is intuitively obvious and
               something you've known for years.

Proof by Non-existent Reference - Claim the proof is available in
               Chapter 9 of the excellent well-known book by Prof.
               Schraufen Zieger which should be available in any
               respectable library. The book either has 8 chapters, or
               does not exist.

Proof by Honesty - This really is true. Honest!

Proof by Change of Direction - State A >= B, waffle about until
               everybody has forgotten the theory, then prove A <= B.

Proof by Jumping into Heavens - State premises. Then say "and this
               clearly implies <%ANY MEANINGLESS STATEMENT> which is
               obviously equivalent to ". State conclusion.

Proof by Pure Maths - State that the result is proved in Analysis III/
               Linear Analysis and go to them if you really want to
               know.

Proof by Axiom - We can see that <%EXPRESSION> ... therefore <%SAME
               EXPRESSION>.

Proof by Cross-breeding - Start off with an equation which is patently
               false, and gradually annotate it with more and more
               corrections as the audience points out the mistakes.
               Eventually you will end up with something so completely
               intractable that you can claim "This can be simplified
               to <%WHATEVER YOU WANT>" and nobody will be able to work
               out that it can't.

Proof by Lecturer - "Your lecturer will prove this result later in the
               course."

Proof by Supervisor - "Your supervisor will prove this if you ask him
               about this part of the lecture course."

Proof by Other Supervisor - " Your <%COURSE> supervisor will deal with
               this.

Proof by Introduction of Terminology - "Theorem: Every planet is an
               orange.
                       Proof:
               Define: A round ball is a stellar object.
               Define: A ball is an orange citrus fruit from which one
               can make orange juice.
               Clearly every planet is a round ball, hence surely a
               ball and therefore an orange. []"

Proof by Tedium - this elegant method is was actually used by Dr. XXX
               in Algebra III. He wrote down a theorem and chalked
               underneath it.
               Proof:
                               Boring!                 []

Proof by Blackmail - If you don't believe this is true, then I'll write
               down the long, incredibly excruciatingly boring proof
               after the lecture - does anyone want to see it? (Pause)
               Right, I won't bother.

Proof by Convenience - This isn't strictly speaking true, but that
               remark makes all our calculations meaningless, so we'll
               assume it is. (This requires courage)

Proof by Proximity - Derive some result looking vaguely similar to
               the required one and then say, "which is correct modulo
               a sign or two and a couple of powers of alpha, so it's
               just an algebraic mistake - you can easily do it
               yourself."

Proof by Mendacity - As above, but the rubric goes, "and on the example
               sheet, there's a mistake of a couple of signs and the
               odd power of alpha."

Proof by Construction - a good one for engineers. "This looks wrong, but
               it works if you build it."

Proof by Physical Intuition - Look, I could prove this formally, but you
               can see that it's got to be true physically anyway, and
               since you're a scientist and not a mathematician
               (ouch!), that should be good enough.

Proof by Contradiction - Assume result ... long incomprehensible working
               ... hence a contradiction, hence result.

Proof by Necessity - If this result isn't true, then I've been talking
               balls all week.
               - As an extension, the negation of the proposition is
               unimaginable or meaningless. Popular for proofs of the
               existence of a God.

Proof by Authority - This result is due to <%EMINENT MATHEMATICIAN>
               (preferably a head of department).

Proof by Thinly Veiled Threat - If this isn't obvious to you, you
               shouldn't be taking this course.

Proof by Number-Crunching - This result follows immediately from 1.4
               along with Corollaries 2.4(b) and 2.6(a), using the same
               method we used in 1.6(b), replacing alpha with 4 pi q /
               episolon nought where required, then substitute the
               answer into 3.3 to give the result.

Proof by Prevarication - Has your <%COURSE> lecturer covered <%TOPIC>
               yet?
               Some geek : 'No!'
               Oh well, this is actually a trivial <%TOPIC> problem,
               and the proof will be obvious to you when you cover it.

Proof by Variation of Statement - Ah yes, b was supposed to equal 3
               rather than 1. Just change it all the way through and
               you'll see it comes out all right.

Proof by Reduction and Confusion - We'll omit some of the hypotheses for
               the moment and add them later when we see what they need
               to be to make the result work.

Proof by Simplification - We'll prove continuity: differentiability is
               harder.

Proof by Chronometry - Well, we'll finish the proof next time... Last
               time we proved that...

Proof by Invisible Blue Chalk - We now integrate along the blue contour.

Proof by Erasure - By the lemma which I've just rubbed off...

Proof by Practicality - This isn't true in general, but you'll see by a
               few examples that it's true for all reasonable
               functions.

Proof by "I think that's not on the/your syllabus" - you don't really
               need this to pass the exam, so we won't bother with it.

Proof by Obviousness - This is obvious.

Proof by a Hint - simply write down the hint, followed by the desired
               result. The hint is obviously true, and the result
               follows clearly from it.

Proof by Integration by Inspection - Simply write down the answer with
               no hint as to how you arrived at it. Any fool can
               differentiate it to discover it is right; hence the
               problem is solved.

Proof by Dodgy Assumption - "We assume a generalized version of the
               Riemann Hypothesis..."

Proof by Precognition - Prove the result using an assumption
               proved later in the course. When proving this, assume
               the result previously proven.

Proof by looking like Basil Fawlty - !

Proof by the Clock - Waffle on until the hour is up, and then run out
               the door as you have another supervision to give
               halfway across Cambridge.

Proof by Feynman - Invent a new functional integral and say that it
               takes care of all the problems.

Proof by Illegibility - Use u, v and y in a proof. Give the v a rounded
               bottom and run the tail of the y into the divide by
               line. Do NOT read out the equations and write small.
               - Alternatively, resort to Greek or Cyrilic alphabets.
               Students will never differentiate (nu) and v, or (ioto)
               and i.

Proof by Squaring - Ideal to hide that missing +/- sign. Square the
               equations half way through, and then take the root at
               the end. This allows you to assume whichever sign you
               want.

Proof by OHP - On an overhead projector - now we can write down five
               conditions here (shows transparency) and it is easy to
               verify that Aimpliesbimpliescwhichisequivalenttodandtoe
               (removes transparency while students still writing down
               a).
            - Write down complicated equation with subscripts, then say
               "Oops, there's a mistake there" and proceed to smudge
               out half the equation so that you can write a single
               subscript ridiculously large in thick black pen.

Proof by Hospital - Get taken to hospital suffering from a stress
               related condition before proving anything, but after
               setting the examination questions. Leave the students
               to try to find something in the library.

Proof by Vigorous Handwaving - works well when teaching to a small
               number.

Proof by Obfuscation - A long plotless sequence of true and/or
               meaningless syntactically related statements.

Proof by Wishful Citation - The lecturer cites the negation, converse,
               or generalization of a theorem from the literature.

Proof by Accumulated Evidence - Long and diligent search has not
               revealed a counter-example.

Proof by Metaproof - A method is given to construct a desired proof.

Proof by Picture - More convincing than proof by example.

Proof by Semantic Shift - Some standard but inconvenient definitions are
               changed for the statement of the result.

Proof by Clarity - (fairly controversial) Know you subject. Check your
               proof in several book. Explain the strategy of the proof
               to your audience. Write clearly and speak loudly.
               Justify each step. Use a minimum of notation and draw a
               useful diagram. Reach a conclusion with the theorem
               clearly proved.
               This fails in the long term as the first couple of
               lectures take 90% of the course leaving the lecturer
               sprinting through the rest and leaving everyone
               stranded.
Title: Re: Proof methods
Post by Icarus on Nov 2nd, 2002, 7:57pm
Drat!! Why do I always learn everything only when its too late!!! I figured out a couple of these when I was lecturing, but the rest sure would have been helpful :D
Title: Re: Proof methods
Post by Johno-G on Jan 10th, 2003, 2:14pm
Proof by being-a-cocky-little-s**t - this one is good for students sitting exams. Example:

Q: Show that the statement <some boring jumble of maths garbage> is true for any x in the set of real numbers.

A: The statement must be true, or you wouldn't have asked me to show that it is.


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