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Garzahd
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Proof methods
« on: Oct 30th, 2002, 5:24pm » |
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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.
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Icarus
wu::riddles Moderator
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Re: Proof methods
« Reply #1 on: Nov 2nd, 2002, 7:57pm » |
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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 
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Johno-G
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Could God create a wall that he could not jump?
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Re: Proof methods
« Reply #2 on: Jan 10th, 2003, 2:14pm » |
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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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