Monday, March 12, 2007

36 Methods of Mathematical Proof

Thanks to Steve, you have been saved from another long post by me! (Well, I obviously can't say how long it would have been, but gathering from my previous posts I'd say long).

Here you have 36 methods of mathematical proofs, and I can't really pin down which I like the most. (I've made the ones used by lecturers bold).

Proof by obviousness
"The proof is so clear that it need not be mentioned."

Proof by general agreement
"All in favor?. . . "

Proof by imagination
"Well, we'll pretend it's true. . .

Proof by convenience
"It would be very nice if it were true, so . . .

Proof by necessity
"It had better be true, or the entire structure of mathematics would crumble to the ground."

Proof by plausibility
"It sounds good, so it must be true."

Proof by intimidation
"Don't be stupid; of course it's true."

Proof by lack of sufficient time
"Because of the time constraint, I'll leave the proof to you."

Proof by postponement
"The proof for this is long and arduous, so it is given in the appendix."

Proof by accident
"Hey, what have we here?!"

Proof by insignificance
"Who really cares, anyway?"

Proof by mumbo-jumbo
" (B Ì P ) , $ (C Î W )

Proof by profanity
(example omitted)

Proof by definition
"We define it to be true."

Proof by tautology
'It's true because it's true."

Proof by plagiarism
"As we see on page 289......"

Proof by lost reference
"I know I saw it somewhere......"

Proof by calculus
"This proof requires calculus, so we'll skip it."

Proof by terror
When intimidation fails ...

Proof by lack of interest
"Does anyone really want to see this?"

Proof by illegibility
(scribble, scribble) QED

Proof by logic
"If it is on the problem sheet, then it must be true!"

Proof by majority rule
Only to be used if general agreement is impossible

Proof by clever variable choice
"Let A be the number such that this proof works. . "

Proof by tessellation
"This proof is the same as the last."

Proof by divine word
"And the Lord said, 'Let it be true,' and it was true."

Proof by stubbornness
"I don't care what you say-it is true!"

Proof by simplification
"This proof reduces to the statement 1 + 1 = 2."

Proof by hasty generalization
"Well, it works for 17, so it works for all reals."

Proof by deception
"Now everyone turn their backs. . ."

Proof by supplication
"Oh please, let it be true."

Proof by poor analogy
"Well, it's just like . . . "

Proof by avoidance
Limit of proof by postponement as it approaches infinity

Proof by design
If it's not true in today's math, invent a new system in which it is.

Proof by authority
"Well, Don Knuth says it's true, so it must be!"

Proof by intuition
"I just have this gut feeling. . ."

Seriously, I can't choose a favourite! The mumbo jumbo really hasn't been used, but since that's what it feels like most of the time, it gets a mention. Well this has made my Monday morning significantly better (actually much better), hope you have more fun trying to choose a favourite!


Jake said...

I liked some of the ones Serre mentioned in his talk on mathematical writing e.g.

Proof by impossible reference:

Proof is given in [1]

[1] Private communication between members of the Transylvanian Philological Society c. 1753

Proof by vague reference:

Proof is given in [1]

[1] Complete works of Euler.

beans said...

Lol, I'd found the Euler reference funny as well.

(I'm easily amused!)

Anatoly said...

Excellent list. The longest I ever saw.

Another method:
Proof by similarity: The trivial case works, the rest 255 cases can be proofed similarly.

Beans said...

It is actually quite great (having skimmed through it again just now!)

I just saw this one:

"Proof by illegibility
(scribble, scribble) QED (Black square!)"

and felt slightly embarrassed at having done that in one of my exams! (Algebraic Structures 1) It was unintentional of course, but erm... that explains why I didn't do as well as I should have in that exam.

Anatoly said...

I just remembered another one: Proof by previous knowledge:
If I need to prove you this you are in the wrong course.

Side effect - one student less next week...

Beans said...

Only one student less? ;)

Anatoly said...

One of the professors told something very close to this to a student in the beginning of the current semester... I didn't count, but by the 8 week at least third of the group left - so it works well.
I am staying because I love challenges.. :)

Beans said...

Seriously?! I can't imagine that ever happening in England... (or at Manchester!)

How big does your group tend to be?

Anatoly said...

Yes, it is difficult to believe something like this. But it did happen.
I don't think it can happen on a mathematical course, but this particular course is physics. For some reason on physical courses the professors don't care if the students understand what they say on the same level math professors do. It seems to be an unofficial policy here.

It is hard to answer your question about group size because it depends a lot of how much the students like the professor who teaches it. People "pile up" on those who explain well. The current group size (the group I am in) on this course is about twenty. It was about 40 in the beginning, but some of them were visitors from other groups (I think so at least).

Beans said...

Another reason for not liking Physics then!

Thanks for your answer, but I have a few more question! What's the system like at Hebrew University? Do you always get to choose which modules you want to take, or are there some compulsory ones?

Anatoly said...

There are a lot of choice, but there are also compulsory courses. Also, often to be able to do a course you need to learn another one firstly.
For those who study mathematics, there are only 13 courses you must take (with some choice even for them).
There are also a computer course, a physics course and two seminars which are compulsory. But even combined, all these courses are less than half of the requirements for graduation.