How to Write a 21st Century Proof? If you've gone through proofs in published papers, you'll know it can be hard work to convince yourself that what's being claimed is in fact true. And it's not uncommon to conclude that whilst the claim is indeed correct, the presented proof is not. [3,1]
Mathematician turned Computer Scientist, Leslie Lamport, suggests an hierarchical structured method for writing proofs that both improves their rigour and eases their communication.
At first glance, the Lamport structured method may seem like a radical departure from the usual prose paragraph method. But after giving it a try, I've found it to be refreshingly freeing and have come to prefer it for my own work.
The common prose-style methods in contrast seem to be more appropriately labelled "proof sketch".
Now there's certainly a place for proof sketches. Indeed every non-trivial proof benefits from a prose overview. But the problem with proof sketches being the method of publishing proof is that "authors seem to choose randomly which details to supply and which to omit." [1:p10]
How many errors are there in published mathematical proofs?
"A previous editor of the Mathematical Reviews [remarked] that approximately one half of the proofs published in it were incomplete and/or contained errors, although the theorems they were purported to prove were essentially true." [3, p.71]
That's a claim which suggests Lamport's methods may indeed have something valuable to offer.
Try the Structured Method
To give Lamport's structured method a try, read one of his short papers available from the links given below.
Then I suggest a "test drive" with the following three problems. All three are elementary in the sense that the content of each should hold no trouble for most of you. The issue will be what you think is a sufficiently convincing proof now that you've read his critique.
Test Problems
Declare whether the statement is True or False. If False, produce a counterexample. If True, write down a complete proof.
(1) The intersection of any two intervals is an interval.
(2) There is always a real solution to the equation y^2 - a = 0, when a>0. (For simplicity, think of a=2.)
(3) N(X+Y) = N(X) + N(Y) - N(X.Y), where X and Y are each finite sets, + is set union, . is set intersection, and N( ) is the number of elements contained in the set.
Lamport's Papers
[1995] How to Write a Proof; Leslie Lamport; 1995; Aug-Sep, American Mathematical Monthly, Vol 102 No 7 pp.600-608
PDF here: http://research.microsoft.com/en-us/um/people/lamport/pubs/pubs.html#lamport-how-to-write
[2012] How to Write a 21st Century Proof; Leslie Lamport; 2012; March; Journal of Fixed Point Theory and Applications
PDF here: http://research.microsoft.com/en-us/um/people/lamport/pubs/pubs.html#proof
Other References
[1983] Rigorous Proof in Mathematics Education; Gila Hanna; 1983; OISE Press (University of Toronto)