Wednesday, March 31, 2004

Erd�s and proofs 'from the Book'

It would take too many books to even begin to describe the genius of Paul Erd�s; one such book is " The Man Who Loved Only Numbers: The Story of Paul Erd�s and the Search for Mathematical Truth" by Paul Hoffman.

Apart from his 1500+ research papers, Erd�s numbers (mine is 3!), and a dazzling array of problems in combinatorics, geometry, number theory, and countless other areas, Erd�s also bequeathed to us the beautiful notion of a 'proof from the Book'.

If I see a really nice proof, I say it comes straight from the Book . . . God has a transfinite Book, which contains all theorems and their best proofs, and if He is well intentioned toward those [mathematicians], He shows them the Book for a moment. And you wouldn't even have to believe in God, but you must believe that the Book exists.
(A. Soifer, Mathematics Competitions, vol.4, I, 1991, p.63) (via this link)

The idea of beauty in mathematics is something any mathematician understands intuitively. Although to a layman, the process of proving a theorem may seem mechanical (A => B, B=> C), to a mathematician, the elegance of a proof is sometimes as valuable as the proof itself. Like any aesthetic notion, elegance is hard to define; it lies in the eyes (or in this case the mind's eye) of the beholder. Erd�s 's idea of a 'Book proof' is a poetic way of describing a proof that is elegant, surprising, and hard to achieve.

Here is an example of one such proof, of the Sylvester-Gallai Theorem, taken from the book 'Proofs from THE BOOK' by Aigner and Ziegler:

Theorem. There is no way to arrange n points in the plane (not on a line) so that a line through any two points will always pass through a third.

I will post the proof tomorrow; till then, keep your brain open !

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