A Chess Problem Is Simply

Real Mathematics Must Be Justified As Art If It Can Be Justified At All.

No Mathematician Should Ever Allow Him To Forget That Mathematics, More Than Any Other Art Or Science, Is A Young Man's Game. ... Galois Died At Twenty-one, Abel At Twenty-seven, Ramanujan At Thirty-three, Riemann At Forty. There Have Been Men Who Have Done Great Work Later; ... [but] I Do Not Know Of A Single Instance Of A Major Mathematical Advance Initiated By A Man Past Fifty. ... A Mathematician May Still Be Competent Enough At Sixty, But It Is Useless To Expect Him To Have Original Ideas.

Reductio Ad Absurdum, Which Euclid Loved So Much, Is One Of A Mathematician's Finest Weapons. It Is A Far Finer Gambit Than Any Chess Play: A Chess Player May Offer The Sacrifice Of A Pawn Or Even A Piece, But A Mathematician Offers The Game.

I Do Not Remember Having Felt, As A Boy, Any Passion For Mathematics, And Such Notions As I May Have Had Of The Career Of A Mathematician Were Far From Noble. I Thought Of Mathematics In Terms Of Examinations And Scholarships: I Wanted To Beat Other Boys, And This Seemed To Be The Way In Which I Could Do So Most Decisively.

I Believe That Mathematical Reality Lies Outside Us, That Our Function Is To Discover Or Observe It, And That The Theorems Which We Prove, And Which We Describe Grandiloquently As Our "creations," Are Simply The Notes Of Our Observations.