Young Men Should Prove Theorems,

A Chess Problem Is Genuine Mathematics, But It Is In Some Way "trivial" Mathematics. However, Ingenious And Intricate, However Original And Surprising The Moves, There Is Something Essential Lacking. Chess Problems Are Unimportant. The Best Mathematics Is Serious As Well As Beautiful-"important" If You Like, But The Word Is Very Ambiguous, And "serious" Expresses What I Mean Much Better.

There Is No Scorn More Profound, Or On The Whole More Justifiable, Than That Of The Men Who Make For The Men Who Explain.

I Have Never Done Anything 'useful'. No Discovery Of Mine Has Made, Or Is Likely To Make, Directly Or Indirectly, For Good Or Ill, The Least Difference To The Amenity Of The World... Judged By All Practical Standards, The Value Of My Mathematical Life Is Nil; And Outside Mathematics It Is Trivial Anyhow. I Have Just One Chance Of Escaping A Verdict Of Complete Triviality, That I May Be Judged To Have Created Something Worth Creating. And That I Have Created Something Is Undeniable: The Question Is About Its Value.

I Count Maxwell And Einstein, Eddington And Dirac, Among "real" Mathematicians. The Great Modern Achievements Of Applied Mathematics Have Been In Relativity And Quantum Mechanics, And These Subjects Are At Present At Any Rate, Almost As "useless" As The Theory Of Numbers.

Pure Mathematics Is On The Whole Distinctly More Useful Than Applied... For What Is Useful Above All Is Technique, And Mathematical Technique Is Taught Mainly Through Pure Mathematics.