The Art Of Doing Mathematics Consists In

We Do Not Master A Scientific Theory Until We Have Shelled And Completely Prised Free Its Mathematical Kernel.

In Mathematics ... We Find Two Tendencies Present. On The One Hand, The Tendency Towards Abstraction Seeks To Crystallise The Logical Relations Inherent In The Maze Of Materials ... Being Studied, And To Correlate The Material In A Systematic And Orderly Manner. On The Other Hand, The Tendency Towards Intuitive Understanding Fosters A More Immediate Grasp Of The Objects One Studies, A Live Rapport With Them, So To Speak, Which Stresses The Concrete Meaning Of Their Relations.

A Mathematical Problem Should Be Difficult In Order To Entice Us, Yet Not Completely Inaccessible, Lest It Mock At Our Efforts. It Should Be To Us A Guide Post On The Mazy Paths To Hidden Truths, And Ultimately A Reminder Of Our Pleasure In The Successful Solution.

He Who Seeks For Methods Without Having A Definite Problem In Mind Seeks In The Most Part In Vain.

We Ought Not To Believe Those Who Today, Adopting A Philosophical Air And With A Tone Of Superiority, Prophesy The Decline Of Culter And Are Content With The Unknowable In A Self-satisfied Way. For Us There Is No Unknowable, And In My Opinion There Is Also Non Whatsoever For The Natural Sciences. In Place Of This Foolish Unknowable, Let Our Watchword On The Contrary Be: We Must Know - We Shall Know.