Every Kind Of Science, If It Has

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

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.

I Do Not Want To Presuppose Anything As Known. I See In My Explanation In Section 1 The Definition Of The Concepts Point, Straight Line And Plane, If One Adds To These All The Axioms Of Groups I-v As Characteristics. If One Is Looking For Other Definitions Of Point, Perhaps By Means Of Paraphrase In Terms Of Extensionless, Etc., Then, Of Course, I Would Most Decidedly Have To Oppose Such An Enterprise. One Is Then Looking For Something That Can Never Be Found, For There Is Nothing There, And Everything Gets Lost, Becomes Confused And Vague, And Degenerates Into A Game Of Hide And Seek.

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.

An Old French Mathematician Said: "a Mathematical Theory Is Not To Be Considered Complete Until You Have Made It So Clear That You Can Explain It To The First Man Whom You Meet On The Street." This Clearness And Ease Of Comprehension, Here Insisted On For A Mathematical Theory, I Should Still More Demand For A Mathematical Problem If It Is To Be Perfect; For What Is Clear And Easily Comprehended Attracts, The Complicated Repels Us.