It Is Not Order Only,

Thus, Be It Understood, To Demonstrate A Theorem, It Is Neither Necessary Nor Even Advantageous To Know What It Means.

In One Word, To Draw The Rule From Experience, One Must Generalize; This Is A Necessity That Imposes Itself On The Most Circumspect Observer.

Astronomy Is Useful Because It Raises Us Above Ourselves; It Is Useful Because It Is Grand; .... It Shows Us How Small Is Man's Body, How Great His Mind, Since His Intelligence Can Embrace The Whole Of This Dazzling Immensity, Where His Body Is Only An Obscure Point, And Enjoy Its Silent Harmony.

Thus, Be It Understood, To Demonstrate A Theorem, It Is Neither Necessary Nor Even Advantageous To Know What It Means. The Geometer Might Be Replaced By The "logic Piano" Imagined By Stanley Jevons; Or, If You Choose, A Machine Might Be Imagined Where The Assumptions Were Put In At One End, While The Theorems Came Out At The Other, Like The Legendary Chicago Machine Where The Pigs Go In Alive And Come Out Transformed Into Hams And Sausages. No More Than These Machines Need The Mathematician Know What He Does.

What Is It Indeed That Gives Us The Feeling Of Elegance In A Solution, In A Demonstration?